%------------------------------------------------------------------------------
% File     : ITP214^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Hoare_Triple 00428_012651
% 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_00428_012651 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.2.0, 0.23 v8.1.0
% Syntax   : Number of formulae    : 12112 (5171 unt;1845 typ;   0 def)
%            Number of atoms       : 29629 (11254 equ;   0 cnn)
%            Maximal formula atoms :   48 (   2 avg)
%            Number of connectives : 111866 (2669   ~; 470   |;2086   &;95072   @)
%                                         (   0 <=>;11569  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   25 (   6 avg)
%            Number of types       :  192 ( 191 usr)
%            Number of type conns  : 8544 (8544   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1657 (1654 usr;  99 con; 0-8 aty)
%            Number of variables   : 29524 (3706   ^;25258   !; 560   ?;29524   :)
% 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 15:41:40.043
%------------------------------------------------------------------------------
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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__List__Olist_It__Option__Ooption_It__Num__Onum_J_J,type,
    list_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__Option__Ooption_It__Code____Numeral__Ointeger_J,type,
    option_Code_integer: $tType ).

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

thf(ty_n_t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    heap_Time_Heap_nat: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__Int__Oint_J,type,
    heap_Time_Heap_int: $tType ).

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

thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
    set_list_int: $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__Heap__Oarray_It__Product____Type__Ounit_J,type,
    array_Product_unit: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Ounit_J,type,
    list_Product_unit: $tType ).

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

thf(ty_n_t__Heap__Oref_It__Product____Type__Ounit_J,type,
    ref_Product_unit: $tType ).

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

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    product_prod_b_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    product_prod_a_a: $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__b_J,type,
    heap_Time_Heap_b: $tType ).

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

thf(ty_n_t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    heap_Time_Heap_o: $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__Heap__Oarray_It__Nat__Onat_J,type,
    array_nat: $tType ).

thf(ty_n_t__Heap__Oarray_It__Int__Oint_J,type,
    array_int: $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__Option__Ooption_I_Eo_J,type,
    option_o: $tType ).

thf(ty_n_t__List__Olist_Itf__b_J,type,
    list_b: $tType ).

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

thf(ty_n_t__Heap__Oarray_I_Eo_J,type,
    array_o: $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__String__Ochar,type,
    char: $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__b,type,
    b: $tType ).

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

% Explicit typings (1654)
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_Array__Time_Oalloc_001_Eo,type,
    array_alloc_o: list_o > heap_e7401611519738050253t_unit > produc5011041761010125393t_unit ).

thf(sy_c_Array__Time_Oalloc_001t__Int__Oint,type,
    array_alloc_int: list_int > heap_e7401611519738050253t_unit > produc3407818250607552075t_unit ).

thf(sy_c_Array__Time_Oalloc_001t__Nat__Onat,type,
    array_alloc_nat: list_nat > heap_e7401611519738050253t_unit > produc1730305018825802663t_unit ).

thf(sy_c_Array__Time_Olen_001_Eo,type,
    array_len_o: array_o > heap_Time_Heap_nat ).

thf(sy_c_Array__Time_Olen_001t__Int__Oint,type,
    array_len_int: array_int > heap_Time_Heap_nat ).

thf(sy_c_Array__Time_Olen_001t__Nat__Onat,type,
    array_len_nat: array_nat > heap_Time_Heap_nat ).

thf(sy_c_Array__Time_Omake_001_Eo,type,
    array_make_o: nat > ( nat > $o ) > heap_T5660665574680485309rray_o ).

thf(sy_c_Array__Time_Omake_001t__Int__Oint,type,
    array_make_int: nat > ( nat > int ) > heap_T1346037964561226099ay_int ).

thf(sy_c_Array__Time_Omake_001t__Nat__Onat,type,
    array_make_nat: nat > ( nat > nat ) > heap_T3836121109492952855ay_nat ).

thf(sy_c_Array__Time_Onth_001_Eo,type,
    array_nth_o: array_o > nat > heap_Time_Heap_o ).

thf(sy_c_Array__Time_Onth_001t__Int__Oint,type,
    array_nth_int: array_int > nat > heap_Time_Heap_int ).

thf(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
    array_nth_nat: array_nat > nat > heap_Time_Heap_nat ).

thf(sy_c_Array__Time_Onth_001t__Product____Type__Ounit,type,
    array_7872002506669749220t_unit: array_Product_unit > nat > heap_T5738788834812785303t_unit ).

thf(sy_c_Array__Time_Oof__list_001_Eo,type,
    array_of_list_o: list_o > heap_T5660665574680485309rray_o ).

thf(sy_c_Array__Time_Oof__list_001t__Int__Oint,type,
    array_of_list_int: list_int > heap_T1346037964561226099ay_int ).

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

thf(sy_c_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_Osnga__assn_001_Eo,type,
    snga_assn_o: array_o > list_o > assn ).

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

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

thf(sy_c_Assertions_Osnga__assn_001t__Product____Type__Ounit,type,
    snga_a4522542871529764173t_unit: array_Product_unit > list_Product_unit > assn ).

thf(sy_c_Assertions_Osngr__assn_001t__Product____Type__Ounit,type,
    sngr_a5825115052027484668t_unit: ref_Product_unit > product_unit > assn ).

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

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

thf(sy_c_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__Order__Relation_Ocard__of_001t__Nat__Onat,type,
    bNF_Ca3793111618940312692of_nat: set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001t__Nat__Onat,type,
    bNF_Ca1281551314933786834on_nat: set_nat > set_Pr1261947904930325089at_nat > $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__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Int__Oint,type,
    bNF_Ca1965613569405424510nt_int: set_Pr958786334691620121nt_int > ( int > int ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Nat__Onat,type,
    bNF_Ca1968104039914474786nt_nat: set_Pr958786334691620121nt_int > ( int > nat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Num__Onum,type,
    bNF_Ca7748807862925029228nt_num: set_Pr958786334691620121nt_int > ( int > num ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Rat__Orat,type,
    bNF_Ca1332973979827979050nt_rat: set_Pr958786334691620121nt_int > ( int > rat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
    bNF_Ca583493526879471924et_int: set_Pr958786334691620121nt_int > ( int > set_int ) > $o ).

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

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

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__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_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).

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

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

thf(sy_c_BNF__Def_Orel__fun_001t__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 ).

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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 ).

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    bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).

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    bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).

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    bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).

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    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

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

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thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).

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

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
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thf(sy_c_BNF__Wellorder__Constructions_OordIso_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
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thf(sy_c_Basic__BNFs_Opred__prod_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Basic__BNFs_Orel__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Basic__BNFs_Orel__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Binomial_Obinomial,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > nat > rat ).

thf(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Bit__Operations_Oand__not__num,type,
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thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Oconcat__bit,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Otake__bit__num,type,
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thf(sy_c_Code__Numeral_Obit__cut__integer,type,
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thf(sy_c_Code__Numeral_Odivmod__abs,type,
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thf(sy_c_Code__Numeral_Odivmod__integer,type,
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thf(sy_c_Code__Numeral_Odup,type,
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thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
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thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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thf(sy_c_Code__Numeral_Ointeger__of__num,type,
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thf(sy_c_Code__Numeral_Onat__of__integer,type,
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thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
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thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
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thf(sy_c_Code__Numeral_Onum__of__integer,type,
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thf(sy_c_Code__Numeral_Osub,type,
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thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Int__Oint,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Heap__Time__Monad_Oureturn_001tf__b,type,
    heap_Time_ureturn_b: b > heap_Time_Heap_b ).

thf(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_T5738788834812785303t_unit ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001_Eo,type,
    hoare_hoare_triple_o: assn > heap_Time_Heap_o > ( $o > assn ) > $o ).

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

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Nat__Onat,type,
    hoare_3067605981109127869le_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Product____Type__Ounit,type,
    hoare_8945653483474564448t_unit: assn > heap_T5738788834812785303t_unit > ( product_unit > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001tf__a,type,
    hoare_hoare_triple_a: assn > heap_Time_Heap_a > ( a > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001tf__b,type,
    hoare_hoare_triple_b: assn > heap_Time_Heap_b > ( b > assn ) > $o ).

thf(sy_c_Hoare__Triple_Onew__addrs,type,
    hoare_new_addrs: heap_e7401611519738050253t_unit > set_nat > heap_e7401611519738050253t_unit > set_nat ).

thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).

thf(sy_c_If_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    if_int2409958687428134794nt_int: $o > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > int > product_prod_int_int ).

thf(sy_c_If_001_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_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
    if_Pro5796790085669187537et_nat: $o > ( produc3925858234332021118et_nat > produc2732055786443039994et_nat ) > ( produc3925858234332021118et_nat > produc2732055786443039994et_nat ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat ).

thf(sy_c_If_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J,type,
    if_Pro2760142802149639654atural: $o > ( produc7822875418678951345atural > produc5835291356934675326atural ) > ( produc7822875418678951345atural > produc5835291356934675326atural ) > produc7822875418678951345atural > produc5835291356934675326atural ).

thf(sy_c_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__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,
    if_Pro3578090444564244254it_nat: $o > ( produc6653097349344004940it_nat > produc3260487557148687353it_nat ) > ( produc6653097349344004940it_nat > produc3260487557148687353it_nat ) > produc6653097349344004940it_nat > produc3260487557148687353it_nat ).

thf(sy_c_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    if_Pro4080778131203054751it_nat: $o > ( produc6653097349344004940it_nat > produc7388388658123137530it_nat ) > ( produc6653097349344004940it_nat > produc7388388658123137530it_nat ) > produc6653097349344004940it_nat > produc7388388658123137530it_nat ).

thf(sy_c_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    if_Pro6440128116348121305et_nat: $o > ( produc3658429121746597890et_nat > produc3925858234332021118et_nat ) > ( produc3658429121746597890et_nat > produc3925858234332021118et_nat ) > produc3658429121746597890et_nat > produc3925858234332021118et_nat ).

thf(sy_c_If_001t__Assertions__Oassn,type,
    if_assn: $o > assn > assn > assn ).

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

thf(sy_c_If_001t__Code____Numeral__Onatural,type,
    if_Code_natural: $o > code_natural > code_natural > code_natural ).

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

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

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

thf(sy_c_If_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    if_mul8430962117462786573at_nat: $o > multis2468970476368604999at_nat > multis2468970476368604999at_nat > multis2468970476368604999at_nat ).

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

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

thf(sy_c_If_001t__Option__Ooption_It__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,
    if_opt1729522071442692626it_nat: $o > option8956607266484857688it_nat > option8956607266484857688it_nat > option8956607266484857688it_nat ).

thf(sy_c_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,type,
    if_opt6883606601682554499it_nat: $o > option3562590408128118217it_nat > option3562590408128118217it_nat > option3562590408128118217it_nat ).

thf(sy_c_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    if_opt7386294288321364996it_nat: $o > option4065278094766928714it_nat > option4065278094766928714it_nat > option4065278094766928714it_nat ).

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

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

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

thf(sy_c_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    if_opt7704869406773131885at_nat: $o > option8963830502488799655at_nat > option8963830502488799655at_nat > option8963830502488799655at_nat ).

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

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

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

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

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

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

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

thf(sy_c_Int_OAbs__Integ,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(sy_c_Int_ORep__Integ,type,
    rep_Integ: int > product_prod_nat_nat ).

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

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

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

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

thf(sy_c_Int_Opcr__int,type,
    pcr_int: product_prod_nat_nat > int > $o ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
    ring_11222124179247155820nteger: set_Code_integer ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
    ring_1_Ints_int: set_int ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
    ring_1_Ints_rat: set_rat ).

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

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

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

thf(sy_c_Lattices_Oinf__class_Oinf_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,
    inf_in2641120393918057659_nat_o: ( ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_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,
    inf_in3295504058751909687_nat_o: ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_Eo_J,type,
    inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    inf_inf_int_int_o: ( int > int > $o ) > ( int > int > $o ) > int > int > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Int__Oint_M_Eo_J,type,
    inf_inf_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    inf_inf_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    inf_inf_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
    inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    inf_in3604695632404883862_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    inf_in5163264567034779214_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    inf_inf_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_M_Eo_J,type,
    inf_in5102985939729578038_int_o: ( set_Pr958786334691620121nt_int > $o ) > ( set_Pr958786334691620121nt_int > $o ) > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_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,
    inf_in8051596381320575356_nat_o: ( a > produc6653097349344004940it_nat > $o ) > ( a > produc6653097349344004940it_nat > $o ) > a > produc6653097349344004940it_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__b_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,
    inf_in844129575377303227_nat_o: ( b > produc6653097349344004940it_nat > $o ) > ( b > produc6653097349344004940it_nat > $o ) > b > produc6653097349344004940it_nat > $o ).

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
    inf_inf_int: int > int > int ).

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Option__Ooption_I_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    inf_in7104746047340619750_int_o: option1893999432384633940_int_o > option1893999432384633940_int_o > option1893999432384633940_int_o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Option__Ooption_It__Assertions__Oassn_J,type,
    inf_inf_option_assn: option_assn > option_assn > option_assn ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Option__Ooption_It__Nat__Onat_J,type,
    inf_inf_option_nat: option_nat > option_nat > option_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    inf_in7812914138253463912et_nat: option_set_nat > option_set_nat > option_set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    inf_in777885744494410645at_nat: option8963830502488799655at_nat > option8963830502488799655at_nat > option8963830502488799655at_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Ounit,type,
    inf_inf_Product_unit: product_unit > product_unit > product_unit ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Rat__Orat,type,
    inf_inf_rat: rat > rat > rat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
    inf_inf_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Assertions__Oassn_J,type,
    inf_inf_set_assn: set_assn > set_assn > set_assn ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    inf_in1364745209274528805nteger: set_Code_integer > set_Code_integer > set_Code_integer ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
    inf_inf_set_int: set_int > set_int > set_int ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    inf_inf_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Num__Onum_J,type,
    inf_inf_set_num: set_num > set_num > set_num ).

thf(sy_c_Lattices_Oinf__class_Oinf_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,
    inf_in3088352823822785602et_nat: set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat ).

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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
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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,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    sup_su3714129841433130204it_nat: set_Pr7600907837789447088it_nat > set_Pr7600907837789447088it_nat > set_Pr7600907837789447088it_nat ).

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    sup_su2047564715030645325nt_int: set_se6260736226359567993nt_int > set_se6260736226359567993nt_int > set_se6260736226359567993nt_int ).

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_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    lattic8660852769118194346_int_o: ( ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o ) > ( ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ) > ( ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ) > $o ).

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,
    lattic6006661108824415698et_int: ( int > int > int ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

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__Num__Onum,type,
    lattic2566483365489244608et_num: ( num > num > num ) > ( num > num > $o ) > ( num > num > $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 ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_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,
    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,
    lattic6529551498584149819at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_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,
    lattic6976360593897529633at_nat: ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ) > ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ) > ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ) > $o ).

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

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

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

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: 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_Ofind_001_Eo,type,
    find_o: ( $o > $o ) > list_o > option_o ).

thf(sy_c_List_Ofind_001t__Int__Oint,type,
    find_int: ( int > $o ) > list_int > option_int ).

thf(sy_c_List_Ofind_001t__Nat__Onat,type,
    find_nat: ( nat > $o ) > list_nat > option_nat ).

thf(sy_c_List_Ofind_001t__Num__Onum,type,
    find_num: ( num > $o ) > list_num > option_num ).

thf(sy_c_List_Ofind_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,
    find_P5317947125398478060it_nat: ( produc8664842809031399944it_nat > $o ) > list_P626663023886443800it_nat > option8956607266484857688it_nat ).

thf(sy_c_List_Ofind_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,
    find_P4815471733760996061it_nat: ( produc3260487557148687353it_nat > $o ) > list_P6935614879863011209it_nat > option3562590408128118217it_nat ).

thf(sy_c_List_Ofind_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    find_P8943372834735446238it_nat: ( produc7388388658123137530it_nat > $o ) > list_P7438302566501821706it_nat > option4065278094766928714it_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_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    cons_P6219271836124797827_nat_o: ( produc3658429121746597890et_nat > $o ) > list_P7985473006766602707_nat_o > list_P7985473006766602707_nat_o ).

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

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

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

thf(sy_c_List_Olist_OCons_001t__Num__Onum,type,
    cons_num: num > list_num > list_num ).

thf(sy_c_List_Olist_OCons_001t__Option__Ooption_It__Num__Onum_J,type,
    cons_option_num: option_num > list_option_num > list_option_num ).

thf(sy_c_List_Olist_OCons_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,
    cons_P6329468272379687876et_nat: produc2732055786443039994et_nat > list_P362550909693114634et_nat > list_P362550909693114634et_nat ).

thf(sy_c_List_Olist_OCons_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,
    cons_P3419706843779801160et_nat: produc3925858234332021118et_nat > list_P2321686559999237006et_nat > list_P2321686559999237006et_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    cons_P7512249878480867347nt_nat: product_prod_int_nat > list_P8198026277950538467nt_nat > list_P8198026277950538467nt_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    cons_P4136552807766418642it_nat: produc8664842809031399944it_nat > list_P626663023886443800it_nat > list_P626663023886443800it_nat ).

thf(sy_c_List_Olist_OCons_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,
    cons_P2339423585217329347it_nat: produc3260487557148687353it_nat > list_P6935614879863011209it_nat > list_P6935614879863011209it_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    cons_P6467324686191779524it_nat: produc7388388658123137530it_nat > list_P7438302566501821706it_nat > list_P7438302566501821706it_nat ).

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

thf(sy_c_List_Olist_OCons_001tf__a,type,
    cons_a: a > list_a > list_a ).

thf(sy_c_List_Olist_OCons_001tf__b,type,
    cons_b: b > list_b > list_b ).

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_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

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__Option__Ooption_It__Num__Onum_J,type,
    nth_option_num: list_option_num > nat > option_num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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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,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    nth_Pr2956344619228612545it_nat: list_P626663023886443800it_nat > nat > produc8664842809031399944it_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__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    nth_Pr3591576263217728691it_nat: list_P7438302566501821706it_nat > nat > produc7388388658123137530it_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Ounit,type,
    nth_Product_unit: list_Product_unit > nat > product_unit ).

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

thf(sy_c_List_Onth_001tf__b,type,
    nth_b: list_b > nat > b ).

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__b_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc1870533552098665926it_nat: list_b > list_P131111800688179804it_nat > list_P7438302566501821706it_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_Orotate_001_Eo,type,
    rotate_o: nat > list_o > list_o ).

thf(sy_c_List_Orotate_001t__Int__Oint,type,
    rotate_int: nat > list_int > list_int ).

thf(sy_c_List_Orotate_001t__Nat__Onat,type,
    rotate_nat: 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_List_Ozip_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    zip_Pr8136144321567152340et_nat: list_P7985473006766602707_nat_o > list_P2321686559999237006et_nat > list_P362550909693114634et_nat ).

thf(sy_c_List_Ozip_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    zip_Pr7134870689397686104et_nat: list_P7985473006766602707_nat_o > list_P9062070895058802706et_nat > list_P2321686559999237006et_nat ).

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Ozip_001tf__a_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    zip_a_1859587264247984595it_nat: list_a > list_P131111800688179804it_nat > list_P6935614879863011209it_nat ).

thf(sy_c_List_Ozip_001tf__b_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    zip_b_5987488365222434772it_nat: list_b > list_P131111800688179804it_nat > list_P7438302566501821706it_nat ).

thf(sy_c_Misc_OEps__Opt_001t__Int__Oint,type,
    eps_Opt_int: ( int > $o ) > option_int ).

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__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_OEps__Opt_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    eps_Op5869321083069970014it_nat: ( produc7388388658123137530it_nat > $o ) > option4065278094766928714it_nat ).

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

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

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

thf(sy_c_Misc_Obijective_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    biject5714339216877808333at_nat: set_Pr8551490117392284871at_nat > $o ).

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

thf(sy_c_Misc_Obijective_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    biject3503750217840948301at_nat: set_Pr8693737435421807431at_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_Obijective_001tf__b_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    biject3496835315794560510it_nat: set_Pr7600907837789447088it_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_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_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_Orel__of_001tf__b_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    rel_of6889123638293104083it_nat: ( b > option233860712434008220it_nat ) > ( produc7388388658123137530it_nat > $o ) > set_Pr7600907837789447088it_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_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_001t__Int__Oint_001t__Int__Oint,type,
    set_to_map_int_int: set_Pr958786334691620121nt_int > int > option_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_Oset__to__map_001tf__b_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    set_to6136816377707935252it_nat: set_Pr7600907837789447088it_nat > b > 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__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_de4670738602788725795it_nat: produc7388388658123137530it_nat > option4065278094766928714it_nat > produc7388388658123137530it_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__Int__Oint,type,
    case_nat_int: int > ( nat > int ) > nat > int ).

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,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Code____Numeral__Ointeger,type,
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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,
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thf(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
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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_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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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,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_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,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_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,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Ounit_J,type,
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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__b_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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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,
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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,
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thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: nat > list_nat ).

thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: nat > nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: nat > product_prod_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_Oprod__encode,type,
    nat_prod_encode: product_prod_nat_nat > nat ).

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

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

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

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

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: int > int ).

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

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

thf(sy_c_Num_Onum_OBit1,type,
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thf(sy_c_Num_Onum_OOne,type,
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thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Num_Onum_Osize__num,type,
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thf(sy_c_Num_Onum__of__nat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
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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,
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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_Num_Osqr,type,
    sqr: num > num ).

thf(sy_c_Option_Ocombine__options_001t__Num__Onum,type,
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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,
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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,
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thf(sy_c_Option_Ocombine__options_001t__Product____Type__Oprod_Itf__b_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_ONone_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Assertions__Oassn,type,
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thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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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,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_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_ONone_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_OSome_001_Eo,type,
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thf(sy_c_Option_Ooption_OSome_001t__Assertions__Oassn,type,
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thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
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thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oarray_It__Int__Oint_J_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_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
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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,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
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    some_s287724117700012716at_nat: set_Pr8551490117392284871at_nat > option2498585697089621389at_nat ).

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    some_s147305329494351046at_nat: set_Pr1261947904930325089at_nat > option8963830502488799655at_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_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    case_o1441893360019914891nteger: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > option8057788054806935849nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Option_Ooption_Ocase__option_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    case_o7134296353695833103nteger: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > option_Code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

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__Rat__Orat,type,
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thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Code____Numeral__Onatural_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_I_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    case_o670596351732880613_int_o: option1893999432384633940_int_o > ( ( product_prod_int_int > $o ) > option1893999432384633940_int_o ) > option1893999432384633940_int_o > option1893999432384633940_int_o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Assertions__Oassn_J_001t__Assertions__Oassn,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Int__Oint_J_001t__Int__Oint,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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    case_o7727800614283616222it_nat: option8956607266484857688it_nat > ( produc3260487557148687353it_nat > option8956607266484857688it_nat ) > option3562590408128118217it_nat > option8956607266484857688it_nat ).

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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_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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    case_o5901184372446974929it_nat: option4065278094766928714it_nat > ( produc7388388658123137530it_nat > option4065278094766928714it_nat ) > option4065278094766928714it_nat > option4065278094766928714it_nat ).

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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    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__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__Int__Oint_001t__Nat__Onat,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_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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    member_list_o: list_o > set_list_o > $o ).

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_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,
    member8781333585448626064_nat_o: produc4928098042776334183_nat_o > set_Pr2161125870931222855_nat_o > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__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_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
    member6341495586645257982et_nat: produc5657529347773406293et_nat > set_Pr3444600963470892981et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member4763271486408492550et_nat: produc6830853553727218525et_nat > set_Pr7928877670098842301et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    member5617269971687963298it_nat: produc8961450480463052793it_nat > set_Pr4389693562480114009it_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_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,
    member6099555550032318734et_nat: produc8111630337999740517et_nat > set_Pr719794911490849221et_nat > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__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,
    member5335690527091456380it_nat: produc9011797661310329043it_nat > set_Pr2819221443900773171it_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    member6820296301096096254it_nat: produc1273031398460193109it_nat > set_Pr5508209795250834101it_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_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__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    member8682313912305727889it_nat: produc7388388658123137530it_nat > set_Pr7600907837789447088it_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $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_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member2340774599025711042nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int > $o ).

thf(sy_v_P,type,
    p: assn ).

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

thf(sy_v_R,type,
    r: a > assn ).

thf(sy_v_as____,type,
    as: set_nat ).

thf(sy_v_f,type,
    f: heap_Time_Heap_a ).

thf(sy_v_g,type,
    g: a > heap_Time_Heap_b ).

thf(sy_v_h_H_H____,type,
    h: heap_e7401611519738050253t_unit ).

thf(sy_v_h_H____,type,
    h2: heap_e7401611519738050253t_unit ).

thf(sy_v_h____,type,
    h3: heap_e7401611519738050253t_unit ).

thf(sy_v_rf____,type,
    rf: a ).

thf(sy_v_rg____,type,
    rg: b ).

thf(sy_v_t_H_H____,type,
    t: nat ).

thf(sy_v_t_H____,type,
    t2: nat ).

% Relevant facts (10207)
thf(fact_0__092_060open_062new__addrs_Ah_H_A_Inew__addrs_Ah_Aas_Ah_H_J_Ah_H_H_A_061_Anew__addrs_Ah_Aas_Ah_H_H_092_060close_062,axiom,
    ( ( hoare_new_addrs @ h2 @ ( hoare_new_addrs @ h3 @ as @ h2 ) @ h )
    = ( hoare_new_addrs @ h3 @ as @ h ) ) ).

% \<open>new_addrs h' (new_addrs h as h') h'' = new_addrs h as h''\<close>
thf(fact_1__092_060open_062_Ih_M_Aas_J_A_092_060Turnstile_062_AP_092_060close_062,axiom,
    rep_assn @ p @ ( produc7507926704131184380et_nat @ h3 @ as ) ).

% \<open>(h, as) \<Turnstile> P\<close>
thf(fact_2_new__addr__refl,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_new_addrs @ H @ As @ H )
      = As ) ).

% new_addr_refl
thf(fact_3_POST__G,axiom,
    rep_assn @ ( q @ rg ) @ ( produc7507926704131184380et_nat @ h @ ( hoare_new_addrs @ h2 @ ( hoare_new_addrs @ h3 @ as @ h2 ) @ h ) ) ).

% POST_G
thf(fact_4_POST__F,axiom,
    rep_assn @ ( r @ rf ) @ ( produc7507926704131184380et_nat @ h2 @ ( hoare_new_addrs @ h3 @ as @ h2 ) ) ).

% POST_F
thf(fact_5_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( rep_assn @ X )
        = ( rep_assn @ Y ) )
      = ( X = Y ) ) ).

% Rep_assn_inject
thf(fact_6_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( ( product_Pair_int_int @ X1 @ X2 )
        = ( product_Pair_int_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_7_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_8_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_9_prod_Oinject,axiom,
    ! [X1: b,X2: produc6653097349344004940it_nat,Y1: b,Y2: produc6653097349344004940it_nat] :
      ( ( ( produc4082563078715348724it_nat @ X1 @ X2 )
        = ( produc4082563078715348724it_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_10_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_11_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_12_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_13_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_14_old_Oprod_Oinject,axiom,
    ! [A: b,B: produc6653097349344004940it_nat,A2: b,B2: produc6653097349344004940it_nat] :
      ( ( ( produc4082563078715348724it_nat @ A @ B )
        = ( produc4082563078715348724it_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_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_16_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ).

% one_assn_raw.cases
thf(fact_17_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P @ ( produc5001842942810119800et_nat @ Q @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ).

% times_assn_raw.cases
thf(fact_18_prod__induct4,axiom,
    ! [P2: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat > $o,C: heap_e7401611519738050253t_unit,D: set_nat] : ( P2 @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B3 @ ( produc7507926704131184380et_nat @ C @ D ) ) ) )
     => ( P2 @ X ) ) ).

% prod_induct4
thf(fact_19_prod__induct3,axiom,
    ! [P2: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: heap_e7401611519738050253t_unit,C: set_nat] : ( P2 @ ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_20_prod__induct3,axiom,
    ! [P2: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat] : ( P2 @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_21_prod__induct3,axiom,
    ! [P2: produc7388388658123137530it_nat > $o,X: produc7388388658123137530it_nat] :
      ( ! [A3: b,B3: heap_e7401611519738050253t_unit,C: nat] : ( P2 @ ( produc4082563078715348724it_nat @ A3 @ ( produc584006145561248582it_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_22_prod__induct3,axiom,
    ! [P2: produc3260487557148687353it_nat > $o,X: produc3260487557148687353it_nat] :
      ( ! [A3: a,B3: heap_e7401611519738050253t_unit,C: nat] : ( P2 @ ( produc9178034014595674355it_nat @ A3 @ ( produc584006145561248582it_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_23_prod__cases4,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat > $o,C: heap_e7401611519738050253t_unit,D: set_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B3 @ ( produc7507926704131184380et_nat @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_24_prod__cases3,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: heap_e7401611519738050253t_unit,C: set_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_25_prod__cases3,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_26_prod__cases3,axiom,
    ! [Y: produc7388388658123137530it_nat] :
      ~ ! [A3: b,B3: heap_e7401611519738050253t_unit,C: nat] :
          ( Y
         != ( produc4082563078715348724it_nat @ A3 @ ( produc584006145561248582it_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_27_prod__cases3,axiom,
    ! [Y: produc3260487557148687353it_nat] :
      ~ ! [A3: a,B3: heap_e7401611519738050253t_unit,C: nat] :
          ( Y
         != ( produc9178034014595674355it_nat @ A3 @ ( produc584006145561248582it_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_28_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_29_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_30_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_31_Pair__inject,axiom,
    ! [A: b,B: produc6653097349344004940it_nat,A2: b,B2: produc6653097349344004940it_nat] :
      ( ( ( produc4082563078715348724it_nat @ A @ B )
        = ( produc4082563078715348724it_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_32_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_33_prod__cases,axiom,
    ! [P2: product_prod_int_int > $o,P3: product_prod_int_int] :
      ( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_34_prod__cases,axiom,
    ! [P2: produc3925858234332021118et_nat > $o,P3: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] : ( P2 @ ( produc5001842942810119800et_nat @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_35_prod__cases,axiom,
    ! [P2: produc2732055786443039994et_nat > $o,P3: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] : ( P2 @ ( produc2245416461498447860et_nat @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_36_prod__cases,axiom,
    ! [P2: produc7388388658123137530it_nat > $o,P3: produc7388388658123137530it_nat] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] : ( P2 @ ( produc4082563078715348724it_nat @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_37_prod__cases,axiom,
    ! [P2: produc3260487557148687353it_nat > $o,P3: produc3260487557148687353it_nat] :
      ( ! [A3: a,B3: produc6653097349344004940it_nat] : ( P2 @ ( produc9178034014595674355it_nat @ A3 @ B3 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_38_surj__pair,axiom,
    ! [P3: product_prod_int_int] :
    ? [X3: int,Y3: int] :
      ( P3
      = ( product_Pair_int_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_39_surj__pair,axiom,
    ! [P3: produc3925858234332021118et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
      ( P3
      = ( produc5001842942810119800et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_40_surj__pair,axiom,
    ! [P3: produc2732055786443039994et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
      ( P3
      = ( produc2245416461498447860et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_41_surj__pair,axiom,
    ! [P3: produc7388388658123137530it_nat] :
    ? [X3: b,Y3: produc6653097349344004940it_nat] :
      ( P3
      = ( produc4082563078715348724it_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_42_surj__pair,axiom,
    ! [P3: produc3260487557148687353it_nat] :
    ? [X3: a,Y3: produc6653097349344004940it_nat] :
      ( P3
      = ( produc9178034014595674355it_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_43_bex2I,axiom,
    ! [A: int,B: int,S: set_Pr958786334691620121nt_int,P2: int > int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S )
     => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S )
         => ( P2 @ A @ B ) )
       => ? [A3: int,B3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ S )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_44_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat,P2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S )
     => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S )
         => ( P2 @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
            ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A3 @ B3 ) @ S )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_45_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat,P2: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S )
     => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S )
         => ( P2 @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
            ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A3 @ B3 ) @ S )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_46_bex2I,axiom,
    ! [A: b,B: produc6653097349344004940it_nat,S: set_Pr7600907837789447088it_nat,P2: b > produc6653097349344004940it_nat > $o] :
      ( ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ A @ B ) @ S )
     => ( ( ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ A @ B ) @ S )
         => ( P2 @ A @ B ) )
       => ? [A3: b,B3: produc6653097349344004940it_nat] :
            ( ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ A3 @ B3 ) @ S )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_47_bex2I,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,S: set_Pr7098220151150636591it_nat,P2: a > produc6653097349344004940it_nat > $o] :
      ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ A @ B ) @ S )
     => ( ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ A @ B ) @ S )
         => ( P2 @ A @ B ) )
       => ? [A3: a,B3: produc6653097349344004940it_nat] :
            ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ A3 @ B3 ) @ S )
            & ( P2 @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_48_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] :
          ( Y
         != ( product_Pair_int_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_49_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_50_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_51_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7388388658123137530it_nat] :
      ~ ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( Y
         != ( produc4082563078715348724it_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3260487557148687353it_nat] :
      ~ ! [A3: a,B3: produc6653097349344004940it_nat] :
          ( Y
         != ( produc9178034014595674355it_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_53_T1,axiom,
    hoare_hoare_triple_a @ p @ f @ r ).

% T1
thf(fact_54_T2,axiom,
    ! [X: a] : ( hoare_hoare_triple_b @ ( r @ X ) @ ( g @ X ) @ q ) ).

% T2
thf(fact_55_EX__G,axiom,
    ( ( heap_Time_execute_b @ ( g @ rf ) @ h2 )
    = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ rg @ ( produc584006145561248582it_nat @ h @ t ) ) ) ) ).

% EX_G
thf(fact_56_LIM__G,axiom,
    ord_less_eq_nat @ ( lim_Product_unit @ h2 ) @ ( lim_Product_unit @ h ) ).

% LIM_G
thf(fact_57_LIM__F,axiom,
    ord_less_eq_nat @ ( lim_Product_unit @ h3 ) @ ( lim_Product_unit @ h2 ) ).

% LIM_F
thf(fact_58_EX__F,axiom,
    ( ( heap_Time_execute_a @ f @ h3 )
    = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ rf @ ( produc584006145561248582it_nat @ h2 @ t2 ) ) ) ) ).

% EX_F
thf(fact_59_mem__Collect__eq,axiom,
    ! [A: $o,P2: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( member2340774599025711042nt_int @ A @ ( collec5210948495886036740nt_int @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
    ! [A: int,P2: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
    ! [A4: set_o] :
      ( ( collect_o
        @ ^ [X4: $o] : ( member_o @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ( collec5210948495886036740nt_int
        @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
    ! [A4: set_int] :
      ( ( collect_int
        @ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_71_Collect__cong,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_list_nat @ P2 )
        = ( collect_list_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_72_Collect__cong,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec5210948495886036740nt_int @ P2 )
        = ( collec5210948495886036740nt_int @ Q2 ) ) ) ).

% Collect_cong
thf(fact_73_Collect__cong,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_set_nat @ P2 )
        = ( collect_set_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_74_Collect__cong,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_nat @ P2 )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_75_Collect__cong,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_int @ P2 )
        = ( collect_int @ Q2 ) ) ) ).

% Collect_cong
thf(fact_76_bijective__def,axiom,
    ( bijective_int_int
    = ( ^ [R: set_Pr958786334691620121nt_int] :
          ( ! [X4: int,Y4: int,Z: int] :
              ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R )
                & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Z ) @ R ) )
             => ( Y4 = Z ) )
          & ! [X4: int,Y4: int,Z: int] :
              ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Z ) @ R )
                & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y4 @ Z ) @ R ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_77_bijective__def,axiom,
    ( biject2615096655818420098et_nat
    = ( ^ [R: set_Pr3286484037609594932et_nat] :
          ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat,Z: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Z ) @ R ) )
             => ( Y4 = Z ) )
          & ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Z ) @ R )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y4 @ Z ) @ R ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_78_bijective__def,axiom,
    ( biject1468766312547416318et_nat
    = ( ^ [R: set_Pr8536935166611901872et_nat] :
          ( ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat,Z: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Z ) @ R ) )
             => ( Y4 = Z ) )
          & ! [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Z ) @ R )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y4 @ Z ) @ R ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_79_bijective__def,axiom,
    ( biject3496835315794560510it_nat
    = ( ^ [R: set_Pr7600907837789447088it_nat] :
          ( ! [X4: b,Y4: produc6653097349344004940it_nat,Z: produc6653097349344004940it_nat] :
              ( ( ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ R )
                & ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Z ) @ R ) )
             => ( Y4 = Z ) )
          & ! [X4: b,Y4: b,Z: produc6653097349344004940it_nat] :
              ( ( ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Z ) @ R )
                & ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ Y4 @ Z ) @ R ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_80_bijective__def,axiom,
    ( biject8592306251674886141it_nat
    = ( ^ [R: set_Pr7098220151150636591it_nat] :
          ( ! [X4: a,Y4: produc6653097349344004940it_nat,Z: produc6653097349344004940it_nat] :
              ( ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R )
                & ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Z ) @ R ) )
             => ( Y4 = Z ) )
          & ! [X4: a,Y4: a,Z: produc6653097349344004940it_nat] :
              ( ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Z ) @ R )
                & ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ Y4 @ Z ) @ R ) )
             => ( X4 = Y4 ) ) ) ) ) ).

% bijective_def
thf(fact_81_ssubst__Pair__rhs,axiom,
    ! [R2: int,S2: int,R3: set_Pr958786334691620121nt_int,S3: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S2 ) @ R3 )
     => ( ( S3 = S2 )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S3 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_82_ssubst__Pair__rhs,axiom,
    ! [R2: produc3658429121746597890et_nat > $o,S2: produc3658429121746597890et_nat,R3: set_Pr3286484037609594932et_nat,S3: produc3658429121746597890et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S2 ) @ R3 )
     => ( ( S3 = S2 )
       => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S3 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_83_ssubst__Pair__rhs,axiom,
    ! [R2: produc3658429121746597890et_nat > $o,S2: produc3925858234332021118et_nat,R3: set_Pr8536935166611901872et_nat,S3: produc3925858234332021118et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S2 ) @ R3 )
     => ( ( S3 = S2 )
       => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S3 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_84_ssubst__Pair__rhs,axiom,
    ! [R2: b,S2: produc6653097349344004940it_nat,R3: set_Pr7600907837789447088it_nat,S3: produc6653097349344004940it_nat] :
      ( ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ R2 @ S2 ) @ R3 )
     => ( ( S3 = S2 )
       => ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ R2 @ S3 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_85_ssubst__Pair__rhs,axiom,
    ! [R2: a,S2: produc6653097349344004940it_nat,R3: set_Pr7098220151150636591it_nat,S3: produc6653097349344004940it_nat] :
      ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ R2 @ S2 ) @ R3 )
     => ( ( S3 = S2 )
       => ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ R2 @ S3 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_86_curryI,axiom,
    ! [F: product_prod_int_int > $o,A: int,B: int] :
      ( ( F @ ( product_Pair_int_int @ A @ B ) )
     => ( produc175634133007206835_int_o @ F @ A @ B ) ) ).

% curryI
thf(fact_87_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_88_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_89_curryI,axiom,
    ! [F: produc7388388658123137530it_nat > $o,A: b,B: produc6653097349344004940it_nat] :
      ( ( F @ ( produc4082563078715348724it_nat @ A @ B ) )
     => ( produc8043051624866120150_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_90_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_91_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_92_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_93_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_94_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_95_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_96_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D2: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_set_int @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_97_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_rat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_98_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_num @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_99_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_nat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_100_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_int @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_101_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_Time_Heap_a,Q2: a > assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H2 )
         => ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 ) )
     => ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 ) ) ).

% hoare_triple_preI
thf(fact_102_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_Time_Heap_b,Q2: b > assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H2 )
         => ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 ) )
     => ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 ) ) ).

% hoare_triple_preI
thf(fact_103_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H2 )
         => ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 ) )
     => ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 ) ) ).

% hoare_triple_preI
thf(fact_104_curryD,axiom,
    ! [F: product_prod_int_int > $o,A: int,B: int] :
      ( ( produc175634133007206835_int_o @ F @ A @ B )
     => ( F @ ( product_Pair_int_int @ A @ B ) ) ) ).

% curryD
thf(fact_105_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_106_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_107_curryD,axiom,
    ! [F: produc7388388658123137530it_nat > $o,A: b,B: produc6653097349344004940it_nat] :
      ( ( produc8043051624866120150_nat_o @ F @ A @ B )
     => ( F @ ( produc4082563078715348724it_nat @ A @ B ) ) ) ).

% curryD
thf(fact_108_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_109_curryE,axiom,
    ! [F: product_prod_int_int > $o,A: int,B: int] :
      ( ( produc175634133007206835_int_o @ F @ A @ B )
     => ( F @ ( product_Pair_int_int @ A @ B ) ) ) ).

% curryE
thf(fact_110_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_111_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_112_curryE,axiom,
    ! [F: produc7388388658123137530it_nat > $o,A: b,B: produc6653097349344004940it_nat] :
      ( ( produc8043051624866120150_nat_o @ F @ A @ B )
     => ( F @ ( produc4082563078715348724it_nat @ A @ B ) ) ) ).

% curryE
thf(fact_113_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_114_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_115_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_116_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_117_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_118_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_119_option_Oinject,axiom,
    ! [X2: produc7388388658123137530it_nat,Y2: produc7388388658123137530it_nat] :
      ( ( ( some_P2818173045054083285it_nat @ X2 )
        = ( some_P2818173045054083285it_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_120_option_Oinject,axiom,
    ! [X2: produc3260487557148687353it_nat,Y2: produc3260487557148687353it_nat] :
      ( ( ( some_P7913643980934408916it_nat @ X2 )
        = ( some_P7913643980934408916it_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_121_option_Oinject,axiom,
    ! [X2: produc8664842809031399944it_nat,Y2: produc8664842809031399944it_nat] :
      ( ( ( some_P1914260805536162275it_nat @ X2 )
        = ( some_P1914260805536162275it_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_122_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( some_num @ X2 )
        = ( some_num @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_123_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_124_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_125_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_126_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_127_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_128_order__refl,axiom,
    ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).

% order_refl
thf(fact_129_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_130_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_131_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_132_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_133__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rg_Ah_H_H_At_H_H_O_A_092_060lbrakk_062execute_A_Ig_Arf_J_Ah_H_A_061_ASome_A_Irg_M_Ah_H_H_M_At_H_H_J_059_A_Ih_H_H_M_Anew__addrs_Ah_H_A_Inew__addrs_Ah_Aas_Ah_H_J_Ah_H_H_J_A_092_060Turnstile_062_AQ_Arg_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_H_A_092_060and_062_Aa_A_092_060notin_062_Anew__addrs_Ah_Aas_Ah_H_125_Ah_H_Ah_H_H_059_Alim_Ah_H_A_092_060le_062_Alim_Ah_H_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Rg: b,H3: heap_e7401611519738050253t_unit] :
        ( ? [T: nat] :
            ( ( heap_Time_execute_b @ ( g @ rf ) @ h2 )
            = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ Rg @ ( produc584006145561248582it_nat @ H3 @ T ) ) ) )
       => ( ( rep_assn @ ( q @ Rg ) @ ( produc7507926704131184380et_nat @ H3 @ ( hoare_new_addrs @ h2 @ ( hoare_new_addrs @ h3 @ as @ h2 ) @ H3 ) ) )
         => ( ( relH
              @ ( collect_nat
                @ ^ [A5: nat] :
                    ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h2 ) )
                    & ~ ( member_nat @ A5 @ ( hoare_new_addrs @ h3 @ as @ h2 ) ) ) )
              @ h2
              @ H3 )
           => ~ ( ord_less_eq_nat @ ( lim_Product_unit @ h2 ) @ ( lim_Product_unit @ H3 ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>rg h'' t''. \<lbrakk>execute (g rf) h' = Some (rg, h'', t''); (h'', new_addrs h' (new_addrs h as h') h'') \<Turnstile> Q rg; relH {a. a < lim h' \<and> a \<notin> new_addrs h as h'} h' h''; lim h' \<le> lim h''\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_134__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rf_Ah_H_At_H_O_A_092_060lbrakk_062execute_Af_Ah_A_061_ASome_A_Irf_M_Ah_H_M_At_H_J_059_A_Ih_H_M_Anew__addrs_Ah_Aas_Ah_H_J_A_092_060Turnstile_062_AR_Arf_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_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,
    ~ ! [Rf: a,H4: heap_e7401611519738050253t_unit] :
        ( ? [T2: nat] :
            ( ( heap_Time_execute_a @ f @ h3 )
            = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ Rf @ ( produc584006145561248582it_nat @ H4 @ T2 ) ) ) )
       => ( ( rep_assn @ ( r @ Rf ) @ ( produc7507926704131184380et_nat @ H4 @ ( hoare_new_addrs @ h3 @ as @ H4 ) ) )
         => ( ( relH
              @ ( collect_nat
                @ ^ [A5: nat] :
                    ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h3 ) )
                    & ~ ( member_nat @ A5 @ as ) ) )
              @ h3
              @ H4 )
           => ~ ( ord_less_eq_nat @ ( lim_Product_unit @ h3 ) @ ( lim_Product_unit @ H4 ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>rf h' t'. \<lbrakk>execute f h = Some (rf, h', t'); (h', new_addrs h as h') \<Turnstile> R rf; relH {a. a < lim h \<and> a \<notin> as} h h'; lim h \<le> lim h'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_135_the__default_Osimps_I1_J,axiom,
    ! [Uu: produc7388388658123137530it_nat,X: produc7388388658123137530it_nat] :
      ( ( the_de4670738602788725795it_nat @ Uu @ ( some_P2818173045054083285it_nat @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_136_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_137_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_138_the__default_Osimps_I1_J,axiom,
    ! [Uu: num,X: num] :
      ( ( the_default_num @ Uu @ ( some_num @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_139_hoare__triple__effect,axiom,
    ! [P2: assn,C2: heap_Time_Heap_a,Q2: a > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ? [H4: heap_e7401611519738050253t_unit,R4: a,T3: nat] :
            ( ( heap_Time_effect_a @ C2 @ H @ H4 @ R4 @ T3 )
            & ( rep_assn @ ( Q2 @ R4 ) @ ( produc7507926704131184380et_nat @ H4 @ ( hoare_new_addrs @ H @ As @ H4 ) ) ) ) ) ) ).

% hoare_triple_effect
thf(fact_140_hoare__triple__effect,axiom,
    ! [P2: assn,C2: heap_Time_Heap_b,Q2: b > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ? [H4: heap_e7401611519738050253t_unit,R4: b,T3: nat] :
            ( ( heap_Time_effect_b @ C2 @ H @ H4 @ R4 @ T3 )
            & ( rep_assn @ ( Q2 @ R4 ) @ ( produc7507926704131184380et_nat @ H4 @ ( hoare_new_addrs @ H @ As @ H4 ) ) ) ) ) ) ).

% hoare_triple_effect
thf(fact_141_hoare__triple__effect,axiom,
    ! [P2: assn,C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ? [H4: heap_e7401611519738050253t_unit,R4: product_unit,T3: nat] :
            ( ( heap_T6553295506729943825t_unit @ C2 @ H @ H4 @ R4 @ T3 )
            & ( rep_assn @ ( Q2 @ R4 ) @ ( produc7507926704131184380et_nat @ H4 @ ( hoare_new_addrs @ H @ As @ H4 ) ) ) ) ) ) ).

% hoare_triple_effect
thf(fact_142_hoare__triple__success,axiom,
    ! [P2: assn,C2: heap_Time_Heap_a,Q2: a > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ( heap_Time_success_a @ C2 @ H ) ) ) ).

% hoare_triple_success
thf(fact_143_hoare__triple__success,axiom,
    ! [P2: assn,C2: heap_Time_Heap_b,Q2: b > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ( heap_Time_success_b @ C2 @ H ) ) ) ).

% hoare_triple_success
thf(fact_144_hoare__triple__success,axiom,
    ! [P2: assn,C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ( heap_T584514906347983379t_unit @ C2 @ H ) ) ) ).

% hoare_triple_success
thf(fact_145_relChain__def,axiom,
    ( bNF_Ca583493526879471924et_int
    = ( ^ [R5: set_Pr958786334691620121nt_int,As3: int > set_int] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R5 )
         => ( ord_less_eq_set_int @ ( As3 @ I ) @ ( As3 @ J ) ) ) ) ) ).

% relChain_def
thf(fact_146_relChain__def,axiom,
    ( bNF_Ca1332973979827979050nt_rat
    = ( ^ [R5: set_Pr958786334691620121nt_int,As3: int > rat] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R5 )
         => ( ord_less_eq_rat @ ( As3 @ I ) @ ( As3 @ J ) ) ) ) ) ).

% relChain_def
thf(fact_147_relChain__def,axiom,
    ( bNF_Ca7748807862925029228nt_num
    = ( ^ [R5: set_Pr958786334691620121nt_int,As3: int > num] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R5 )
         => ( ord_less_eq_num @ ( As3 @ I ) @ ( As3 @ J ) ) ) ) ) ).

% relChain_def
thf(fact_148_relChain__def,axiom,
    ( bNF_Ca1968104039914474786nt_nat
    = ( ^ [R5: set_Pr958786334691620121nt_int,As3: int > nat] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R5 )
         => ( ord_less_eq_nat @ ( As3 @ I ) @ ( As3 @ J ) ) ) ) ) ).

% relChain_def
thf(fact_149_relChain__def,axiom,
    ( bNF_Ca1965613569405424510nt_int
    = ( ^ [R5: set_Pr958786334691620121nt_int,As3: int > int] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R5 )
         => ( ord_less_eq_int @ ( As3 @ I ) @ ( As3 @ J ) ) ) ) ) ).

% relChain_def
thf(fact_150_Heap__eqI,axiom,
    ! [F: heap_Time_Heap_b,G: heap_Time_Heap_b] :
      ( ! [H2: heap_e7401611519738050253t_unit] :
          ( ( heap_Time_execute_b @ F @ H2 )
          = ( heap_Time_execute_b @ G @ H2 ) )
     => ( F = G ) ) ).

% Heap_eqI
thf(fact_151_Heap__eqI,axiom,
    ! [F: heap_Time_Heap_a,G: heap_Time_Heap_a] :
      ( ! [H2: heap_e7401611519738050253t_unit] :
          ( ( heap_Time_execute_a @ F @ H2 )
          = ( heap_Time_execute_a @ G @ H2 ) )
     => ( F = G ) ) ).

% Heap_eqI
thf(fact_152_RH__G,axiom,
    ( relH
    @ ( collect_nat
      @ ^ [A5: nat] :
          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h2 ) )
          & ~ ( member_nat @ A5 @ ( hoare_new_addrs @ h3 @ as @ h2 ) ) ) )
    @ h2
    @ h ) ).

% RH_G
thf(fact_153_RH__F,axiom,
    ( relH
    @ ( collect_nat
      @ ^ [A5: nat] :
          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ h3 ) )
          & ~ ( member_nat @ A5 @ as ) ) )
    @ h3
    @ h2 ) ).

% RH_F
thf(fact_154_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_155_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_156_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_157_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_158_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_159_relH__subset,axiom,
    ! [Bs: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( relH @ Bs @ H @ H5 )
     => ( ( ord_less_eq_set_nat @ As @ Bs )
       => ( relH @ As @ H @ H5 ) ) ) ).

% relH_subset
thf(fact_160_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).

% lt_ex
thf(fact_161_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_162_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).

% gt_ex
thf(fact_163_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_164_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_165_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_166_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_167_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_168_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_169_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_170_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_171_order_Oasym,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ~ ( ord_less_assn @ B @ A ) ) ).

% order.asym
thf(fact_172_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_173_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_174_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_175_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_176_ord__eq__less__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( A = B )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_177_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_178_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_179_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_180_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_181_ord__less__eq__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_182_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_183_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_184_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_185_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_186_less__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P2 @ Y5 ) )
         => ( P2 @ X3 ) )
     => ( P2 @ A ) ) ).

% less_induct
thf(fact_187_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_188_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_189_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_190_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_191_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_192_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_193_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_194_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_195_dual__order_Oasym,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ~ ( ord_less_assn @ A @ B ) ) ).

% dual_order.asym
thf(fact_196_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_197_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_198_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_199_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_200_dual__order_Oirrefl,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ A ) ).

% dual_order.irrefl
thf(fact_201_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_202_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_203_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_204_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_205_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X5: nat] : ( P4 @ X5 ) )
    = ( ^ [P5: nat > $o] :
        ? [N: nat] :
          ( ( P5 @ N )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N )
             => ~ ( P5 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_206_linorder__less__wlog,axiom,
    ! [P2: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_rat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: rat] : ( P2 @ A3 @ A3 )
       => ( ! [A3: rat,B3: rat] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_207_linorder__less__wlog,axiom,
    ! [P2: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: num] : ( P2 @ A3 @ A3 )
       => ( ! [A3: num,B3: num] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_208_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P2 @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_209_linorder__less__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: int] : ( P2 @ A3 @ A3 )
       => ( ! [A3: int,B3: int] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_210_order_Ostrict__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_211_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_212_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_213_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_214_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_215_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_216_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_217_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_218_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_219_dual__order_Ostrict__trans,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( ord_less_assn @ C2 @ B )
       => ( ord_less_assn @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_220_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_221_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_222_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_223_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_224_order_Ostrict__implies__not__eq,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_225_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_226_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_227_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_228_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_229_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_230_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_231_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_232_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_233_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_234_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_235_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_236_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_237_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_238_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_asym
thf(fact_239_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_240_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_241_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_242_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_243_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_244_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_245_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_246_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_247_order__less__asym_H,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ~ ( ord_less_assn @ B @ A ) ) ).

% order_less_asym'
thf(fact_248_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_249_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_250_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_251_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_252_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ Z3 )
       => ( ord_less_assn @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_253_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z3 )
       => ( ord_less_rat @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_254_order__less__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z3 )
       => ( ord_less_num @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_255_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_256_order__less__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_257_ord__eq__less__subst,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_258_ord__eq__less__subst,axiom,
    ! [A: rat,F: assn > rat,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_259_ord__eq__less__subst,axiom,
    ! [A: num,F: assn > num,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_260_ord__eq__less__subst,axiom,
    ! [A: nat,F: assn > nat,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_261_ord__eq__less__subst,axiom,
    ! [A: int,F: assn > int,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_262_ord__eq__less__subst,axiom,
    ! [A: assn,F: rat > assn,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_263_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_264_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_265_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_266_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_267_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_268_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > rat,C2: rat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_269_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > num,C2: num] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_270_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_271_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > int,C2: int] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_272_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > assn,C2: assn] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_273_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_274_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_275_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_276_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_277_order__less__irrefl,axiom,
    ! [X: assn] :
      ~ ( ord_less_assn @ X @ X ) ).

% order_less_irrefl
thf(fact_278_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_279_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_280_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_281_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_282_order__less__subst1,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_283_order__less__subst1,axiom,
    ! [A: assn,F: rat > assn,B: rat,C2: rat] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_284_order__less__subst1,axiom,
    ! [A: assn,F: num > assn,B: num,C2: num] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_285_order__less__subst1,axiom,
    ! [A: assn,F: nat > assn,B: nat,C2: nat] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_286_order__less__subst1,axiom,
    ! [A: assn,F: int > assn,B: int,C2: int] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_287_order__less__subst1,axiom,
    ! [A: rat,F: assn > rat,B: assn,C2: assn] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_288_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_289_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_290_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_291_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C2: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_292_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_293_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > rat,C2: rat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_294_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > num,C2: num] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_295_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_296_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > int,C2: int] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_297_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > assn,C2: assn] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_298_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_299_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_300_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_301_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_302_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_303_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_304_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_305_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_306_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_307_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,P2: $o] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_308_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P2: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_309_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P2: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_310_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P2: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_311_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P2: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_312_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_313_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_314_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_315_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_316_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_317_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_318_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_319_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_320_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_321_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_322_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_323_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_324_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_325_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_326_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_327_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_328_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_329_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_330_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_331_relH__trans,axiom,
    ! [As: set_nat,H1: heap_e7401611519738050253t_unit,H22: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H1 @ H22 )
     => ( ( relH @ As @ H22 @ H32 )
       => ( relH @ As @ H1 @ H32 ) ) ) ).

% relH_trans
thf(fact_332_relH__sym,axiom,
    ! [As: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H @ H5 )
     => ( relH @ As @ H5 @ H ) ) ).

% relH_sym
thf(fact_333_leD,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ~ ( ord_less_assn @ X @ Y ) ) ).

% leD
thf(fact_334_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ~ ( ord_less_set_int @ X @ Y ) ) ).

% leD
thf(fact_335_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_336_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_337_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_338_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_339_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_340_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_341_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_342_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_343_nless__le,axiom,
    ! [A: assn,B: assn] :
      ( ( ~ ( ord_less_assn @ A @ B ) )
      = ( ~ ( ord_less_eq_assn @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_344_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_345_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_346_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_347_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_348_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_349_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_350_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_351_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_352_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_353_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_354_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_355_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ~ ( ord_less_assn @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_356_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_357_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_358_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_359_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_360_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_361_dense__ge,axiom,
    ! [Z3: rat,Y: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ Z3 @ X3 )
         => ( ord_less_eq_rat @ Y @ X3 ) )
     => ( ord_less_eq_rat @ Y @ Z3 ) ) ).

% dense_ge
thf(fact_362_dense__le,axiom,
    ! [Y: rat,Z3: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Y )
         => ( ord_less_eq_rat @ X3 @ Z3 ) )
     => ( ord_less_eq_rat @ Y @ Z3 ) ) ).

% dense_le
thf(fact_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [A5: assn,B4: assn] :
          ( ( ord_less_assn @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_374_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_set_int @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_375_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_rat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_376_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_num @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_377_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_378_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_int @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_379_order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [A5: assn,B4: assn] :
          ( ( ord_less_eq_assn @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_380_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_381_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_382_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_383_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_384_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_385_order_Ostrict__trans1,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_386_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_set_int @ B @ C2 )
       => ( ord_less_set_int @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_387_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_388_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_389_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_390_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_391_order_Ostrict__trans2,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_392_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ord_less_set_int @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_393_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_394_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_395_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_396_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_397_order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [A5: assn,B4: assn] :
          ( ( ord_less_eq_assn @ A5 @ B4 )
          & ~ ( ord_less_eq_assn @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_398_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B4 )
          & ~ ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_399_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_400_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_401_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_402_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_403_dense__ge__bounded,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z3 @ X )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z3 @ W )
           => ( ( ord_less_rat @ W @ X )
             => ( ord_less_eq_rat @ Y @ W ) ) )
       => ( ord_less_eq_rat @ Y @ Z3 ) ) ) ).

% dense_ge_bounded
thf(fact_404_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X @ W )
           => ( ( ord_less_rat @ W @ Y )
             => ( ord_less_eq_rat @ W @ Z3 ) ) )
       => ( ord_less_eq_rat @ Y @ Z3 ) ) ) ).

% dense_le_bounded
thf(fact_405_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [B4: assn,A5: assn] :
          ( ( ord_less_assn @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_406_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( ( ord_less_set_int @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_407_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_rat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_408_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_num @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_409_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_nat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_410_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_int @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_411_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [B4: assn,A5: assn] :
          ( ( ord_less_eq_assn @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_412_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_413_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_414_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_415_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_416_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_417_dual__order_Ostrict__trans1,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( ord_less_assn @ C2 @ B )
       => ( ord_less_assn @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_418_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_set_int @ C2 @ B )
       => ( ord_less_set_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_419_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_420_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_421_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_422_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_423_dual__order_Ostrict__trans2,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( ord_less_eq_assn @ C2 @ B )
       => ( ord_less_assn @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_424_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C2 @ B )
       => ( ord_less_set_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_425_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_426_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_427_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_428_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_429_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [B4: assn,A5: assn] :
          ( ( ord_less_eq_assn @ B4 @ A5 )
          & ~ ( ord_less_eq_assn @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_430_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A5 )
          & ~ ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_431_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_432_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_433_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_434_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_435_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_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_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_444_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_445_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_446_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_447_order__le__less,axiom,
    ( ord_less_eq_assn
    = ( ^ [X4: assn,Y4: assn] :
          ( ( ord_less_assn @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_448_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_449_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_rat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_450_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_num @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_451_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_nat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_452_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_int @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_453_order__less__le,axiom,
    ( ord_less_assn
    = ( ^ [X4: assn,Y4: assn] :
          ( ( ord_less_eq_assn @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_454_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_455_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_456_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_457_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_458_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_459_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_460_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_461_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_462_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_463_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_464_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_465_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_466_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ Z3 )
       => ( ord_less_assn @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_486_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ Y @ Z3 )
       => ( ord_less_set_int @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_487_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z3 )
       => ( ord_less_rat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_488_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z3 )
       => ( ord_less_num @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_489_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_490_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_491_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ Z3 )
       => ( ord_less_assn @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_492_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z3 )
       => ( ord_less_set_int @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_493_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z3 )
       => ( ord_less_rat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_494_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z3 )
       => ( ord_less_num @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_495_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_496_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_497_order__le__less__subst1,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_498_order__le__less__subst1,axiom,
    ! [A: assn,F: rat > assn,B: rat,C2: rat] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_499_order__le__less__subst1,axiom,
    ! [A: assn,F: num > assn,B: num,C2: num] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_500_order__le__less__subst1,axiom,
    ! [A: assn,F: nat > assn,B: nat,C2: nat] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_501_order__le__less__subst1,axiom,
    ! [A: assn,F: int > assn,B: int,C2: int] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_502_order__le__less__subst1,axiom,
    ! [A: rat,F: assn > rat,B: assn,C2: assn] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_503_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_504_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_505_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_506_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C2: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_507_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > assn,C2: assn] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_508_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_509_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_510_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_511_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_512_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > assn,C2: assn] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_513_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_514_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_515_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_516_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_517_order__less__le__subst1,axiom,
    ! [A: assn,F: rat > assn,B: rat,C2: rat] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_518_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_519_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C2: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_520_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C2: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_521_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C2: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_522_order__less__le__subst1,axiom,
    ! [A: assn,F: num > assn,B: num,C2: num] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_523_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_524_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C2: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_525_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C2: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_526_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C2: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_527_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_528_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > assn,C2: assn] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_529_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > assn,C2: assn] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_530_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_531_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > assn,C2: assn] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_532_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: assn > rat,C2: rat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_533_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_534_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C2: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_535_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C2: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_536_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C2: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_exists__leI,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ N2 )
           => ~ ( P2 @ N3 ) )
       => ( P2 @ N2 ) )
     => ? [N4: nat] :
          ( ( ord_less_eq_nat @ N4 @ N2 )
          & ( P2 @ N4 ) ) ) ).

% exists_leI
thf(fact_548_mod__relH,axiom,
    ! [As: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,P2: assn] :
      ( ( relH @ As @ H @ H5 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
        = ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H5 @ As ) ) ) ) ).

% mod_relH
thf(fact_549_hoare__triple__def,axiom,
    ( hoare_hoare_triple_a
    = ( ^ [P5: assn,C3: heap_Time_Heap_a,Q3: a > assn] :
        ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( ( rep_assn @ P5 @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
         => ? [R5: a,H7: heap_e7401611519738050253t_unit] :
              ( ? [T4: nat] :
                  ( ( heap_Time_execute_a @ C3 @ H6 )
                  = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ T4 ) ) ) )
              & ( rep_assn @ ( Q3 @ R5 ) @ ( produc7507926704131184380et_nat @ H7 @ ( hoare_new_addrs @ H6 @ As4 @ H7 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H6 ) )
                      & ~ ( member_nat @ A5 @ As4 ) ) )
                @ H6
                @ H7 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H6 ) @ ( lim_Product_unit @ H7 ) ) ) ) ) ) ).

% hoare_triple_def
thf(fact_550_hoare__triple__def,axiom,
    ( hoare_hoare_triple_b
    = ( ^ [P5: assn,C3: heap_Time_Heap_b,Q3: b > assn] :
        ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( ( rep_assn @ P5 @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
         => ? [R5: b,H7: heap_e7401611519738050253t_unit] :
              ( ? [T4: nat] :
                  ( ( heap_Time_execute_b @ C3 @ H6 )
                  = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ T4 ) ) ) )
              & ( rep_assn @ ( Q3 @ R5 ) @ ( produc7507926704131184380et_nat @ H7 @ ( hoare_new_addrs @ H6 @ As4 @ H7 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H6 ) )
                      & ~ ( member_nat @ A5 @ As4 ) ) )
                @ H6
                @ H7 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H6 ) @ ( lim_Product_unit @ H7 ) ) ) ) ) ) ).

% hoare_triple_def
thf(fact_551_hoare__triple__def,axiom,
    ( hoare_8945653483474564448t_unit
    = ( ^ [P5: assn,C3: heap_T5738788834812785303t_unit,Q3: product_unit > assn] :
        ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( ( rep_assn @ P5 @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
         => ? [R5: product_unit,H7: heap_e7401611519738050253t_unit] :
              ( ? [T4: nat] :
                  ( ( heap_T875086893843062177t_unit @ C3 @ H6 )
                  = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ T4 ) ) ) )
              & ( rep_assn @ ( Q3 @ R5 ) @ ( produc7507926704131184380et_nat @ H7 @ ( hoare_new_addrs @ H6 @ As4 @ H7 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H6 ) )
                      & ~ ( member_nat @ A5 @ As4 ) ) )
                @ H6
                @ H7 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H6 ) @ ( lim_Product_unit @ H7 ) ) ) ) ) ) ).

% hoare_triple_def
thf(fact_552_hoare__tripleI,axiom,
    ! [P2: assn,C2: heap_Time_Heap_a,Q2: a > assn] :
      ( ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
          ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
         => ? [R6: a,H8: heap_e7401611519738050253t_unit,T5: nat] :
              ( ( ( heap_Time_execute_a @ C2 @ H2 )
                = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R6 @ ( produc584006145561248582it_nat @ H8 @ T5 ) ) ) )
              & ( rep_assn @ ( Q2 @ R6 ) @ ( produc7507926704131184380et_nat @ H8 @ ( hoare_new_addrs @ H2 @ As2 @ H8 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H2 ) )
                      & ~ ( member_nat @ A5 @ As2 ) ) )
                @ H2
                @ H8 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H2 ) @ ( lim_Product_unit @ H8 ) ) ) )
     => ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 ) ) ).

% hoare_tripleI
thf(fact_553_hoare__tripleI,axiom,
    ! [P2: assn,C2: heap_Time_Heap_b,Q2: b > assn] :
      ( ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
          ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
         => ? [R6: b,H8: heap_e7401611519738050253t_unit,T5: nat] :
              ( ( ( heap_Time_execute_b @ C2 @ H2 )
                = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R6 @ ( produc584006145561248582it_nat @ H8 @ T5 ) ) ) )
              & ( rep_assn @ ( Q2 @ R6 ) @ ( produc7507926704131184380et_nat @ H8 @ ( hoare_new_addrs @ H2 @ As2 @ H8 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H2 ) )
                      & ~ ( member_nat @ A5 @ As2 ) ) )
                @ H2
                @ H8 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H2 ) @ ( lim_Product_unit @ H8 ) ) ) )
     => ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 ) ) ).

% hoare_tripleI
thf(fact_554_hoare__tripleI,axiom,
    ! [P2: assn,C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn] :
      ( ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
          ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
         => ? [R6: product_unit,H8: heap_e7401611519738050253t_unit,T5: nat] :
              ( ( ( heap_T875086893843062177t_unit @ C2 @ H2 )
                = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R6 @ ( produc584006145561248582it_nat @ H8 @ T5 ) ) ) )
              & ( rep_assn @ ( Q2 @ R6 ) @ ( produc7507926704131184380et_nat @ H8 @ ( hoare_new_addrs @ H2 @ As2 @ H8 ) ) )
              & ( relH
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H2 ) )
                      & ~ ( member_nat @ A5 @ As2 ) ) )
                @ H2
                @ H8 )
              & ( ord_less_eq_nat @ ( lim_Product_unit @ H2 ) @ ( lim_Product_unit @ H8 ) ) ) )
     => ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 ) ) ).

% hoare_tripleI
thf(fact_555_hoare__tripleE,axiom,
    ! [P2: assn,C2: heap_Time_Heap_a,Q2: a > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ~ ! [R4: a,H4: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] :
                  ( ( heap_Time_execute_a @ C2 @ H )
                  = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H4 @ T3 ) ) ) )
             => ( ( rep_assn @ ( Q2 @ R4 ) @ ( produc7507926704131184380et_nat @ H4 @ ( hoare_new_addrs @ H @ As @ H4 ) ) )
               => ( ( relH
                    @ ( collect_nat
                      @ ^ [A5: nat] :
                          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H ) )
                          & ~ ( member_nat @ A5 @ As ) ) )
                    @ H
                    @ H4 )
                 => ~ ( ord_less_eq_nat @ ( lim_Product_unit @ H ) @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% hoare_tripleE
thf(fact_556_hoare__tripleE,axiom,
    ! [P2: assn,C2: heap_Time_Heap_b,Q2: b > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ~ ! [R4: b,H4: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] :
                  ( ( heap_Time_execute_b @ C2 @ H )
                  = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R4 @ ( produc584006145561248582it_nat @ H4 @ T3 ) ) ) )
             => ( ( rep_assn @ ( Q2 @ R4 ) @ ( produc7507926704131184380et_nat @ H4 @ ( hoare_new_addrs @ H @ As @ H4 ) ) )
               => ( ( relH
                    @ ( collect_nat
                      @ ^ [A5: nat] :
                          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H ) )
                          & ~ ( member_nat @ A5 @ As ) ) )
                    @ H
                    @ H4 )
                 => ~ ( ord_less_eq_nat @ ( lim_Product_unit @ H ) @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% hoare_tripleE
thf(fact_557_hoare__tripleE,axiom,
    ! [P2: assn,C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
       => ~ ! [R4: product_unit,H4: heap_e7401611519738050253t_unit] :
              ( ? [T3: nat] :
                  ( ( heap_T875086893843062177t_unit @ C2 @ H )
                  = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H4 @ T3 ) ) ) )
             => ( ( rep_assn @ ( Q2 @ R4 ) @ ( produc7507926704131184380et_nat @ H4 @ ( hoare_new_addrs @ H @ As @ H4 ) ) )
               => ( ( relH
                    @ ( collect_nat
                      @ ^ [A5: nat] :
                          ( ( ord_less_nat @ A5 @ ( lim_Product_unit @ H ) )
                          & ~ ( member_nat @ A5 @ As ) ) )
                    @ H
                    @ H4 )
                 => ~ ( ord_less_eq_nat @ ( lim_Product_unit @ H ) @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% hoare_tripleE
thf(fact_558_successE,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
      ( ( heap_T584514906347983379t_unit @ F @ H )
     => ~ ! [R4: product_unit,H4: produc6653097349344004940it_nat] :
            ( ( heap_T875086893843062177t_unit @ F @ H )
           != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ H4 ) ) ) ) ).

% successE
thf(fact_559_successE,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_success_b @ F @ H )
     => ~ ! [R4: b,H4: produc6653097349344004940it_nat] :
            ( ( heap_Time_execute_b @ F @ H )
           != ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R4 @ H4 ) ) ) ) ).

% successE
thf(fact_560_successE,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_success_a @ F @ H )
     => ~ ! [R4: a,H4: produc6653097349344004940it_nat] :
            ( ( heap_Time_execute_a @ F @ H )
           != ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ H4 ) ) ) ) ).

% successE
thf(fact_561_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_562_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_563_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_564_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_565_le__cases3,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z3 ) )
       => ( ( ( ord_less_eq_rat @ X @ Z3 )
           => ~ ( ord_less_eq_rat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z3 @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z3 )
               => ~ ( ord_less_eq_rat @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z3 @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_566_le__cases3,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z3 ) )
       => ( ( ( ord_less_eq_num @ X @ Z3 )
           => ~ ( ord_less_eq_num @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_num @ Z3 @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z3 )
               => ~ ( ord_less_eq_num @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_num @ Z3 @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_567_le__cases3,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_568_le__cases3,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z3 ) )
       => ( ( ( ord_less_eq_int @ X @ Z3 )
           => ~ ( ord_less_eq_int @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_int @ Z3 @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z3 )
               => ~ ( ord_less_eq_int @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_int @ Z3 @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_569_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
    = ( ^ [X4: set_int,Y4: set_int] :
          ( ( ord_less_eq_set_int @ X4 @ Y4 )
          & ( ord_less_eq_set_int @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_570_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_571_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_572_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_573_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_574_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ord_less_eq_set_int @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_575_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_576_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_577_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_578_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_579_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_set_int @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_580_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_581_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_582_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_583_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_584_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_585_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_586_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_587_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_588_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_589_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ord_less_eq_set_int @ A @ C2 ) ) ) ).

% order.trans
thf(fact_590_order_Otrans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_591_order_Otrans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% order.trans
thf(fact_592_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_593_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% order.trans
thf(fact_594_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z3 )
       => ( ord_less_eq_set_int @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_595_order__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z3 )
       => ( ord_less_eq_rat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_596_order__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z3 )
       => ( ord_less_eq_num @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_597_order__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_598_order__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z3 )
       => ( ord_less_eq_int @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_599_linorder__wlog,axiom,
    ! [P2: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: rat,B3: rat] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_600_linorder__wlog,axiom,
    ! [P2: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: num,B3: num] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_601_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_602_linorder__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: int,B3: int] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_603_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A5 )
          & ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_604_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_605_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_606_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_607_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_608_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_609_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_610_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_611_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_612_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_613_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C2 @ B )
       => ( ord_less_eq_set_int @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_614_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_615_dual__order_Otrans,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ B )
       => ( ord_less_eq_num @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_616_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_617_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_618_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_619_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_620_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_621_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_622_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_623_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B4 )
          & ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_624_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_625_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_626_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_627_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_628_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_629_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_630_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_631_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C2: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_632_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_633_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_634_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_635_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C2: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_636_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_637_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C2: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_638_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_639_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_640_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_641_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_642_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_643_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_644_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_645_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_646_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_647_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C2: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_648_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_649_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_650_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_651_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_652_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_653_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_654_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_655_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_656_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_657_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_658_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_659_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_660_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C2: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_661_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C2: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_662_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C2: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_663_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C2: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_664_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C2: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_665_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_666_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_667_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_668_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_669_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_670_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_671_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_672_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_673_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_674_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_675_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_676_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C2: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_effect__def,axiom,
    ( heap_T6553295506729943825t_unit
    = ( ^ [C3: heap_T5738788834812785303t_unit,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: product_unit,N: nat] :
          ( ( heap_T875086893843062177t_unit @ C3 @ H6 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ N ) ) ) ) ) ) ).

% effect_def
thf(fact_687_effect__def,axiom,
    ( heap_Time_effect_b
    = ( ^ [C3: heap_Time_Heap_b,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: b,N: nat] :
          ( ( heap_Time_execute_b @ C3 @ H6 )
          = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ N ) ) ) ) ) ) ).

% effect_def
thf(fact_688_effect__def,axiom,
    ( heap_Time_effect_a
    = ( ^ [C3: heap_Time_Heap_a,H6: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: a,N: nat] :
          ( ( heap_Time_execute_a @ C3 @ H6 )
          = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ N ) ) ) ) ) ) ).

% effect_def
thf(fact_689_effectI,axiom,
    ! [C2: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,R2: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( ( heap_T875086893843062177t_unit @ C2 @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( heap_T6553295506729943825t_unit @ C2 @ H @ H5 @ R2 @ N2 ) ) ).

% effectI
thf(fact_690_effectI,axiom,
    ! [C2: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit,R2: b,H5: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( ( heap_Time_execute_b @ C2 @ H )
        = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R2 @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( heap_Time_effect_b @ C2 @ H @ H5 @ R2 @ N2 ) ) ).

% effectI
thf(fact_691_effectI,axiom,
    ! [C2: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,R2: a,H5: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( ( heap_Time_execute_a @ C2 @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( heap_Time_effect_a @ C2 @ H @ H5 @ R2 @ N2 ) ) ).

% effectI
thf(fact_692_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
          & ( M3 != N ) ) ) ) ).

% nat_less_le
thf(fact_693_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_694_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_695_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_696_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_697_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J2: nat] :
      ( ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_698_nat__descend__induct,axiom,
    ! [N2: nat,P2: nat > $o,M: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N2 @ K )
         => ( P2 @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N2 )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K @ I4 )
                 => ( P2 @ I4 ) )
             => ( P2 @ K ) ) )
       => ( P2 @ M ) ) ) ).

% nat_descend_induct
thf(fact_699_minf_I8_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z2 )
     => ~ ( ord_less_eq_rat @ T6 @ X6 ) ) ).

% minf(8)
thf(fact_700_minf_I8_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z2 )
     => ~ ( ord_less_eq_num @ T6 @ X6 ) ) ).

% minf(8)
thf(fact_701_minf_I8_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ~ ( ord_less_eq_nat @ T6 @ X6 ) ) ).

% minf(8)
thf(fact_702_minf_I8_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z2 )
     => ~ ( ord_less_eq_int @ T6 @ X6 ) ) ).

% minf(8)
thf(fact_703_minf_I6_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z2 )
     => ( ord_less_eq_rat @ X6 @ T6 ) ) ).

% minf(6)
thf(fact_704_minf_I6_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z2 )
     => ( ord_less_eq_num @ X6 @ T6 ) ) ).

% minf(6)
thf(fact_705_minf_I6_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ( ord_less_eq_nat @ X6 @ T6 ) ) ).

% minf(6)
thf(fact_706_minf_I6_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z2 )
     => ( ord_less_eq_int @ X6 @ T6 ) ) ).

% minf(6)
thf(fact_707_pinf_I8_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z2 @ X6 )
     => ( ord_less_eq_rat @ T6 @ X6 ) ) ).

% pinf(8)
thf(fact_708_pinf_I8_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z2 @ X6 )
     => ( ord_less_eq_num @ T6 @ X6 ) ) ).

% pinf(8)
thf(fact_709_pinf_I8_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ( ord_less_eq_nat @ T6 @ X6 ) ) ).

% pinf(8)
thf(fact_710_pinf_I8_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z2 @ X6 )
     => ( ord_less_eq_int @ T6 @ X6 ) ) ).

% pinf(8)
thf(fact_711_subset__Collect__conv,axiom,
    ! [S: set_list_nat,P2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ S @ ( collect_list_nat @ P2 ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ S )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_712_subset__Collect__conv,axiom,
    ! [S: set_se6260736226359567993nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( ord_le483042692224249369nt_int @ S @ ( collec5210948495886036740nt_int @ P2 ) )
      = ( ! [X4: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X4 @ S )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_713_subset__Collect__conv,axiom,
    ! [S: set_set_nat,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ S @ ( collect_set_nat @ P2 ) )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ S )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_714_subset__Collect__conv,axiom,
    ! [S: set_nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P2 ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ S )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_715_subset__Collect__conv,axiom,
    ! [S: set_int,P2: int > $o] :
      ( ( ord_less_eq_set_int @ S @ ( collect_int @ P2 ) )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ S )
           => ( P2 @ X4 ) ) ) ) ).

% subset_Collect_conv
thf(fact_716_Collect__restrict,axiom,
    ! [X7: set_o,P2: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ X7 )
            & ( P2 @ X4 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_717_Collect__restrict,axiom,
    ! [X7: set_list_nat,P2: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ X7 )
            & ( P2 @ X4 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_718_Collect__restrict,axiom,
    ! [X7: set_se6260736226359567993nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ord_le483042692224249369nt_int
      @ ( collec5210948495886036740nt_int
        @ ^ [X4: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X4 @ X7 )
            & ( P2 @ X4 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_719_Collect__restrict,axiom,
    ! [X7: set_set_nat,P2: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ X7 )
            & ( P2 @ X4 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_720_Collect__restrict,axiom,
    ! [X7: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ X7 )
            & ( P2 @ X4 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_721_Collect__restrict,axiom,
    ! [X7: set_int,P2: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ X7 )
            & ( P2 @ X4 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_722_prop__restrict,axiom,
    ! [X: $o,Z5: set_o,X7: set_o,P2: $o > $o] :
      ( ( member_o @ X @ Z5 )
     => ( ( ord_less_eq_set_o @ Z5
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ X7 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_723_prop__restrict,axiom,
    ! [X: list_nat,Z5: set_list_nat,X7: set_list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ X @ Z5 )
     => ( ( ord_le6045566169113846134st_nat @ Z5
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ X7 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_724_prop__restrict,axiom,
    ! [X: set_Pr958786334691620121nt_int,Z5: set_se6260736226359567993nt_int,X7: set_se6260736226359567993nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( member2340774599025711042nt_int @ X @ Z5 )
     => ( ( ord_le483042692224249369nt_int @ Z5
          @ ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] :
                ( ( member2340774599025711042nt_int @ X4 @ X7 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_725_prop__restrict,axiom,
    ! [X: set_nat,Z5: set_set_nat,X7: set_set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ X @ Z5 )
     => ( ( ord_le6893508408891458716et_nat @ Z5
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ X7 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_726_prop__restrict,axiom,
    ! [X: nat,Z5: set_nat,X7: set_nat,P2: nat > $o] :
      ( ( member_nat @ X @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ X7 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_727_prop__restrict,axiom,
    ! [X: int,Z5: set_int,X7: set_int,P2: int > $o] :
      ( ( member_int @ X @ Z5 )
     => ( ( ord_less_eq_set_int @ Z5
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ X7 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_728_pinf_I1_J,axiom,
    ! [P2: rat > $o,P6: rat > $o,Q2: rat > $o,Q4: rat > $o] :
      ( ? [Z6: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z6 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z6 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z2 @ X6 )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q4 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_729_pinf_I1_J,axiom,
    ! [P2: num > $o,P6: num > $o,Q2: num > $o,Q4: num > $o] :
      ( ? [Z6: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z6 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z6 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z2 @ X6 )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q4 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_730_pinf_I1_J,axiom,
    ! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
      ( ? [Z6: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z6 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z6 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z2 @ X6 )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q4 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_731_pinf_I1_J,axiom,
    ! [P2: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
      ( ? [Z6: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z6 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z6 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z2 @ X6 )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q4 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_732_pinf_I2_J,axiom,
    ! [P2: rat > $o,P6: rat > $o,Q2: rat > $o,Q4: rat > $o] :
      ( ? [Z6: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z6 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z6 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z2 @ X6 )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q4 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_733_pinf_I2_J,axiom,
    ! [P2: num > $o,P6: num > $o,Q2: num > $o,Q4: num > $o] :
      ( ? [Z6: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z6 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z6 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z2 @ X6 )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q4 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_734_pinf_I2_J,axiom,
    ! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
      ( ? [Z6: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z6 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z6 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z2 @ X6 )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q4 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_735_pinf_I2_J,axiom,
    ! [P2: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
      ( ? [Z6: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z6 @ X3 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z6 @ X3 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z2 @ X6 )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q4 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_736_pinf_I3_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z2 @ X6 )
     => ( X6 != T6 ) ) ).

% pinf(3)
thf(fact_737_pinf_I3_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z2 @ X6 )
     => ( X6 != T6 ) ) ).

% pinf(3)
thf(fact_738_pinf_I3_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ( X6 != T6 ) ) ).

% pinf(3)
thf(fact_739_pinf_I3_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z2 @ X6 )
     => ( X6 != T6 ) ) ).

% pinf(3)
thf(fact_740_pinf_I4_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z2 @ X6 )
     => ( X6 != T6 ) ) ).

% pinf(4)
thf(fact_741_pinf_I4_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z2 @ X6 )
     => ( X6 != T6 ) ) ).

% pinf(4)
thf(fact_742_pinf_I4_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ( X6 != T6 ) ) ).

% pinf(4)
thf(fact_743_pinf_I4_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z2 @ X6 )
     => ( X6 != T6 ) ) ).

% pinf(4)
thf(fact_744_pinf_I5_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z2 @ X6 )
     => ~ ( ord_less_rat @ X6 @ T6 ) ) ).

% pinf(5)
thf(fact_745_pinf_I5_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z2 @ X6 )
     => ~ ( ord_less_num @ X6 @ T6 ) ) ).

% pinf(5)
thf(fact_746_pinf_I5_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ~ ( ord_less_nat @ X6 @ T6 ) ) ).

% pinf(5)
thf(fact_747_pinf_I5_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z2 @ X6 )
     => ~ ( ord_less_int @ X6 @ T6 ) ) ).

% pinf(5)
thf(fact_748_pinf_I7_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z2 @ X6 )
     => ( ord_less_rat @ T6 @ X6 ) ) ).

% pinf(7)
thf(fact_749_pinf_I7_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z2 @ X6 )
     => ( ord_less_num @ T6 @ X6 ) ) ).

% pinf(7)
thf(fact_750_pinf_I7_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ( ord_less_nat @ T6 @ X6 ) ) ).

% pinf(7)
thf(fact_751_pinf_I7_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z2 @ X6 )
     => ( ord_less_int @ T6 @ X6 ) ) ).

% pinf(7)
thf(fact_752_minf_I1_J,axiom,
    ! [P2: rat > $o,P6: rat > $o,Q2: rat > $o,Q4: rat > $o] :
      ( ? [Z6: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z6 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z6 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z2 )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q4 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_753_minf_I1_J,axiom,
    ! [P2: num > $o,P6: num > $o,Q2: num > $o,Q4: num > $o] :
      ( ? [Z6: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z6 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z6 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z2 )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q4 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_754_minf_I1_J,axiom,
    ! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
      ( ? [Z6: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z6 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z6 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z2 )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q4 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_755_minf_I1_J,axiom,
    ! [P2: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
      ( ? [Z6: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z6 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z6 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z2 )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q4 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_756_minf_I2_J,axiom,
    ! [P2: rat > $o,P6: rat > $o,Q2: rat > $o,Q4: rat > $o] :
      ( ? [Z6: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z6 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z6 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z2 )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q4 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_757_minf_I2_J,axiom,
    ! [P2: num > $o,P6: num > $o,Q2: num > $o,Q4: num > $o] :
      ( ? [Z6: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z6 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z6 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z2 )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q4 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_758_minf_I2_J,axiom,
    ! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q4: nat > $o] :
      ( ? [Z6: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z6 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z6 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z2 )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q4 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_759_minf_I2_J,axiom,
    ! [P2: int > $o,P6: int > $o,Q2: int > $o,Q4: int > $o] :
      ( ? [Z6: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z6 )
         => ( ( P2 @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z6: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z6 )
           => ( ( Q2 @ X3 )
              = ( Q4 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z2 )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q4 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_760_minf_I3_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z2 )
     => ( X6 != T6 ) ) ).

% minf(3)
thf(fact_761_minf_I3_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z2 )
     => ( X6 != T6 ) ) ).

% minf(3)
thf(fact_762_minf_I3_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ( X6 != T6 ) ) ).

% minf(3)
thf(fact_763_minf_I3_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z2 )
     => ( X6 != T6 ) ) ).

% minf(3)
thf(fact_764_minf_I4_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z2 )
     => ( X6 != T6 ) ) ).

% minf(4)
thf(fact_765_minf_I4_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z2 )
     => ( X6 != T6 ) ) ).

% minf(4)
thf(fact_766_minf_I4_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ( X6 != T6 ) ) ).

% minf(4)
thf(fact_767_minf_I4_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z2 )
     => ( X6 != T6 ) ) ).

% minf(4)
thf(fact_768_minf_I5_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z2 )
     => ( ord_less_rat @ X6 @ T6 ) ) ).

% minf(5)
thf(fact_769_minf_I5_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z2 )
     => ( ord_less_num @ X6 @ T6 ) ) ).

% minf(5)
thf(fact_770_minf_I5_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ( ord_less_nat @ X6 @ T6 ) ) ).

% minf(5)
thf(fact_771_minf_I5_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z2 )
     => ( ord_less_int @ X6 @ T6 ) ) ).

% minf(5)
thf(fact_772_minf_I7_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z2 )
     => ~ ( ord_less_rat @ T6 @ X6 ) ) ).

% minf(7)
thf(fact_773_minf_I7_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z2 )
     => ~ ( ord_less_num @ T6 @ X6 ) ) ).

% minf(7)
thf(fact_774_minf_I7_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ~ ( ord_less_nat @ T6 @ X6 ) ) ).

% minf(7)
thf(fact_775_minf_I7_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z2 )
     => ~ ( ord_less_int @ T6 @ X6 ) ) ).

% minf(7)
thf(fact_776_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_777_infinite__descent,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ! [N5: nat] :
          ( ~ ( P2 @ N5 )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
              & ~ ( P2 @ M4 ) ) )
     => ( P2 @ N2 ) ) ).

% infinite_descent
thf(fact_778_nat__less__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
             => ( P2 @ M4 ) )
         => ( P2 @ N5 ) )
     => ( P2 @ N2 ) ) ).

% nat_less_induct
thf(fact_779_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_780_less__not__refl3,axiom,
    ! [S2: nat,T6: nat] :
      ( ( ord_less_nat @ S2 @ T6 )
     => ( S2 != T6 ) ) ).

% less_not_refl3
thf(fact_781_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_782_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_783_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_784_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K2: nat,B: nat] :
      ( ( P2 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P2 @ X3 )
            & ! [Y5: nat] :
                ( ( P2 @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_785_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_786_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_787_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_788_le__trans,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K2 )
       => ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).

% le_trans
thf(fact_789_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_790_pinf_I6_J,axiom,
    ! [T6: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z2 @ X6 )
     => ~ ( ord_less_eq_rat @ X6 @ T6 ) ) ).

% pinf(6)
thf(fact_791_pinf_I6_J,axiom,
    ! [T6: num] :
    ? [Z2: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z2 @ X6 )
     => ~ ( ord_less_eq_num @ X6 @ T6 ) ) ).

% pinf(6)
thf(fact_792_pinf_I6_J,axiom,
    ! [T6: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T6 ) ) ).

% pinf(6)
thf(fact_793_pinf_I6_J,axiom,
    ! [T6: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z2 @ X6 )
     => ~ ( ord_less_eq_int @ X6 @ T6 ) ) ).

% pinf(6)
thf(fact_794_execute__bind_I1_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,X: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_Time_Heap_b] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_Time_execute_b @ ( heap_T757603679106148409unit_b @ F @ G ) @ H )
        = ( heap_T7616092557645711336rame_b @ N2 @ ( heap_Time_execute_b @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_795_execute__bind_I1_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,X: product_unit,H5: 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 @ H5 @ N2 ) ) ) )
     => ( ( heap_Time_execute_a @ ( heap_T757603679106148408unit_a @ F @ G ) @ H )
        = ( heap_T7616092557645711335rame_a @ N2 @ ( heap_Time_execute_a @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_796_execute__bind_I1_J,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit,X: b,H5: heap_e7401611519738050253t_unit,N2: nat,G: b > heap_Time_Heap_b] :
      ( ( ( heap_Time_execute_b @ F @ H )
        = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_Time_execute_b @ ( heap_Time_bind_b_b @ F @ G ) @ H )
        = ( heap_T7616092557645711336rame_b @ N2 @ ( heap_Time_execute_b @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_797_execute__bind_I1_J,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit,X: b,H5: heap_e7401611519738050253t_unit,N2: nat,G: b > heap_Time_Heap_a] :
      ( ( ( heap_Time_execute_b @ F @ H )
        = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_Time_execute_a @ ( heap_Time_bind_b_a @ F @ G ) @ H )
        = ( heap_T7616092557645711335rame_a @ N2 @ ( heap_Time_execute_a @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_798_execute__bind_I1_J,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,X: a,H5: heap_e7401611519738050253t_unit,N2: nat,G: a > heap_Time_Heap_b] :
      ( ( ( heap_Time_execute_a @ F @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_Time_execute_b @ ( heap_Time_bind_a_b @ F @ G ) @ H )
        = ( heap_T7616092557645711336rame_b @ N2 @ ( heap_Time_execute_b @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_799_execute__bind_I1_J,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,X: a,H5: 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 @ H5 @ N2 ) ) ) )
     => ( ( heap_Time_execute_a @ ( heap_Time_bind_a_a @ F @ G ) @ H )
        = ( heap_T7616092557645711335rame_a @ N2 @ ( heap_Time_execute_a @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_800_subset__antisym,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( A4 = B5 ) ) ) ).

% subset_antisym
thf(fact_801_psubsetI,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( A4 != B5 )
       => ( ord_less_set_int @ A4 @ B5 ) ) ) ).

% psubsetI
thf(fact_802_subsetI,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( member_o @ X3 @ B5 ) )
     => ( ord_less_eq_set_o @ A4 @ B5 ) ) ).

% subsetI
thf(fact_803_subsetI,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ X3 @ A4 )
         => ( member2340774599025711042nt_int @ X3 @ B5 ) )
     => ( ord_le483042692224249369nt_int @ A4 @ B5 ) ) ).

% subsetI
thf(fact_804_subsetI,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A4 )
         => ( member_set_nat @ X3 @ B5 ) )
     => ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ).

% subsetI
thf(fact_805_subsetI,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( member_nat @ X3 @ B5 ) )
     => ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).

% subsetI
thf(fact_806_subsetI,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( member_int @ X3 @ B5 ) )
     => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).

% subsetI
thf(fact_807_complete__interval,axiom,
    ! [A: nat,B: nat,P2: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: nat] :
              ( ( ord_less_eq_nat @ A @ C )
              & ( ord_less_eq_nat @ C @ B )
              & ! [X6: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X6 )
                    & ( ord_less_nat @ X6 @ C ) )
                 => ( P2 @ X6 ) )
              & ! [D3: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D3 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_nat @ D3 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_808_complete__interval,axiom,
    ! [A: int,B: int,P2: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: int] :
              ( ( ord_less_eq_int @ A @ C )
              & ( ord_less_eq_int @ C @ B )
              & ! [X6: int] :
                  ( ( ( ord_less_eq_int @ A @ X6 )
                    & ( ord_less_int @ X6 @ C ) )
                 => ( P2 @ X6 ) )
              & ! [D3: int] :
                  ( ! [X3: int] :
                      ( ( ( ord_less_eq_int @ A @ X3 )
                        & ( ord_less_int @ X3 @ D3 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_int @ D3 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_809_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_810_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_811_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_812_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_813_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,X: product_unit,H5: 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 @ H5 @ N2 ) ) ) )
     => ( ( ( heap_T875086893843062177t_unit @ ( G @ X ) @ H5 )
          = ( 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_814_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,X: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_Time_Heap_b,Y: b,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute_b @ ( G @ X ) @ H5 )
          = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_Time_execute_b @ ( heap_T757603679106148409unit_b @ F @ G ) @ H )
          = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_815_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,X: product_unit,H5: 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 @ H5 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute_a @ ( G @ X ) @ H5 )
          = ( 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_816_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit,X: b,H5: heap_e7401611519738050253t_unit,N2: nat,G: b > heap_T5738788834812785303t_unit,Y: product_unit,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_Time_execute_b @ F @ H )
        = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_T875086893843062177t_unit @ ( G @ X ) @ H5 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_T875086893843062177t_unit @ ( heap_T3413130826733729493t_unit @ F @ G ) @ H )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_817_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit,X: b,H5: heap_e7401611519738050253t_unit,N2: nat,G: b > heap_Time_Heap_b,Y: b,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_Time_execute_b @ F @ H )
        = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute_b @ ( G @ X ) @ H5 )
          = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_Time_execute_b @ ( heap_Time_bind_b_b @ F @ G ) @ H )
          = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_818_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit,X: b,H5: heap_e7401611519738050253t_unit,N2: nat,G: b > heap_Time_Heap_a,Y: a,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_Time_execute_b @ F @ H )
        = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute_a @ ( G @ X ) @ H5 )
          = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_Time_execute_a @ ( heap_Time_bind_b_a @ F @ G ) @ H )
          = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_819_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,X: a,H5: 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 @ H5 @ N2 ) ) ) )
     => ( ( ( heap_T875086893843062177t_unit @ ( G @ X ) @ H5 )
          = ( 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_820_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,X: a,H5: heap_e7401611519738050253t_unit,N2: nat,G: a > heap_Time_Heap_b,Y: b,H9: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( ( heap_Time_execute_a @ F @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute_b @ ( G @ X ) @ H5 )
          = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ N6 ) ) ) )
       => ( ( heap_Time_execute_b @ ( heap_Time_bind_a_b @ F @ G ) @ H )
          = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ Y @ ( produc584006145561248582it_nat @ H9 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_821_execute__bind__eq__SomeI,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,X: a,H5: 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 @ H5 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute_a @ ( G @ X ) @ H5 )
          = ( 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_822_new__addrs__def,axiom,
    ( hoare_new_addrs
    = ( ^ [H6: heap_e7401611519738050253t_unit,As4: set_nat,H7: heap_e7401611519738050253t_unit] :
          ( sup_sup_set_nat @ As4
          @ ( 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_823_subset__Collect__iff,axiom,
    ! [B5: set_o,A4: set_o,P2: $o > $o] :
      ( ( ord_less_eq_set_o @ B5 @ A4 )
     => ( ( ord_less_eq_set_o @ B5
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: $o] :
              ( ( member_o @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_824_subset__Collect__iff,axiom,
    ! [B5: set_list_nat,A4: set_list_nat,P2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B5 @ A4 )
     => ( ( ord_le6045566169113846134st_nat @ B5
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: list_nat] :
              ( ( member_list_nat @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_825_subset__Collect__iff,axiom,
    ! [B5: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( ord_le483042692224249369nt_int @ B5 @ A4 )
     => ( ( ord_le483042692224249369nt_int @ B5
          @ ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] :
                ( ( member2340774599025711042nt_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: set_Pr958786334691620121nt_int] :
              ( ( member2340774599025711042nt_int @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_826_subset__Collect__iff,axiom,
    ! [B5: set_set_nat,A4: set_set_nat,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
     => ( ( ord_le6893508408891458716et_nat @ B5
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: set_nat] :
              ( ( member_set_nat @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_827_subset__Collect__iff,axiom,
    ! [B5: set_nat,A4: set_nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( ord_less_eq_set_nat @ B5
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_828_subset__Collect__iff,axiom,
    ! [B5: set_int,A4: set_int,P2: int > $o] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( ord_less_eq_set_int @ B5
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_829_subset__CollectI,axiom,
    ! [B5: set_o,A4: set_o,Q2: $o > $o,P2: $o > $o] :
      ( ( ord_less_eq_set_o @ B5 @ A4 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_830_subset__CollectI,axiom,
    ! [B5: set_list_nat,A4: set_list_nat,Q2: list_nat > $o,P2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B5 @ A4 )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le6045566169113846134st_nat
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( member_list_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_831_subset__CollectI,axiom,
    ! [B5: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int,Q2: set_Pr958786334691620121nt_int > $o,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( ord_le483042692224249369nt_int @ B5 @ A4 )
     => ( ! [X3: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le483042692224249369nt_int
          @ ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] :
                ( ( member2340774599025711042nt_int @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] :
                ( ( member2340774599025711042nt_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_832_subset__CollectI,axiom,
    ! [B5: set_set_nat,A4: set_set_nat,Q2: set_nat > $o,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le6893508408891458716et_nat
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( member_set_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_833_subset__CollectI,axiom,
    ! [B5: set_nat,A4: set_nat,Q2: nat > $o,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_834_subset__CollectI,axiom,
    ! [B5: set_int,A4: set_int,Q2: int > $o,P2: int > $o] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_835_conj__subset__def,axiom,
    ! [A4: set_list_nat,P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A4
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) )
      = ( ( ord_le6045566169113846134st_nat @ A4 @ ( collect_list_nat @ P2 ) )
        & ( ord_le6045566169113846134st_nat @ A4 @ ( collect_list_nat @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_836_conj__subset__def,axiom,
    ! [A4: set_se6260736226359567993nt_int,P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ( ord_le483042692224249369nt_int @ A4
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) )
      = ( ( ord_le483042692224249369nt_int @ A4 @ ( collec5210948495886036740nt_int @ P2 ) )
        & ( ord_le483042692224249369nt_int @ A4 @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_837_conj__subset__def,axiom,
    ! [A4: set_set_nat,P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A4
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A4 @ ( collect_set_nat @ P2 ) )
        & ( ord_le6893508408891458716et_nat @ A4 @ ( collect_set_nat @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_838_conj__subset__def,axiom,
    ! [A4: set_nat,P2: nat > $o,Q2: nat > $o] :
      ( ( ord_less_eq_set_nat @ A4
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) )
      = ( ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ P2 ) )
        & ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_839_conj__subset__def,axiom,
    ! [A4: set_int,P2: int > $o,Q2: int > $o] :
      ( ( ord_less_eq_set_int @ A4
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) )
      = ( ( ord_less_eq_set_int @ A4 @ ( collect_int @ P2 ) )
        & ( ord_less_eq_set_int @ A4 @ ( collect_int @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_840_UnCI,axiom,
    ! [C2: $o,B5: set_o,A4: set_o] :
      ( ( ~ ( member_o @ C2 @ B5 )
       => ( member_o @ C2 @ A4 ) )
     => ( member_o @ C2 @ ( sup_sup_set_o @ A4 @ B5 ) ) ) ).

% UnCI
thf(fact_841_UnCI,axiom,
    ! [C2: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( ~ ( member2340774599025711042nt_int @ C2 @ B5 )
       => ( member2340774599025711042nt_int @ C2 @ A4 ) )
     => ( member2340774599025711042nt_int @ C2 @ ( sup_su2047564715030645325nt_int @ A4 @ B5 ) ) ) ).

% UnCI
thf(fact_842_UnCI,axiom,
    ! [C2: set_nat,B5: set_set_nat,A4: set_set_nat] :
      ( ( ~ ( member_set_nat @ C2 @ B5 )
       => ( member_set_nat @ C2 @ A4 ) )
     => ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A4 @ B5 ) ) ) ).

% UnCI
thf(fact_843_UnCI,axiom,
    ! [C2: int,B5: set_int,A4: set_int] :
      ( ( ~ ( member_int @ C2 @ B5 )
       => ( member_int @ C2 @ A4 ) )
     => ( member_int @ C2 @ ( sup_sup_set_int @ A4 @ B5 ) ) ) ).

% UnCI
thf(fact_844_UnCI,axiom,
    ! [C2: produc4166570645942440679at_nat,B5: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( ( ~ ( member6689249552917799696at_nat @ C2 @ B5 )
       => ( member6689249552917799696at_nat @ C2 @ A4 ) )
     => ( member6689249552917799696at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) ) ).

% UnCI
thf(fact_845_UnCI,axiom,
    ! [C2: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ C2 @ B5 )
       => ( member8757157785044589968at_nat @ C2 @ A4 ) )
     => ( member8757157785044589968at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) ) ).

% UnCI
thf(fact_846_UnCI,axiom,
    ! [C2: nat,B5: set_nat,A4: set_nat] :
      ( ( ~ ( member_nat @ C2 @ B5 )
       => ( member_nat @ C2 @ A4 ) )
     => ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).

% UnCI
thf(fact_847_Un__iff,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( sup_sup_set_o @ A4 @ B5 ) )
      = ( ( member_o @ C2 @ A4 )
        | ( member_o @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_848_Un__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( sup_su2047564715030645325nt_int @ A4 @ B5 ) )
      = ( ( member2340774599025711042nt_int @ C2 @ A4 )
        | ( member2340774599025711042nt_int @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_849_Un__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A4 @ B5 ) )
      = ( ( member_set_nat @ C2 @ A4 )
        | ( member_set_nat @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_850_Un__iff,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( sup_sup_set_int @ A4 @ B5 ) )
      = ( ( member_int @ C2 @ A4 )
        | ( member_int @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_851_Un__iff,axiom,
    ! [C2: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( member6689249552917799696at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
      = ( ( member6689249552917799696at_nat @ C2 @ A4 )
        | ( member6689249552917799696at_nat @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_852_Un__iff,axiom,
    ! [C2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
      = ( ( member8757157785044589968at_nat @ C2 @ A4 )
        | ( member8757157785044589968at_nat @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_853_Un__iff,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
      = ( ( member_nat @ C2 @ A4 )
        | ( member_nat @ C2 @ B5 ) ) ) ).

% Un_iff
thf(fact_854_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_855_Un__subset__iff,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) @ C4 )
      = ( ( ord_le8081472938463900775at_nat @ A4 @ C4 )
        & ( ord_le8081472938463900775at_nat @ B5 @ C4 ) ) ) ).

% Un_subset_iff
thf(fact_856_Un__subset__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) @ C4 )
      = ( ( ord_le1268244103169919719at_nat @ A4 @ C4 )
        & ( ord_le1268244103169919719at_nat @ B5 @ C4 ) ) ) ).

% Un_subset_iff
thf(fact_857_Un__subset__iff,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ C4 )
      = ( ( ord_less_eq_set_nat @ A4 @ C4 )
        & ( ord_less_eq_set_nat @ B5 @ C4 ) ) ) ).

% Un_subset_iff
thf(fact_858_Un__subset__iff,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A4 @ B5 ) @ C4 )
      = ( ( ord_less_eq_set_int @ A4 @ C4 )
        & ( ord_less_eq_set_int @ B5 @ C4 ) ) ) ).

% Un_subset_iff
thf(fact_859_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_860_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_861_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_862_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_863_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_864_sup__Some,axiom,
    ! [X: int,Y: int] :
      ( ( sup_sup_option_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( some_int @ ( sup_sup_int @ X @ Y ) ) ) ).

% sup_Some
thf(fact_865_relH__dist__union,axiom,
    ! [As: set_nat,As5: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ ( sup_sup_set_nat @ As @ As5 ) @ H @ H5 )
      = ( ( relH @ As @ H @ H5 )
        & ( relH @ As5 @ H @ H5 ) ) ) ).

% relH_dist_union
thf(fact_866_psubsetD,axiom,
    ! [A4: set_o,B5: set_o,C2: $o] :
      ( ( ord_less_set_o @ A4 @ B5 )
     => ( ( member_o @ C2 @ A4 )
       => ( member_o @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_867_psubsetD,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le1924305788584680229nt_int @ A4 @ B5 )
     => ( ( member2340774599025711042nt_int @ C2 @ A4 )
       => ( member2340774599025711042nt_int @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_868_psubsetD,axiom,
    ! [A4: set_set_nat,B5: set_set_nat,C2: set_nat] :
      ( ( ord_less_set_set_nat @ A4 @ B5 )
     => ( ( member_set_nat @ C2 @ A4 )
       => ( member_set_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_869_psubsetD,axiom,
    ! [A4: set_nat,B5: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A4 @ B5 )
     => ( ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_870_psubsetD,axiom,
    ! [A4: set_int,B5: set_int,C2: int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ( ( member_int @ C2 @ A4 )
       => ( member_int @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_871_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_872_less__set__def,axiom,
    ( ord_le1924305788584680229nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
          ( ord_le2688692977766382584_int_o
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ A6 )
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_873_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_874_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_875_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_876_UnE,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( sup_sup_set_o @ A4 @ B5 ) )
     => ( ~ ( member_o @ C2 @ A4 )
       => ( member_o @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_877_UnE,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( sup_su2047564715030645325nt_int @ A4 @ B5 ) )
     => ( ~ ( member2340774599025711042nt_int @ C2 @ A4 )
       => ( member2340774599025711042nt_int @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_878_UnE,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A4 @ B5 ) )
     => ( ~ ( member_set_nat @ C2 @ A4 )
       => ( member_set_nat @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_879_UnE,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( sup_sup_set_int @ A4 @ B5 ) )
     => ( ~ ( member_int @ C2 @ A4 )
       => ( member_int @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_880_UnE,axiom,
    ! [C2: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( member6689249552917799696at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
     => ( ~ ( member6689249552917799696at_nat @ C2 @ A4 )
       => ( member6689249552917799696at_nat @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_881_UnE,axiom,
    ! [C2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
     => ( ~ ( member8757157785044589968at_nat @ C2 @ A4 )
       => ( member8757157785044589968at_nat @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_882_UnE,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
     => ( ~ ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B5 ) ) ) ).

% UnE
thf(fact_883_UnI1,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ A4 )
     => ( member_o @ C2 @ ( sup_sup_set_o @ A4 @ B5 ) ) ) ).

% UnI1
thf(fact_884_UnI1,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ A4 )
     => ( member2340774599025711042nt_int @ C2 @ ( sup_su2047564715030645325nt_int @ A4 @ B5 ) ) ) ).

% UnI1
thf(fact_885_UnI1,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ A4 )
     => ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A4 @ B5 ) ) ) ).

% UnI1
thf(fact_886_UnI1,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ A4 )
     => ( member_int @ C2 @ ( sup_sup_set_int @ A4 @ B5 ) ) ) ).

% UnI1
thf(fact_887_UnI1,axiom,
    ! [C2: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( member6689249552917799696at_nat @ C2 @ A4 )
     => ( member6689249552917799696at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) ) ).

% UnI1
thf(fact_888_UnI1,axiom,
    ! [C2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C2 @ A4 )
     => ( member8757157785044589968at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) ) ).

% UnI1
thf(fact_889_UnI1,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ A4 )
     => ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).

% UnI1
thf(fact_890_UnI2,axiom,
    ! [C2: $o,B5: set_o,A4: set_o] :
      ( ( member_o @ C2 @ B5 )
     => ( member_o @ C2 @ ( sup_sup_set_o @ A4 @ B5 ) ) ) ).

% UnI2
thf(fact_891_UnI2,axiom,
    ! [C2: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ B5 )
     => ( member2340774599025711042nt_int @ C2 @ ( sup_su2047564715030645325nt_int @ A4 @ B5 ) ) ) ).

% UnI2
thf(fact_892_UnI2,axiom,
    ! [C2: set_nat,B5: set_set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ C2 @ B5 )
     => ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A4 @ B5 ) ) ) ).

% UnI2
thf(fact_893_UnI2,axiom,
    ! [C2: int,B5: set_int,A4: set_int] :
      ( ( member_int @ C2 @ B5 )
     => ( member_int @ C2 @ ( sup_sup_set_int @ A4 @ B5 ) ) ) ).

% UnI2
thf(fact_894_UnI2,axiom,
    ! [C2: produc4166570645942440679at_nat,B5: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( ( member6689249552917799696at_nat @ C2 @ B5 )
     => ( member6689249552917799696at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) ) ).

% UnI2
thf(fact_895_UnI2,axiom,
    ! [C2: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C2 @ B5 )
     => ( member8757157785044589968at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) ) ).

% UnI2
thf(fact_896_UnI2,axiom,
    ! [C2: nat,B5: set_nat,A4: set_nat] :
      ( ( member_nat @ C2 @ B5 )
     => ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).

% UnI2
thf(fact_897_bex__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,P2: produc4166570645942440679at_nat > $o] :
      ( ( ? [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ A4 )
            & ( P2 @ X4 ) )
        | ? [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ B5 )
            & ( P2 @ X4 ) ) ) ) ).

% bex_Un
thf(fact_898_bex__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A4 )
            & ( P2 @ X4 ) )
        | ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ B5 )
            & ( P2 @ X4 ) ) ) ) ).

% bex_Un
thf(fact_899_bex__Un,axiom,
    ! [A4: set_nat,B5: set_nat,P2: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( P2 @ X4 ) )
        | ? [X4: nat] :
            ( ( member_nat @ X4 @ B5 )
            & ( P2 @ X4 ) ) ) ) ).

% bex_Un
thf(fact_900_ball__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,P2: produc4166570645942440679at_nat > $o] :
      ( ( ! [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ A4 )
           => ( P2 @ X4 ) )
        & ! [X4: produc4166570645942440679at_nat] :
            ( ( member6689249552917799696at_nat @ X4 @ B5 )
           => ( P2 @ X4 ) ) ) ) ).

% ball_Un
thf(fact_901_ball__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A4 )
           => ( P2 @ X4 ) )
        & ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ B5 )
           => ( P2 @ X4 ) ) ) ) ).

% ball_Un
thf(fact_902_ball__Un,axiom,
    ! [A4: set_nat,B5: set_nat,P2: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( sup_sup_set_nat @ A4 @ B5 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( P2 @ X4 ) )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ B5 )
           => ( P2 @ X4 ) ) ) ) ).

% ball_Un
thf(fact_903_Un__assoc,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) @ C4 )
      = ( sup_su3035147773818789531at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) ) ) ).

% Un_assoc
thf(fact_904_Un__assoc,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) @ C4 )
      = ( sup_su5525570899277871387at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) ) ) ).

% Un_assoc
thf(fact_905_Un__assoc,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ C4 )
      = ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C4 ) ) ) ).

% Un_assoc
thf(fact_906_Un__absorb,axiom,
    ! [A4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A4 @ A4 )
      = A4 ) ).

% Un_absorb
thf(fact_907_Un__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ A4 )
      = A4 ) ).

% Un_absorb
thf(fact_908_Un__absorb,axiom,
    ! [A4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ A4 )
      = A4 ) ).

% Un_absorb
thf(fact_909_Un__commute,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [A6: set_Pr8551490117392284871at_nat,B6: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_910_Un__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_911_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_912_Un__left__absorb,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
      = ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) ).

% Un_left_absorb
thf(fact_913_Un__left__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
      = ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) ).

% Un_left_absorb
thf(fact_914_Un__left__absorb,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ A4 @ B5 ) )
      = ( sup_sup_set_nat @ A4 @ B5 ) ) ).

% Un_left_absorb
thf(fact_915_Un__left__commute,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) )
      = ( sup_su3035147773818789531at_nat @ B5 @ ( sup_su3035147773818789531at_nat @ A4 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_916_Un__left__commute,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) )
      = ( sup_su5525570899277871387at_nat @ B5 @ ( sup_su5525570899277871387at_nat @ A4 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_917_Un__left__commute,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C4 ) )
      = ( sup_sup_set_nat @ B5 @ ( sup_sup_set_nat @ A4 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_918_Collect__disj__eq,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P2 @ X4 )
            | ( Q2 @ X4 ) ) )
      = ( sup_sup_set_list_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_919_Collect__disj__eq,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ( collec5210948495886036740nt_int
        @ ^ [X4: set_Pr958786334691620121nt_int] :
            ( ( P2 @ X4 )
            | ( Q2 @ X4 ) ) )
      = ( sup_su2047564715030645325nt_int @ ( collec5210948495886036740nt_int @ P2 ) @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_920_Collect__disj__eq,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P2 @ X4 )
            | ( Q2 @ X4 ) ) )
      = ( sup_sup_set_set_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_921_Collect__disj__eq,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( P2 @ X4 )
            | ( Q2 @ X4 ) ) )
      = ( sup_sup_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_922_Collect__disj__eq,axiom,
    ! [P2: produc4166570645942440679at_nat > $o,Q2: produc4166570645942440679at_nat > $o] :
      ( ( collec5204685387357076818at_nat
        @ ^ [X4: produc4166570645942440679at_nat] :
            ( ( P2 @ X4 )
            | ( Q2 @ X4 ) ) )
      = ( sup_su3035147773818789531at_nat @ ( collec5204685387357076818at_nat @ P2 ) @ ( collec5204685387357076818at_nat @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_923_Collect__disj__eq,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,Q2: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( P2 @ X4 )
            | ( Q2 @ X4 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( collec6321179662152712658at_nat @ P2 ) @ ( collec6321179662152712658at_nat @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_924_Collect__disj__eq,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P2 @ X4 )
            | ( Q2 @ X4 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_925_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_926_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_927_Un__def,axiom,
    ( sup_su2047564715030645325nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] :
              ( ( member2340774599025711042nt_int @ X4 @ A6 )
              | ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_928_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_929_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_930_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_931_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_932_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_933_Un__mono,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,D4: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A4 @ C4 )
     => ( ( ord_le8081472938463900775at_nat @ B5 @ D4 )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) @ ( sup_su3035147773818789531at_nat @ C4 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_934_Un__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,D4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ C4 )
     => ( ( ord_le1268244103169919719at_nat @ B5 @ D4 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) @ ( sup_su5525570899277871387at_nat @ C4 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_935_Un__mono,axiom,
    ! [A4: set_nat,C4: set_nat,B5: set_nat,D4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C4 )
     => ( ( ord_less_eq_set_nat @ B5 @ D4 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ ( sup_sup_set_nat @ C4 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_936_Un__mono,axiom,
    ! [A4: set_int,C4: set_int,B5: set_int,D4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ C4 )
     => ( ( ord_less_eq_set_int @ B5 @ D4 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A4 @ B5 ) @ ( sup_sup_set_int @ C4 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_937_Un__least,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A4 @ C4 )
     => ( ( ord_le8081472938463900775at_nat @ B5 @ C4 )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) @ C4 ) ) ) ).

% Un_least
thf(fact_938_Un__least,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ C4 )
     => ( ( ord_le1268244103169919719at_nat @ B5 @ C4 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) @ C4 ) ) ) ).

% Un_least
thf(fact_939_Un__least,axiom,
    ! [A4: set_nat,C4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C4 )
     => ( ( ord_less_eq_set_nat @ B5 @ C4 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ C4 ) ) ) ).

% Un_least
thf(fact_940_Un__least,axiom,
    ! [A4: set_int,C4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ C4 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A4 @ B5 ) @ C4 ) ) ) ).

% Un_least
thf(fact_941_Un__upper1,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) ).

% Un_upper1
thf(fact_942_Un__upper1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) ).

% Un_upper1
thf(fact_943_Un__upper1,axiom,
    ! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ A4 @ B5 ) ) ).

% Un_upper1
thf(fact_944_Un__upper1,axiom,
    ! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ A4 @ ( sup_sup_set_int @ A4 @ B5 ) ) ).

% Un_upper1
thf(fact_945_Un__upper2,axiom,
    ! [B5: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ B5 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) ).

% Un_upper2
thf(fact_946_Un__upper2,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B5 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) ).

% Un_upper2
thf(fact_947_Un__upper2,axiom,
    ! [B5: set_nat,A4: set_nat] : ( ord_less_eq_set_nat @ B5 @ ( sup_sup_set_nat @ A4 @ B5 ) ) ).

% Un_upper2
thf(fact_948_Un__upper2,axiom,
    ! [B5: set_int,A4: set_int] : ( ord_less_eq_set_int @ B5 @ ( sup_sup_set_int @ A4 @ B5 ) ) ).

% Un_upper2
thf(fact_949_Un__absorb1,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A4 @ B5 )
     => ( ( sup_su3035147773818789531at_nat @ A4 @ B5 )
        = B5 ) ) ).

% Un_absorb1
thf(fact_950_Un__absorb1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B5 )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ B5 )
        = B5 ) ) ).

% Un_absorb1
thf(fact_951_Un__absorb1,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( sup_sup_set_nat @ A4 @ B5 )
        = B5 ) ) ).

% Un_absorb1
thf(fact_952_Un__absorb1,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( sup_sup_set_int @ A4 @ B5 )
        = B5 ) ) ).

% Un_absorb1
thf(fact_953_Un__absorb2,axiom,
    ! [B5: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ B5 @ A4 )
     => ( ( sup_su3035147773818789531at_nat @ A4 @ B5 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_954_Un__absorb2,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B5 @ A4 )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ B5 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_955_Un__absorb2,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( sup_sup_set_nat @ A4 @ B5 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_956_Un__absorb2,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( sup_sup_set_int @ A4 @ B5 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_957_subset__UnE,axiom,
    ! [C4: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ C4 @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
     => ~ ! [A7: set_Pr8551490117392284871at_nat] :
            ( ( ord_le8081472938463900775at_nat @ A7 @ A4 )
           => ! [B7: set_Pr8551490117392284871at_nat] :
                ( ( ord_le8081472938463900775at_nat @ B7 @ B5 )
               => ( C4
                 != ( sup_su3035147773818789531at_nat @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_958_subset__UnE,axiom,
    ! [C4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C4 @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
     => ~ ! [A7: set_Pr4329608150637261639at_nat] :
            ( ( ord_le1268244103169919719at_nat @ A7 @ A4 )
           => ! [B7: set_Pr4329608150637261639at_nat] :
                ( ( ord_le1268244103169919719at_nat @ B7 @ B5 )
               => ( C4
                 != ( sup_su5525570899277871387at_nat @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_959_subset__UnE,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ C4 @ ( sup_sup_set_nat @ A4 @ B5 ) )
     => ~ ! [A7: set_nat] :
            ( ( ord_less_eq_set_nat @ A7 @ A4 )
           => ! [B7: set_nat] :
                ( ( ord_less_eq_set_nat @ B7 @ B5 )
               => ( C4
                 != ( sup_sup_set_nat @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_960_subset__UnE,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ C4 @ ( sup_sup_set_int @ A4 @ B5 ) )
     => ~ ! [A7: set_int] :
            ( ( ord_less_eq_set_int @ A7 @ A4 )
           => ! [B7: set_int] :
                ( ( ord_less_eq_set_int @ B7 @ B5 )
               => ( C4
                 != ( sup_sup_set_int @ A7 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_961_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_962_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_963_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_964_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_965_add__lessD1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K2 )
     => ( ord_less_nat @ I2 @ K2 ) ) ).

% add_lessD1
thf(fact_966_add__less__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ K2 @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_less_mono
thf(fact_967_not__add__less1,axiom,
    ! [I2: nat,J2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).

% not_add_less1
thf(fact_968_not__add__less2,axiom,
    ! [J2: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_969_add__less__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).

% add_less_mono1
thf(fact_970_trans__less__add1,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_less_add1
thf(fact_971_trans__less__add2,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_less_add2
thf(fact_972_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K2 @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K2 @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_973_add__leE,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M @ N2 )
         => ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).

% add_leE
thf(fact_974_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_975_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_976_add__leD1,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_977_add__leD2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N2 )
     => ( ord_less_eq_nat @ K2 @ N2 ) ) ).

% add_leD2
thf(fact_978_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ K2 @ L )
     => ? [N5: nat] :
          ( L
          = ( plus_plus_nat @ K2 @ N5 ) ) ) ).

% le_Suc_ex
thf(fact_979_add__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_le_mono
thf(fact_980_add__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).

% add_le_mono1
thf(fact_981_trans__le__add1,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_le_add1
thf(fact_982_trans__le__add2,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_le_add2
thf(fact_983_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_984_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R2: b,H: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( heap_T7616092557645711336rame_b @ N2 @ ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R2 @ ( produc584006145561248582it_nat @ H @ N6 ) ) ) )
      = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R2 @ ( produc584006145561248582it_nat @ H @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ).

% timeFrame.simps(1)
thf(fact_985_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R2: a,H: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( heap_T7616092557645711335rame_a @ N2 @ ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H @ N6 ) ) ) )
      = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R2 @ ( produc584006145561248582it_nat @ H @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ).

% timeFrame.simps(1)
thf(fact_986_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R2: product_unit,H: heap_e7401611519738050253t_unit,N6: nat] :
      ( ( heap_T3616969660504097270t_unit @ N2 @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H @ N6 ) ) ) )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ) ).

% timeFrame.simps(1)
thf(fact_987_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K2: 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 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_988_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_989_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_990_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_991_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_992_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_993_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_994_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_995_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_996_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_997_verit__comp__simplify1_I1_J,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_998_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_999_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1000_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1001_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1002_in__mono,axiom,
    ! [A4: set_o,B5: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A4 @ B5 )
     => ( ( member_o @ X @ A4 )
       => ( member_o @ X @ B5 ) ) ) ).

% in_mono
thf(fact_1003_in__mono,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int] :
      ( ( ord_le483042692224249369nt_int @ A4 @ B5 )
     => ( ( member2340774599025711042nt_int @ X @ A4 )
       => ( member2340774599025711042nt_int @ X @ B5 ) ) ) ).

% in_mono
thf(fact_1004_in__mono,axiom,
    ! [A4: set_set_nat,B5: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
     => ( ( member_set_nat @ X @ A4 )
       => ( member_set_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_1005_in__mono,axiom,
    ! [A4: set_nat,B5: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( member_nat @ X @ A4 )
       => ( member_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_1006_in__mono,axiom,
    ! [A4: set_int,B5: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( member_int @ X @ A4 )
       => ( member_int @ X @ B5 ) ) ) ).

% in_mono
thf(fact_1007_subsetD,axiom,
    ! [A4: set_o,B5: set_o,C2: $o] :
      ( ( ord_less_eq_set_o @ A4 @ B5 )
     => ( ( member_o @ C2 @ A4 )
       => ( member_o @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_1008_subsetD,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int,C2: set_Pr958786334691620121nt_int] :
      ( ( ord_le483042692224249369nt_int @ A4 @ B5 )
     => ( ( member2340774599025711042nt_int @ C2 @ A4 )
       => ( member2340774599025711042nt_int @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_1009_subsetD,axiom,
    ! [A4: set_set_nat,B5: set_set_nat,C2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
     => ( ( member_set_nat @ C2 @ A4 )
       => ( member_set_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_1010_subsetD,axiom,
    ! [A4: set_nat,B5: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_1011_subsetD,axiom,
    ! [A4: set_int,B5: set_int,C2: int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( member_int @ C2 @ A4 )
       => ( member_int @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_1012_psubsetE,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
         => ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).

% psubsetE
thf(fact_1013_equalityE,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( A4 = B5 )
     => ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
         => ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).

% equalityE
thf(fact_1014_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_1015_subset__eq,axiom,
    ( ord_le483042692224249369nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
        ! [X4: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ X4 @ A6 )
         => ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1016_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_1017_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_1018_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_1019_equalityD1,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( A4 = B5 )
     => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).

% equalityD1
thf(fact_1020_equalityD2,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( A4 = B5 )
     => ( ord_less_eq_set_int @ B5 @ A4 ) ) ).

% equalityD2
thf(fact_1021_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_1022_subset__iff,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
        ! [T4: $o] :
          ( ( member_o @ T4 @ A6 )
         => ( member_o @ T4 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1023_subset__iff,axiom,
    ( ord_le483042692224249369nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
        ! [T4: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ T4 @ A6 )
         => ( member2340774599025711042nt_int @ T4 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1024_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [T4: set_nat] :
          ( ( member_set_nat @ T4 @ A6 )
         => ( member_set_nat @ T4 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1025_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T4: nat] :
          ( ( member_nat @ T4 @ A6 )
         => ( member_nat @ T4 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1026_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T4: int] :
          ( ( member_int @ T4 @ A6 )
         => ( member_int @ T4 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1027_subset__refl,axiom,
    ! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ A4 ) ).

% subset_refl
thf(fact_1028_Collect__mono,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1029_Collect__mono,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_le483042692224249369nt_int @ ( collec5210948495886036740nt_int @ P2 ) @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1030_Collect__mono,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1031_Collect__mono,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1032_Collect__mono,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ( P2 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q2 ) ) ) ).

% Collect_mono
thf(fact_1033_subset__trans,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_eq_set_int @ A4 @ C4 ) ) ) ).

% subset_trans
thf(fact_1034_set__eq__subset,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_1035_Collect__mono__iff,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q2 ) )
      = ( ! [X4: list_nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1036_Collect__mono__iff,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ( ord_le483042692224249369nt_int @ ( collec5210948495886036740nt_int @ P2 ) @ ( collec5210948495886036740nt_int @ Q2 ) )
      = ( ! [X4: set_Pr958786334691620121nt_int] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1037_Collect__mono__iff,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q2 ) )
      = ( ! [X4: set_nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1038_Collect__mono__iff,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
      = ( ! [X4: nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1039_Collect__mono__iff,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q2 ) )
      = ( ! [X4: int] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1040_psubset__imp__subset,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).

% psubset_imp_subset
thf(fact_1041_psubset__subset__trans,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_set_int @ A4 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_1042_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_1043_subset__psubset__trans,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( ord_less_set_int @ B5 @ C4 )
       => ( ord_less_set_int @ A4 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_1044_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_1045_Collect__subset,axiom,
    ! [A4: set_o,P2: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_1046_Collect__subset,axiom,
    ! [A4: set_list_nat,P2: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_1047_Collect__subset,axiom,
    ! [A4: set_se6260736226359567993nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ord_le483042692224249369nt_int
      @ ( collec5210948495886036740nt_int
        @ ^ [X4: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_1048_Collect__subset,axiom,
    ! [A4: set_set_nat,P2: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_1049_Collect__subset,axiom,
    ! [A4: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_1050_Collect__subset,axiom,
    ! [A4: set_int,P2: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ A4 )
            & ( P2 @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_1051_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_1052_less__eq__set__def,axiom,
    ( ord_le483042692224249369nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
          ( ord_le8334417538754933252_int_o
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ A6 )
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_1053_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_1054_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_1055_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_1056_le__sup__iff,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z3 )
      = ( ( ord_le8081472938463900775at_nat @ X @ Z3 )
        & ( ord_le8081472938463900775at_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1057_le__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z3 )
      = ( ( ord_le1268244103169919719at_nat @ X @ Z3 )
        & ( ord_le1268244103169919719at_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1058_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_set_nat @ X @ Z3 )
        & ( ord_less_eq_set_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1059_le__sup__iff,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_set_int @ X @ Z3 )
        & ( ord_less_eq_set_int @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1060_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_rat @ X @ Z3 )
        & ( ord_less_eq_rat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1061_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_nat @ X @ Z3 )
        & ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1062_le__sup__iff,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_int @ X @ Z3 )
        & ( ord_less_eq_int @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_1063_sup_Obounded__iff,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C2 ) @ A )
      = ( ( ord_le8081472938463900775at_nat @ B @ A )
        & ( ord_le8081472938463900775at_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1064_sup_Obounded__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C2 ) @ A )
      = ( ( ord_le1268244103169919719at_nat @ B @ A )
        & ( ord_le1268244103169919719at_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1065_sup_Obounded__iff,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_set_nat @ B @ A )
        & ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1066_sup_Obounded__iff,axiom,
    ! [B: set_int,C2: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C2 ) @ A )
      = ( ( ord_less_eq_set_int @ B @ A )
        & ( ord_less_eq_set_int @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1067_sup_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1068_sup_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1069_sup_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1070_add__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1071_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1072_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1073_add__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1074_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1075_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1076_add__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1077_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1078_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1079_add__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1080_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1081_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1082_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_1083_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_1084_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_1085_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_1086_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_1087_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_1088_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J2 )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1089_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J2 )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1090_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J2 )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1091_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J2 )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1092_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1093_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J2 )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1094_sup_Oidem,axiom,
    ! [A: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_1095_sup_Oidem,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_1096_sup_Oidem,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_1097_sup_Oidem,axiom,
    ! [A: nat] :
      ( ( sup_sup_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_1098_sup_Oidem,axiom,
    ! [A: int] :
      ( ( sup_sup_int @ A @ A )
      = A ) ).

% sup.idem
thf(fact_1099_add__right__cancel,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B @ A )
        = ( plus_p7104986032573967614at_nat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_1100_add__right__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_1101_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_1102_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_1103_add__left__cancel,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A @ B )
        = ( plus_p7104986032573967614at_nat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_1104_add__left__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_1105_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_1106_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_1107_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_1108_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_1109_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_1110_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_1111_sup_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ B )
      = ( sup_sup_int @ A @ B ) ) ).

% sup.right_idem
thf(fact_1112_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_1113_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_1114_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_1115_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_1116_sup__left__idem,axiom,
    ! [X: int,Y: int] :
      ( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
      = ( sup_sup_int @ X @ Y ) ) ).

% sup_left_idem
thf(fact_1117_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_1118_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_1119_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_1120_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_1121_sup_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( sup_sup_int @ A @ ( sup_sup_int @ A @ B ) )
      = ( sup_sup_int @ A @ B ) ) ).

% sup.left_idem
thf(fact_1122_sup__idem,axiom,
    ! [X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_1123_sup__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_1124_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_1125_sup__idem,axiom,
    ! [X: nat] :
      ( ( sup_sup_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_1126_sup__idem,axiom,
    ! [X: int] :
      ( ( sup_sup_int @ X @ X )
      = X ) ).

% sup_idem
thf(fact_1127_mod__or__dist,axiom,
    ! [P2: assn,Q2: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( sup_sup_assn @ P2 @ Q2 ) @ H )
      = ( ( rep_assn @ P2 @ H )
        | ( rep_assn @ Q2 @ H ) ) ) ).

% mod_or_dist
thf(fact_1128_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_1129_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_1130_sup__set__def,axiom,
    ( sup_su2047564715030645325nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ( sup_su1852724690005176016_int_o
            @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ A6 )
            @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1131_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_1132_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_1133_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_1134_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_1135_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_1136_add__right__imp__eq,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B @ A )
        = ( plus_p7104986032573967614at_nat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1137_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1138_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1139_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1140_add__left__imp__eq,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A @ B )
        = ( plus_p7104986032573967614at_nat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1141_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1142_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1143_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1144_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ B @ ( plus_p7104986032573967614at_nat @ A @ C2 ) )
      = ( plus_p7104986032573967614at_nat @ A @ ( plus_p7104986032573967614at_nat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1145_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C2 ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1146_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1147_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1148_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_p7104986032573967614at_nat
    = ( ^ [A5: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat] : ( plus_p7104986032573967614at_nat @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1149_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1150_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1151_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1152_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_1153_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_1154_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_1155_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_1156_add_Oassoc,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ C2 )
      = ( plus_p7104986032573967614at_nat @ A @ ( plus_p7104986032573967614at_nat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1157_add_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1158_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1159_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1160_group__cancel_Oadd2,axiom,
    ! [B5: multis2468970476368604999at_nat,K2: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( B5
        = ( plus_p7104986032573967614at_nat @ K2 @ B ) )
     => ( ( plus_p7104986032573967614at_nat @ A @ B5 )
        = ( plus_p7104986032573967614at_nat @ K2 @ ( plus_p7104986032573967614at_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1161_group__cancel_Oadd2,axiom,
    ! [B5: rat,K2: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K2 @ B ) )
     => ( ( plus_plus_rat @ A @ B5 )
        = ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1162_group__cancel_Oadd2,axiom,
    ! [B5: nat,K2: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K2 @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1163_group__cancel_Oadd2,axiom,
    ! [B5: int,K2: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( plus_plus_int @ A @ B5 )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1164_group__cancel_Oadd1,axiom,
    ! [A4: multis2468970476368604999at_nat,K2: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A4
        = ( plus_p7104986032573967614at_nat @ K2 @ A ) )
     => ( ( plus_p7104986032573967614at_nat @ A4 @ B )
        = ( plus_p7104986032573967614at_nat @ K2 @ ( plus_p7104986032573967614at_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1165_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( plus_plus_rat @ A4 @ B )
        = ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1166_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( plus_plus_nat @ A4 @ B )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1167_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( plus_plus_int @ A4 @ B )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1168_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( ( plus_plus_rat @ I2 @ K2 )
        = ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1169_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( ( plus_plus_nat @ I2 @ K2 )
        = ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1170_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( ( plus_plus_int @ I2 @ K2 )
        = ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1171_sup__left__commute,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) )
      = ( sup_su3035147773818789531at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Z3 ) ) ) ).

% sup_left_commute
thf(fact_1172_sup__left__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) )
      = ( sup_su5525570899277871387at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Z3 ) ) ) ).

% sup_left_commute
thf(fact_1173_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ).

% sup_left_commute
thf(fact_1174_sup__left__commute,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) )
      = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z3 ) ) ) ).

% sup_left_commute
thf(fact_1175_sup__left__commute,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z3 ) )
      = ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z3 ) ) ) ).

% sup_left_commute
thf(fact_1176_sup_Oleft__commute,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ B @ ( sup_su3035147773818789531at_nat @ A @ C2 ) )
      = ( sup_su3035147773818789531at_nat @ A @ ( sup_su3035147773818789531at_nat @ B @ C2 ) ) ) ).

% sup.left_commute
thf(fact_1177_sup_Oleft__commute,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) ) ) ).

% sup.left_commute
thf(fact_1178_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% sup.left_commute
thf(fact_1179_sup_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C2 ) )
      = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C2 ) ) ) ).

% sup.left_commute
thf(fact_1180_sup_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( sup_sup_int @ B @ ( sup_sup_int @ A @ C2 ) )
      = ( sup_sup_int @ A @ ( sup_sup_int @ B @ C2 ) ) ) ).

% sup.left_commute
thf(fact_1181_sup__commute,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [X4: set_Pr8551490117392284871at_nat,Y4: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_1182_sup__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_1183_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_1184_sup__commute,axiom,
    ( sup_sup_nat
    = ( ^ [X4: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_1185_sup__commute,axiom,
    ( sup_sup_int
    = ( ^ [X4: int,Y4: int] : ( sup_sup_int @ Y4 @ X4 ) ) ) ).

% sup_commute
thf(fact_1186_sup_Ocommute,axiom,
    ( sup_su3035147773818789531at_nat
    = ( ^ [A5: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] : ( sup_su3035147773818789531at_nat @ B4 @ A5 ) ) ) ).

% sup.commute
thf(fact_1187_sup_Ocommute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B4 @ A5 ) ) ) ).

% sup.commute
thf(fact_1188_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A5 ) ) ) ).

% sup.commute
thf(fact_1189_sup_Ocommute,axiom,
    ( sup_sup_nat
    = ( ^ [A5: nat,B4: nat] : ( sup_sup_nat @ B4 @ A5 ) ) ) ).

% sup.commute
thf(fact_1190_sup_Ocommute,axiom,
    ( sup_sup_int
    = ( ^ [A5: int,B4: int] : ( sup_sup_int @ B4 @ A5 ) ) ) ).

% sup.commute
thf(fact_1191_sup__assoc,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z3 )
      = ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) ) ) ).

% sup_assoc
thf(fact_1192_sup__assoc,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z3 )
      = ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) ) ) ).

% sup_assoc
thf(fact_1193_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z3 )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) ) ) ).

% sup_assoc
thf(fact_1194_sup__assoc,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
      = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).

% sup_assoc
thf(fact_1195_sup__assoc,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z3 )
      = ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z3 ) ) ) ).

% sup_assoc
thf(fact_1196_sup_Oassoc,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ C2 )
      = ( sup_su3035147773818789531at_nat @ A @ ( sup_su3035147773818789531at_nat @ B @ C2 ) ) ) ).

% sup.assoc
thf(fact_1197_sup_Oassoc,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ A @ ( sup_su5525570899277871387at_nat @ B @ C2 ) ) ) ).

% sup.assoc
thf(fact_1198_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% sup.assoc
thf(fact_1199_sup_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C2 )
      = ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C2 ) ) ) ).

% sup.assoc
thf(fact_1200_sup_Oassoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ C2 )
      = ( sup_sup_int @ A @ ( sup_sup_int @ B @ C2 ) ) ) ).

% sup.assoc
thf(fact_1201_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_1202_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_1203_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_1204_inf__sup__aci_I5_J,axiom,
    ( sup_sup_nat
    = ( ^ [X4: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_1205_inf__sup__aci_I5_J,axiom,
    ( sup_sup_int
    = ( ^ [X4: int,Y4: int] : ( sup_sup_int @ Y4 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_1206_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ Z3 )
      = ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) ) ) ).

% inf_sup_aci(6)
thf(fact_1207_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z3 )
      = ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) ) ) ).

% inf_sup_aci(6)
thf(fact_1208_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z3 )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) ) ) ).

% inf_sup_aci(6)
thf(fact_1209_inf__sup__aci_I6_J,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
      = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).

% inf_sup_aci(6)
thf(fact_1210_inf__sup__aci_I6_J,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z3 )
      = ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z3 ) ) ) ).

% inf_sup_aci(6)
thf(fact_1211_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) )
      = ( sup_su3035147773818789531at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Z3 ) ) ) ).

% inf_sup_aci(7)
thf(fact_1212_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) )
      = ( sup_su5525570899277871387at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Z3 ) ) ) ).

% inf_sup_aci(7)
thf(fact_1213_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ).

% inf_sup_aci(7)
thf(fact_1214_inf__sup__aci_I7_J,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) )
      = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z3 ) ) ) ).

% inf_sup_aci(7)
thf(fact_1215_inf__sup__aci_I7_J,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z3 ) )
      = ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z3 ) ) ) ).

% inf_sup_aci(7)
thf(fact_1216_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_1217_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_1218_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_1219_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_1220_inf__sup__aci_I8_J,axiom,
    ! [X: int,Y: int] :
      ( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
      = ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_1221_add__le__imp__le__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1222_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1223_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1224_add__le__imp__le__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1225_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1226_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1227_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
        ? [C3: nat] :
          ( B4
          = ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_1228_add__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1229_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1230_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1231_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C: nat] :
            ( B
           != ( plus_plus_nat @ A @ C ) ) ) ).

% less_eqE
thf(fact_1232_add__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1233_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1234_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1235_add__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_1236_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_1237_add__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_1238_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J2 )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1239_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1240_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J2 )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1241_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1242_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1243_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1244_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1245_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1246_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1247_add__less__imp__less__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1248_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1249_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1250_add__less__imp__less__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1251_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1252_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1253_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1254_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1255_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1256_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1257_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1258_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1259_add__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_1260_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_1261_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_1262_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1263_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1264_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1265_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1266_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1267_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1268_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J2 )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1269_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J2 )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1270_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J2 )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1271_sup_OcoboundedI2,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ C2 @ B )
     => ( ord_le8081472938463900775at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1272_sup_OcoboundedI2,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ B )
     => ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1273_sup_OcoboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ B )
     => ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1274_sup_OcoboundedI2,axiom,
    ! [C2: set_int,B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ C2 @ B )
     => ( ord_less_eq_set_int @ C2 @ ( sup_sup_set_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1275_sup_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C2 @ B )
     => ( ord_less_eq_rat @ C2 @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1276_sup_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1277_sup_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ B )
     => ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1278_sup_OcoboundedI1,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ C2 @ A )
     => ( ord_le8081472938463900775at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1279_sup_OcoboundedI1,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ A )
     => ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1280_sup_OcoboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A )
     => ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1281_sup_OcoboundedI1,axiom,
    ! [C2: set_int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ C2 @ A )
     => ( ord_less_eq_set_int @ C2 @ ( sup_sup_set_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1282_sup_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ord_less_eq_rat @ C2 @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1283_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1284_sup_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1285_sup_Oabsorb__iff2,axiom,
    ( ord_le8081472938463900775at_nat
    = ( ^ [A5: set_Pr8551490117392284871at_nat,B4: set_Pr8551490117392284871at_nat] :
          ( ( sup_su3035147773818789531at_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1286_sup_Oabsorb__iff2,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1287_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1288_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( sup_sup_set_int @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1289_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( sup_sup_rat @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1290_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( sup_sup_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1291_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( sup_sup_int @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1292_sup_Oabsorb__iff1,axiom,
    ( ord_le8081472938463900775at_nat
    = ( ^ [B4: set_Pr8551490117392284871at_nat,A5: set_Pr8551490117392284871at_nat] :
          ( ( sup_su3035147773818789531at_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_1293_sup_Oabsorb__iff1,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B4: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_1294_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_1295_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( ( sup_sup_set_int @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_1296_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( sup_sup_rat @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_1297_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( sup_sup_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_1298_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( sup_sup_int @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_1299_sup_Ocobounded2,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ B @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1300_sup_Ocobounded2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1301_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_1302_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_1303_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1304_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1305_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1306_sup_Ocobounded1,axiom,
    ! [A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ A @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1307_sup_Ocobounded1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1308_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_1309_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_1310_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1311_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1312_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1313_sup_Oorder__iff,axiom,
    ( ord_le8081472938463900775at_nat
    = ( ^ [B4: set_Pr8551490117392284871at_nat,A5: set_Pr8551490117392284871at_nat] :
          ( A5
          = ( sup_su3035147773818789531at_nat @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1314_sup_Oorder__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B4: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat] :
          ( A5
          = ( sup_su5525570899277871387at_nat @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1315_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( A5
          = ( sup_sup_set_nat @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1316_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( A5
          = ( sup_sup_set_int @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1317_sup_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( A5
          = ( sup_sup_rat @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1318_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( A5
          = ( sup_sup_nat @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1319_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( A5
          = ( sup_sup_int @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1320_sup_OboundedI,axiom,
    ! [B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ B @ A )
     => ( ( ord_le8081472938463900775at_nat @ C2 @ A )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1321_sup_OboundedI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B @ A )
     => ( ( ord_le1268244103169919719at_nat @ C2 @ A )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1322_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C2 @ A )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1323_sup_OboundedI,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C2 @ A )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1324_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ A )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1325_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1326_sup_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ A )
       => ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1327_sup_OboundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_le8081472938463900775at_nat @ B @ A )
         => ~ ( ord_le8081472938463900775at_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_1328_sup_OboundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_le1268244103169919719at_nat @ B @ A )
         => ~ ( ord_le1268244103169919719at_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_1329_sup_OboundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_set_nat @ B @ A )
         => ~ ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_1330_sup_OboundedE,axiom,
    ! [B: set_int,C2: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_set_int @ B @ A )
         => ~ ( ord_less_eq_set_int @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_1331_sup_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_1332_sup_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_1333_sup_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_1334_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_1335_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_1336_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_1337_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_1338_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( sup_sup_rat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1339_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1340_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( sup_sup_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1341_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_1342_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_1343_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_1344_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_1345_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( sup_sup_rat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1346_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1347_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( sup_sup_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1348_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_1349_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_1350_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_1351_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_1352_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1353_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1354_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1355_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_1356_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_1357_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_1358_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_1359_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1360_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1361_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1362_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_1363_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_1364_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_1365_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_1366_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_1367_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_1368_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_1369_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_1370_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_1371_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_1372_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_1373_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( sup_sup_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% sup.orderI
thf(fact_1374_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_1375_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( sup_sup_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% sup.orderI
thf(fact_1376_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_1377_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_1378_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_1379_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_1380_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( sup_sup_rat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1381_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1382_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( sup_sup_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1383_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_1384_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_1385_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_1386_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_1387_le__iff__sup,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( sup_sup_rat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_1388_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( sup_sup_nat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_1389_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( sup_sup_int @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_1390_sup__least,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ Y @ X )
     => ( ( ord_le8081472938463900775at_nat @ Z3 @ X )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_1391_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ Y @ X )
     => ( ( ord_le1268244103169919719at_nat @ Z3 @ X )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_1392_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ Z3 @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_1393_sup__least,axiom,
    ! [Y: set_int,X: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_less_eq_set_int @ Z3 @ X )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_1394_sup__least,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ Z3 @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_1395_sup__least,axiom,
    ! [Y: nat,X: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z3 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_1396_sup__least,axiom,
    ! [Y: int,X: int,Z3: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ Z3 @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_1397_sup__mono,axiom,
    ! [A: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A @ C2 )
     => ( ( ord_le8081472938463900775at_nat @ B @ D2 )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ A @ B ) @ ( sup_su3035147773818789531at_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1398_sup__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B @ D2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ ( sup_su5525570899277871387at_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1399_sup__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1400_sup__mono,axiom,
    ! [A: set_int,C2: set_int,B: set_int,D2: set_int] :
      ( ( ord_less_eq_set_int @ A @ C2 )
     => ( ( ord_less_eq_set_int @ B @ D2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ ( sup_sup_set_int @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1401_sup__mono,axiom,
    ! [A: rat,C2: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ C2 )
     => ( ( ord_less_eq_rat @ B @ D2 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ ( sup_sup_rat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1402_sup__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1403_sup__mono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1404_sup_Omono,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,D2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ C2 @ A )
     => ( ( ord_le8081472938463900775at_nat @ D2 @ B )
       => ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ C2 @ D2 ) @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1405_sup_Omono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ A )
     => ( ( ord_le1268244103169919719at_nat @ D2 @ B )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C2 @ D2 ) @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1406_sup_Omono,axiom,
    ! [C2: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A )
     => ( ( ord_less_eq_set_nat @ D2 @ B )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1407_sup_Omono,axiom,
    ! [C2: set_int,A: set_int,D2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ C2 @ A )
     => ( ( ord_less_eq_set_int @ D2 @ B )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ C2 @ D2 ) @ ( sup_sup_set_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1408_sup_Omono,axiom,
    ! [C2: rat,A: rat,D2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ( ord_less_eq_rat @ D2 @ B )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ C2 @ D2 ) @ ( sup_sup_rat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1409_sup_Omono,axiom,
    ! [C2: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1410_sup_Omono,axiom,
    ! [C2: int,A: int,D2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ( ord_less_eq_int @ D2 @ B )
       => ( ord_less_eq_int @ ( sup_sup_int @ C2 @ D2 ) @ ( sup_sup_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1411_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_1412_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_1413_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_1414_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_1415_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_1416_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_1417_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_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_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_1425_sup__ge2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ Y @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1426_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1427_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_1428_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_1429_sup__ge2,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1430_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1431_sup__ge2,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_1432_sup__ge1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1433_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1434_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_1435_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_1436_sup__ge1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1437_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1438_sup__ge1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_1439_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_1440_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_1441_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_1442_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_1443_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_1444_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_1445_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_1446_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_1447_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_1448_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_1449_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_1450_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_1451_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_1452_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_1453_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_1454_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_1455_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_1456_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_1457_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_1458_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_1459_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_1460_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_1461_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_1462_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_1463_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_1464_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_1465_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_1466_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_1467_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ C2 @ B )
     => ( ord_le7642048601412989811at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1468_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C2 @ B )
     => ( ord_le2604355607129572851at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1469_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ C2 @ B )
     => ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1470_sup_Ostrict__coboundedI2,axiom,
    ! [C2: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ C2 @ B )
     => ( ord_less_assn @ C2 @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1471_sup_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ B )
     => ( ord_less_rat @ C2 @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1472_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C2 @ B )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1473_sup_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_int @ C2 @ B )
     => ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1474_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ C2 @ A )
     => ( ord_le7642048601412989811at_nat @ C2 @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1475_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C2 @ A )
     => ( ord_le2604355607129572851at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1476_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ C2 @ A )
     => ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1477_sup_Ostrict__coboundedI1,axiom,
    ! [C2: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ C2 @ A )
     => ( ord_less_assn @ C2 @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1478_sup_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ A )
     => ( ord_less_rat @ C2 @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1479_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C2 @ A )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1480_sup_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ A )
     => ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1481_sup_Ostrict__order__iff,axiom,
    ( ord_le7642048601412989811at_nat
    = ( ^ [B4: set_Pr8551490117392284871at_nat,A5: set_Pr8551490117392284871at_nat] :
          ( ( A5
            = ( sup_su3035147773818789531at_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1482_sup_Ostrict__order__iff,axiom,
    ( ord_le2604355607129572851at_nat
    = ( ^ [B4: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat] :
          ( ( A5
            = ( sup_su5525570899277871387at_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1483_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( ( A5
            = ( sup_sup_set_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1484_sup_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [B4: assn,A5: assn] :
          ( ( A5
            = ( sup_sup_assn @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1485_sup_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( A5
            = ( sup_sup_rat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1486_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( A5
            = ( sup_sup_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1487_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( A5
            = ( sup_sup_int @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1488_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr8551490117392284871at_nat,C2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( ord_le7642048601412989811at_nat @ ( sup_su3035147773818789531at_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_le7642048601412989811at_nat @ B @ A )
         => ~ ( ord_le7642048601412989811at_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1489_sup_Ostrict__boundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ ( sup_su5525570899277871387at_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_le2604355607129572851at_nat @ B @ A )
         => ~ ( ord_le2604355607129572851at_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1490_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_set_nat @ B @ A )
         => ~ ( ord_less_set_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1491_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( ord_less_assn @ ( sup_sup_assn @ B @ C2 ) @ A )
     => ~ ( ( ord_less_assn @ B @ A )
         => ~ ( ord_less_assn @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1492_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_rat @ ( sup_sup_rat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1493_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1494_sup_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1495_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_1496_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_1497_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_1498_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( sup_sup_assn @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1499_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1500_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1501_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1502_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_1503_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_1504_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_1505_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( sup_sup_assn @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1506_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1507_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1508_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1509_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_1510_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_1511_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_1512_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_1513_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_1514_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_1515_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_1516_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_1517_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_1518_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_1519_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_1520_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_1521_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_1522_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_1523_pred__subset__eq,axiom,
    ! [R3: set_o,S: set_o] :
      ( ( ord_less_eq_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R3 )
        @ ^ [X4: $o] : ( member_o @ X4 @ S ) )
      = ( ord_less_eq_set_o @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_1524_pred__subset__eq,axiom,
    ! [R3: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int] :
      ( ( ord_le8334417538754933252_int_o
        @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ R3 )
        @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ S ) )
      = ( ord_le483042692224249369nt_int @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_1525_pred__subset__eq,axiom,
    ! [R3: set_set_nat,S: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R3 )
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S ) )
      = ( ord_le6893508408891458716et_nat @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_1526_pred__subset__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R3 )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
      = ( ord_less_eq_set_nat @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_1527_pred__subset__eq,axiom,
    ! [R3: set_int,S: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R3 )
        @ ^ [X4: int] : ( member_int @ X4 @ S ) )
      = ( ord_less_eq_set_int @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_1528_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option4065278094766928714it_nat,Y: option4065278094766928714it_nat] :
      ( ( ( heap_T7616092557645711336rame_b @ X @ Xa )
        = Y )
     => ( ! [R4: b,H2: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa
              = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) )
           => ( Y
             != ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N4 ) ) ) ) ) )
       => ~ ( ( Xa = none_P6779099274072355161it_nat )
           => ( Y != none_P6779099274072355161it_nat ) ) ) ) ).

% timeFrame.elims
thf(fact_1529_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option3562590408128118217it_nat,Y: option3562590408128118217it_nat] :
      ( ( ( heap_T7616092557645711335rame_a @ X @ Xa )
        = Y )
     => ( ! [R4: a,H2: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa
              = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) )
           => ( Y
             != ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N4 ) ) ) ) ) )
       => ~ ( ( Xa = none_P2651198173097904984it_nat )
           => ( Y != none_P2651198173097904984it_nat ) ) ) ) ).

% timeFrame.elims
thf(fact_1530_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( ( heap_T3616969660504097270t_unit @ X @ Xa )
        = Y )
     => ( ! [R4: product_unit,H2: heap_e7401611519738050253t_unit,N4: nat] :
            ( ( Xa
              = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) )
           => ( Y
             != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N4 ) ) ) ) ) )
       => ~ ( ( Xa = none_P9117596204409417319it_nat )
           => ( Y != none_P9117596204409417319it_nat ) ) ) ) ).

% timeFrame.elims
thf(fact_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_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
              @ ^ [Z: set_Pr8551490117392284871at_nat] : ( some_s287724117700012716at_nat @ ( sup_su3035147773818789531at_nat @ X8 @ Z ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_1540_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
              @ ^ [Z: set_Pr4329608150637261639at_nat] : ( some_s5890477192898017836at_nat @ ( sup_su5525570899277871387at_nat @ X8 @ Z ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_1541_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
              @ ^ [Z: set_nat] : ( some_set_nat @ ( sup_sup_set_nat @ X8 @ Z ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_1542_sup__option__def,axiom,
    ( sup_sup_option_nat
    = ( ^ [X4: option_nat,Y4: option_nat] :
          ( case_o7429725398727453821at_nat @ Y4
          @ ^ [X8: nat] :
              ( case_o7429725398727453821at_nat @ X4
              @ ^ [Z: nat] : ( some_nat @ ( sup_sup_nat @ X8 @ Z ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_1543_sup__option__def,axiom,
    ( sup_sup_option_int
    = ( ^ [X4: option_int,Y4: option_int] :
          ( case_o390784466056649525nt_int @ Y4
          @ ^ [X8: int] :
              ( case_o390784466056649525nt_int @ X4
              @ ^ [Z: int] : ( some_int @ ( sup_sup_int @ X8 @ Z ) )
              @ Y4 )
          @ X4 ) ) ) ).

% sup_option_def
thf(fact_1544_sup__Un__eq,axiom,
    ! [R3: set_o,S: set_o] :
      ( ( sup_sup_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R3 )
        @ ^ [X4: $o] : ( member_o @ X4 @ S ) )
      = ( ^ [X4: $o] : ( member_o @ X4 @ ( sup_sup_set_o @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1545_sup__Un__eq,axiom,
    ! [R3: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int] :
      ( ( sup_su1852724690005176016_int_o
        @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ R3 )
        @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ S ) )
      = ( ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ ( sup_su2047564715030645325nt_int @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1546_sup__Un__eq,axiom,
    ! [R3: set_set_nat,S: set_set_nat] :
      ( ( sup_sup_set_nat_o
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R3 )
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S ) )
      = ( ^ [X4: set_nat] : ( member_set_nat @ X4 @ ( sup_sup_set_set_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1547_sup__Un__eq,axiom,
    ! [R3: set_int,S: set_int] :
      ( ( sup_sup_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R3 )
        @ ^ [X4: int] : ( member_int @ X4 @ S ) )
      = ( ^ [X4: int] : ( member_int @ X4 @ ( sup_sup_set_int @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1548_sup__Un__eq,axiom,
    ! [R3: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
      ( ( sup_su6256023009775730178_nat_o
        @ ^ [X4: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X4 @ R3 )
        @ ^ [X4: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X4 @ S ) )
      = ( ^ [X4: produc4166570645942440679at_nat] : ( member6689249552917799696at_nat @ X4 @ ( sup_su3035147773818789531at_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1549_sup__Un__eq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su2080679670758317954_nat_o
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ R3 )
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ S ) )
      = ( ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1550_sup__Un__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R3 )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
      = ( ^ [X4: nat] : ( member_nat @ X4 @ ( sup_sup_set_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1551_sup__Un__eq2,axiom,
    ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( sup_sup_int_int_o
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ ( sup_su6024340866399070445nt_int @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1552_sup__Un__eq2,axiom,
    ! [R3: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( sup_su1630790145277462993_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1553_sup__Un__eq2,axiom,
    ! [R3: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( sup_su6535292691877529429_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1554_sup__Un__eq2,axiom,
    ! [R3: set_Pr7600907837789447088it_nat,S: set_Pr7600907837789447088it_nat] :
      ( ( sup_su4207681470386032469_nat_o
        @ ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ ( sup_su3714129841433130204it_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1555_sup__Un__eq2,axiom,
    ! [R3: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ( sup_su2191776239474528790_nat_o
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1556_sup__Un__eq2,axiom,
    ! [R3: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
      ( ( sup_su2200014604384089602_nat_o
        @ ^ [X4: multis2468970476368604999at_nat,Y4: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1557_sup__Un__eq2,axiom,
    ! [R3: 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 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1558_pred__subset__eq2,axiom,
    ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ S ) )
      = ( ord_le2843351958646193337nt_int @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_1559_pred__subset__eq2,axiom,
    ! [R3: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( ord_le8000401564054156549_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le5966269811547037844et_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_1560_pred__subset__eq2,axiom,
    ! [R3: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( ord_le6753239538765779593_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le4763372923235995152et_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_1561_pred__subset__eq2,axiom,
    ! [R3: set_Pr7600907837789447088it_nat,S: set_Pr7600907837789447088it_nat] :
      ( ( ord_le785348814064375433_nat_o
        @ ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le4980869671742669840it_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_1562_pred__subset__eq2,axiom,
    ! [R3: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ( ord_le7992815620007647562_nat_o
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le4478181985103859343it_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_1563_pred__equals__eq2,axiom,
    ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R3 ) )
        = ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_1564_pred__equals__eq2,axiom,
    ! [R3: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R3 ) )
        = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_1565_pred__equals__eq2,axiom,
    ! [R3: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R3 ) )
        = ( ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_1566_pred__equals__eq2,axiom,
    ! [R3: set_Pr7600907837789447088it_nat,S: set_Pr7600907837789447088it_nat] :
      ( ( ( ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ R3 ) )
        = ( ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_1567_pred__equals__eq2,axiom,
    ! [R3: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ( ( ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R3 ) )
        = ( ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_1568_execute__guard_I2_J,axiom,
    ! [P2: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
      ( ( P2 @ H )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T8440541562793052209t_unit @ P2 @ F ) @ H )
        = ( some_P1914260805536162275it_nat @ ( F @ H ) ) ) ) ).

% execute_guard(2)
thf(fact_1569_execute__guard_I2_J,axiom,
    ! [P2: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc7388388658123137530it_nat] :
      ( ( P2 @ H )
     => ( ( heap_Time_execute_b @ ( heap_Time_guard_b @ P2 @ F ) @ H )
        = ( some_P2818173045054083285it_nat @ ( F @ H ) ) ) ) ).

% execute_guard(2)
thf(fact_1570_execute__guard_I2_J,axiom,
    ! [P2: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc3260487557148687353it_nat] :
      ( ( P2 @ H )
     => ( ( heap_Time_execute_a @ ( heap_Time_guard_a @ P2 @ F ) @ H )
        = ( some_P7913643980934408916it_nat @ ( F @ H ) ) ) ) ).

% execute_guard(2)
thf(fact_1571_some__opt__eq__trivial,axiom,
    ! [X: produc7388388658123137530it_nat] :
      ( ( eps_Op5869321083069970014it_nat
        @ ^ [Y4: produc7388388658123137530it_nat] : Y4 = X )
      = ( some_P2818173045054083285it_nat @ X ) ) ).

% some_opt_eq_trivial
thf(fact_1572_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_1573_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_1574_some__opt__eq__trivial,axiom,
    ! [X: num] :
      ( ( eps_Opt_num
        @ ^ [Y4: num] : Y4 = X )
      = ( some_num @ X ) ) ).

% some_opt_eq_trivial
thf(fact_1575_not__None__eq,axiom,
    ! [X: option4065278094766928714it_nat] :
      ( ( X != none_P6779099274072355161it_nat )
      = ( ? [Y4: produc7388388658123137530it_nat] :
            ( X
            = ( some_P2818173045054083285it_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_1576_not__None__eq,axiom,
    ! [X: option3562590408128118217it_nat] :
      ( ( X != none_P2651198173097904984it_nat )
      = ( ? [Y4: produc3260487557148687353it_nat] :
            ( X
            = ( some_P7913643980934408916it_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_1577_not__None__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( X != none_P9117596204409417319it_nat )
      = ( ? [Y4: produc8664842809031399944it_nat] :
            ( X
            = ( some_P1914260805536162275it_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_1578_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y4: num] :
            ( X
            = ( some_num @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_1579_not__Some__eq,axiom,
    ! [X: option4065278094766928714it_nat] :
      ( ( ! [Y4: produc7388388658123137530it_nat] :
            ( X
           != ( some_P2818173045054083285it_nat @ Y4 ) ) )
      = ( X = none_P6779099274072355161it_nat ) ) ).

% not_Some_eq
thf(fact_1580_not__Some__eq,axiom,
    ! [X: option3562590408128118217it_nat] :
      ( ( ! [Y4: produc3260487557148687353it_nat] :
            ( X
           != ( some_P7913643980934408916it_nat @ Y4 ) ) )
      = ( X = none_P2651198173097904984it_nat ) ) ).

% not_Some_eq
thf(fact_1581_not__Some__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( ! [Y4: produc8664842809031399944it_nat] :
            ( X
           != ( some_P1914260805536162275it_nat @ Y4 ) ) )
      = ( X = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq
thf(fact_1582_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y4: num] :
            ( X
           != ( some_num @ Y4 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_1583_some__opt__sym__eq__trivial,axiom,
    ! [X: produc7388388658123137530it_nat] :
      ( ( eps_Op5869321083069970014it_nat
        @ ( ^ [Y6: produc7388388658123137530it_nat,Z4: produc7388388658123137530it_nat] : Y6 = Z4
          @ X ) )
      = ( some_P2818173045054083285it_nat @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_1584_some__opt__sym__eq__trivial,axiom,
    ! [X: produc3260487557148687353it_nat] :
      ( ( eps_Op1741419982095519837it_nat
        @ ( ^ [Y6: produc3260487557148687353it_nat,Z4: produc3260487557148687353it_nat] : Y6 = Z4
          @ X ) )
      = ( some_P7913643980934408916it_nat @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_1585_some__opt__sym__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] :
      ( ( eps_Op3393321821070424684it_nat
        @ ( ^ [Y6: produc8664842809031399944it_nat,Z4: produc8664842809031399944it_nat] : Y6 = Z4
          @ X ) )
      = ( some_P1914260805536162275it_nat @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_1586_some__opt__sym__eq__trivial,axiom,
    ! [X: num] :
      ( ( eps_Opt_num
        @ ( ^ [Y6: num,Z4: num] : Y6 = Z4
          @ X ) )
      = ( some_num @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_1587_less__eq__option__None__code,axiom,
    ! [X: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X ) ).

% less_eq_option_None_code
thf(fact_1588_less__option__None,axiom,
    ! [X: option_num] :
      ~ ( ord_less_option_num @ X @ none_num ) ).

% less_option_None
thf(fact_1589_some__opt__false__trivial,axiom,
    ( ( eps_Opt_num
      @ ^ [Uu2: num] : $false )
    = none_num ) ).

% some_opt_false_trivial
thf(fact_1590_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ( ! [X4: int,Y4: int] :
            ( V
           != ( some_P4184893108420464158nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) ) ) )
      = ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
thf(fact_1591_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_1592_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_1593_not__Some__eq2,axiom,
    ! [V: option4065278094766928714it_nat] :
      ( ( ! [X4: b,Y4: produc6653097349344004940it_nat] :
            ( V
           != ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) ) ) )
      = ( V = none_P6779099274072355161it_nat ) ) ).

% not_Some_eq2
thf(fact_1594_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_1595_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_1596_less__eq__option__Some__None,axiom,
    ! [X: num] :
      ~ ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ none_num ) ).

% less_eq_option_Some_None
thf(fact_1597_less__option__None__Some__code,axiom,
    ! [X: num] : ( ord_less_option_num @ none_num @ ( some_num @ X ) ) ).

% less_option_None_Some_code
thf(fact_1598_option_Osimps_I4_J,axiom,
    ! [F1: int,F2: num > int] :
      ( ( case_option_int_num @ F1 @ F2 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_1599_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F2: num > option_num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F2 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_1600_option_Osimps_I4_J,axiom,
    ! [F1: num,F2: num > num] :
      ( ( case_option_num_num @ F1 @ F2 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_1601_execute__guard_I1_J,axiom,
    ! [P2: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc7388388658123137530it_nat] :
      ( ~ ( P2 @ H )
     => ( ( heap_Time_execute_b @ ( heap_Time_guard_b @ P2 @ F ) @ H )
        = none_P6779099274072355161it_nat ) ) ).

% execute_guard(1)
thf(fact_1602_execute__guard_I1_J,axiom,
    ! [P2: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc3260487557148687353it_nat] :
      ( ~ ( P2 @ H )
     => ( ( heap_Time_execute_a @ ( heap_Time_guard_a @ P2 @ F ) @ H )
        = none_P2651198173097904984it_nat ) ) ).

% execute_guard(1)
thf(fact_1603_execute__raise,axiom,
    ! [S2: list_char] :
      ( ( heap_Time_execute_b @ ( heap_Time_raise_b @ S2 ) )
      = ( ^ [Uu2: heap_e7401611519738050253t_unit] : none_P6779099274072355161it_nat ) ) ).

% execute_raise
thf(fact_1604_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_1605_option_Ocase__distrib,axiom,
    ! [H: int > int,F1: int,F2: num > int,Option: option_num] :
      ( ( H @ ( case_option_int_num @ F1 @ F2 @ Option ) )
      = ( case_option_int_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1606_option_Ocase__distrib,axiom,
    ! [H: int > option_num,F1: int,F2: num > int,Option: option_num] :
      ( ( H @ ( case_option_int_num @ F1 @ F2 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1607_option_Ocase__distrib,axiom,
    ! [H: int > num,F1: int,F2: num > int,Option: option_num] :
      ( ( H @ ( case_option_int_num @ F1 @ F2 @ Option ) )
      = ( case_option_num_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1608_option_Ocase__distrib,axiom,
    ! [H: option_num > int,F1: option_num,F2: num > option_num,Option: option_num] :
      ( ( H @ ( case_o6005452278849405969um_num @ F1 @ F2 @ Option ) )
      = ( case_option_int_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1609_option_Ocase__distrib,axiom,
    ! [H: option_num > option_num,F1: option_num,F2: num > option_num,Option: option_num] :
      ( ( H @ ( case_o6005452278849405969um_num @ F1 @ F2 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1610_option_Ocase__distrib,axiom,
    ! [H: option_num > num,F1: option_num,F2: num > option_num,Option: option_num] :
      ( ( H @ ( case_o6005452278849405969um_num @ F1 @ F2 @ Option ) )
      = ( case_option_num_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1611_option_Ocase__distrib,axiom,
    ! [H: num > int,F1: num,F2: num > num,Option: option_num] :
      ( ( H @ ( case_option_num_num @ F1 @ F2 @ Option ) )
      = ( case_option_int_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1612_option_Ocase__distrib,axiom,
    ! [H: num > option_num,F1: num,F2: num > num,Option: option_num] :
      ( ( H @ ( case_option_num_num @ F1 @ F2 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1613_option_Ocase__distrib,axiom,
    ! [H: num > num,F1: num,F2: num > num,Option: option_num] :
      ( ( H @ ( case_option_num_num @ F1 @ F2 @ Option ) )
      = ( case_option_num_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_1614_option_Odistinct_I1_J,axiom,
    ! [X2: produc7388388658123137530it_nat] :
      ( none_P6779099274072355161it_nat
     != ( some_P2818173045054083285it_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_1615_option_Odistinct_I1_J,axiom,
    ! [X2: produc3260487557148687353it_nat] :
      ( none_P2651198173097904984it_nat
     != ( some_P7913643980934408916it_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_1616_option_Odistinct_I1_J,axiom,
    ! [X2: produc8664842809031399944it_nat] :
      ( none_P9117596204409417319it_nat
     != ( some_P1914260805536162275it_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_1617_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_1618_option_OdiscI,axiom,
    ! [Option: option4065278094766928714it_nat,X2: produc7388388658123137530it_nat] :
      ( ( Option
        = ( some_P2818173045054083285it_nat @ X2 ) )
     => ( Option != none_P6779099274072355161it_nat ) ) ).

% option.discI
thf(fact_1619_option_OdiscI,axiom,
    ! [Option: option3562590408128118217it_nat,X2: produc3260487557148687353it_nat] :
      ( ( Option
        = ( some_P7913643980934408916it_nat @ X2 ) )
     => ( Option != none_P2651198173097904984it_nat ) ) ).

% option.discI
thf(fact_1620_option_OdiscI,axiom,
    ! [Option: option8956607266484857688it_nat,X2: produc8664842809031399944it_nat] :
      ( ( Option
        = ( some_P1914260805536162275it_nat @ X2 ) )
     => ( Option != none_P9117596204409417319it_nat ) ) ).

% option.discI
thf(fact_1621_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_1622_option_Oexhaust,axiom,
    ! [Y: option4065278094766928714it_nat] :
      ( ( Y != none_P6779099274072355161it_nat )
     => ~ ! [X22: produc7388388658123137530it_nat] :
            ( Y
           != ( some_P2818173045054083285it_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_1623_option_Oexhaust,axiom,
    ! [Y: option3562590408128118217it_nat] :
      ( ( Y != none_P2651198173097904984it_nat )
     => ~ ! [X22: produc3260487557148687353it_nat] :
            ( Y
           != ( some_P7913643980934408916it_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_1624_option_Oexhaust,axiom,
    ! [Y: option8956607266484857688it_nat] :
      ( ( Y != none_P9117596204409417319it_nat )
     => ~ ! [X22: produc8664842809031399944it_nat] :
            ( Y
           != ( some_P1914260805536162275it_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_1625_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] :
            ( Y
           != ( some_num @ X22 ) ) ) ).

% option.exhaust
thf(fact_1626_split__option__ex,axiom,
    ( ( ^ [P4: option4065278094766928714it_nat > $o] :
        ? [X5: option4065278094766928714it_nat] : ( P4 @ X5 ) )
    = ( ^ [P5: option4065278094766928714it_nat > $o] :
          ( ( P5 @ none_P6779099274072355161it_nat )
          | ? [X4: produc7388388658123137530it_nat] : ( P5 @ ( some_P2818173045054083285it_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_1627_split__option__ex,axiom,
    ( ( ^ [P4: option3562590408128118217it_nat > $o] :
        ? [X5: option3562590408128118217it_nat] : ( P4 @ X5 ) )
    = ( ^ [P5: option3562590408128118217it_nat > $o] :
          ( ( P5 @ none_P2651198173097904984it_nat )
          | ? [X4: produc3260487557148687353it_nat] : ( P5 @ ( some_P7913643980934408916it_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_1628_split__option__ex,axiom,
    ( ( ^ [P4: option8956607266484857688it_nat > $o] :
        ? [X5: option8956607266484857688it_nat] : ( P4 @ X5 ) )
    = ( ^ [P5: option8956607266484857688it_nat > $o] :
          ( ( P5 @ none_P9117596204409417319it_nat )
          | ? [X4: produc8664842809031399944it_nat] : ( P5 @ ( some_P1914260805536162275it_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_1629_split__option__ex,axiom,
    ( ( ^ [P4: option_num > $o] :
        ? [X5: option_num] : ( P4 @ X5 ) )
    = ( ^ [P5: option_num > $o] :
          ( ( P5 @ none_num )
          | ? [X4: num] : ( P5 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_1630_split__option__all,axiom,
    ( ( ^ [P4: option4065278094766928714it_nat > $o] :
        ! [X5: option4065278094766928714it_nat] : ( P4 @ X5 ) )
    = ( ^ [P5: option4065278094766928714it_nat > $o] :
          ( ( P5 @ none_P6779099274072355161it_nat )
          & ! [X4: produc7388388658123137530it_nat] : ( P5 @ ( some_P2818173045054083285it_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_1631_split__option__all,axiom,
    ( ( ^ [P4: option3562590408128118217it_nat > $o] :
        ! [X5: option3562590408128118217it_nat] : ( P4 @ X5 ) )
    = ( ^ [P5: option3562590408128118217it_nat > $o] :
          ( ( P5 @ none_P2651198173097904984it_nat )
          & ! [X4: produc3260487557148687353it_nat] : ( P5 @ ( some_P7913643980934408916it_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_1632_split__option__all,axiom,
    ( ( ^ [P4: option8956607266484857688it_nat > $o] :
        ! [X5: option8956607266484857688it_nat] : ( P4 @ X5 ) )
    = ( ^ [P5: option8956607266484857688it_nat > $o] :
          ( ( P5 @ none_P9117596204409417319it_nat )
          & ! [X4: produc8664842809031399944it_nat] : ( P5 @ ( some_P1914260805536162275it_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_1633_split__option__all,axiom,
    ( ( ^ [P4: option_num > $o] :
        ! [X5: option_num] : ( P4 @ X5 ) )
    = ( ^ [P5: option_num > $o] :
          ( ( P5 @ none_num )
          & ! [X4: num] : ( P5 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_1634_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: num,B3: num] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1635_combine__options__cases,axiom,
    ! [X: option4065278094766928714it_nat,P2: option4065278094766928714it_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P6779099274072355161it_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: produc7388388658123137530it_nat,B3: num] :
              ( ( X
                = ( some_P2818173045054083285it_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1636_combine__options__cases,axiom,
    ! [X: option3562590408128118217it_nat,P2: option3562590408128118217it_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P2651198173097904984it_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: produc3260487557148687353it_nat,B3: num] :
              ( ( X
                = ( some_P7913643980934408916it_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1637_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,P2: option8956607266484857688it_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: produc8664842809031399944it_nat,B3: num] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1638_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option4065278094766928714it_nat > $o,Y: option4065278094766928714it_nat] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P6779099274072355161it_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: num,B3: produc7388388658123137530it_nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_P2818173045054083285it_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1639_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option3562590408128118217it_nat > $o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: num,B3: produc3260487557148687353it_nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_P7913643980934408916it_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1640_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option8956607266484857688it_nat > $o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: num,B3: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1641_combine__options__cases,axiom,
    ! [X: option4065278094766928714it_nat,P2: option4065278094766928714it_nat > option4065278094766928714it_nat > $o,Y: option4065278094766928714it_nat] :
      ( ( ( X = none_P6779099274072355161it_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P6779099274072355161it_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: produc7388388658123137530it_nat,B3: produc7388388658123137530it_nat] :
              ( ( X
                = ( some_P2818173045054083285it_nat @ A3 ) )
             => ( ( Y
                  = ( some_P2818173045054083285it_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1642_combine__options__cases,axiom,
    ! [X: option4065278094766928714it_nat,P2: option4065278094766928714it_nat > option3562590408128118217it_nat > $o,Y: option3562590408128118217it_nat] :
      ( ( ( X = none_P6779099274072355161it_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P2651198173097904984it_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: produc7388388658123137530it_nat,B3: produc3260487557148687353it_nat] :
              ( ( X
                = ( some_P2818173045054083285it_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7913643980934408916it_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1643_combine__options__cases,axiom,
    ! [X: option4065278094766928714it_nat,P2: option4065278094766928714it_nat > option8956607266484857688it_nat > $o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_P6779099274072355161it_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: produc7388388658123137530it_nat,B3: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_P2818173045054083285it_nat @ A3 ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1644_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_1645_less__eq__option__None,axiom,
    ! [X: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X ) ).

% less_eq_option_None
thf(fact_1646_option_Osimps_I5_J,axiom,
    ! [F1: int,F2: num > int,X2: num] :
      ( ( case_option_int_num @ F1 @ F2 @ ( some_num @ X2 ) )
      = ( F2 @ X2 ) ) ).

% option.simps(5)
thf(fact_1647_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F2: num > option_num,X2: num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F2 @ ( some_num @ X2 ) )
      = ( F2 @ X2 ) ) ).

% option.simps(5)
thf(fact_1648_option_Osimps_I5_J,axiom,
    ! [F1: num,F2: num > num,X2: num] :
      ( ( case_option_num_num @ F1 @ F2 @ ( some_num @ X2 ) )
      = ( F2 @ X2 ) ) ).

% option.simps(5)
thf(fact_1649_the__default_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( the_default_num @ X @ none_num )
      = X ) ).

% the_default.simps(2)
thf(fact_1650_Eps__Opt__eq__Some__implies,axiom,
    ! [P2: produc7388388658123137530it_nat > $o,X: produc7388388658123137530it_nat] :
      ( ( ( eps_Op5869321083069970014it_nat @ P2 )
        = ( some_P2818173045054083285it_nat @ X ) )
     => ( P2 @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_1651_Eps__Opt__eq__Some__implies,axiom,
    ! [P2: produc3260487557148687353it_nat > $o,X: produc3260487557148687353it_nat] :
      ( ( ( eps_Op1741419982095519837it_nat @ P2 )
        = ( some_P7913643980934408916it_nat @ X ) )
     => ( P2 @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_1652_Eps__Opt__eq__Some__implies,axiom,
    ! [P2: produc8664842809031399944it_nat > $o,X: produc8664842809031399944it_nat] :
      ( ( ( eps_Op3393321821070424684it_nat @ P2 )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( P2 @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_1653_Eps__Opt__eq__Some__implies,axiom,
    ! [P2: num > $o,X: num] :
      ( ( ( eps_Opt_num @ P2 )
        = ( some_num @ X ) )
     => ( P2 @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_1654_Eps__Opt__eq__Some,axiom,
    ! [P2: produc7388388658123137530it_nat > $o,X: produc7388388658123137530it_nat] :
      ( ! [X9: produc7388388658123137530it_nat] :
          ( ( P2 @ X )
         => ( ( P2 @ X9 )
           => ( X9 = X ) ) )
     => ( ( ( eps_Op5869321083069970014it_nat @ P2 )
          = ( some_P2818173045054083285it_nat @ X ) )
        = ( P2 @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_1655_Eps__Opt__eq__Some,axiom,
    ! [P2: produc3260487557148687353it_nat > $o,X: produc3260487557148687353it_nat] :
      ( ! [X9: produc3260487557148687353it_nat] :
          ( ( P2 @ X )
         => ( ( P2 @ X9 )
           => ( X9 = X ) ) )
     => ( ( ( eps_Op1741419982095519837it_nat @ P2 )
          = ( some_P7913643980934408916it_nat @ X ) )
        = ( P2 @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_1656_Eps__Opt__eq__Some,axiom,
    ! [P2: produc8664842809031399944it_nat > $o,X: produc8664842809031399944it_nat] :
      ( ! [X9: produc8664842809031399944it_nat] :
          ( ( P2 @ X )
         => ( ( P2 @ X9 )
           => ( X9 = X ) ) )
     => ( ( ( eps_Op3393321821070424684it_nat @ P2 )
          = ( some_P1914260805536162275it_nat @ X ) )
        = ( P2 @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_1657_Eps__Opt__eq__Some,axiom,
    ! [P2: num > $o,X: num] :
      ( ! [X9: num] :
          ( ( P2 @ X )
         => ( ( P2 @ X9 )
           => ( X9 = X ) ) )
     => ( ( ( eps_Opt_num @ P2 )
          = ( some_num @ X ) )
        = ( P2 @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_1658_execute__bind_I2_J,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit,G: b > heap_Time_Heap_b] :
      ( ( ( heap_Time_execute_b @ F @ H )
        = none_P6779099274072355161it_nat )
     => ( ( heap_Time_execute_b @ ( heap_Time_bind_b_b @ F @ G ) @ H )
        = none_P6779099274072355161it_nat ) ) ).

% execute_bind(2)
thf(fact_1659_execute__bind_I2_J,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit,G: b > heap_Time_Heap_a] :
      ( ( ( heap_Time_execute_b @ F @ H )
        = none_P6779099274072355161it_nat )
     => ( ( heap_Time_execute_a @ ( heap_Time_bind_b_a @ F @ G ) @ H )
        = none_P2651198173097904984it_nat ) ) ).

% execute_bind(2)
thf(fact_1660_execute__bind_I2_J,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit,G: a > heap_Time_Heap_b] :
      ( ( ( heap_Time_execute_a @ F @ H )
        = none_P2651198173097904984it_nat )
     => ( ( heap_Time_execute_b @ ( heap_Time_bind_a_b @ F @ G ) @ H )
        = none_P6779099274072355161it_nat ) ) ).

% execute_bind(2)
thf(fact_1661_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_1662_success__def,axiom,
    ( heap_Time_success_b
    = ( ^ [F3: heap_Time_Heap_b,H6: heap_e7401611519738050253t_unit] :
          ( ( heap_Time_execute_b @ F3 @ H6 )
         != none_P6779099274072355161it_nat ) ) ) ).

% success_def
thf(fact_1663_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_1664_successI,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit] :
      ( ( ( heap_Time_execute_b @ F @ H )
       != none_P6779099274072355161it_nat )
     => ( heap_Time_success_b @ F @ H ) ) ).

% successI
thf(fact_1665_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_1666_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_1667_less__option__None__Some,axiom,
    ! [X: num] : ( ord_less_option_num @ none_num @ ( some_num @ X ) ) ).

% less_option_None_Some
thf(fact_1668_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R2 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1669_subrelI,axiom,
    ! [R2: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
          ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R2 )
         => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le5966269811547037844et_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1670_subrelI,axiom,
    ! [R2: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
          ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R2 )
         => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le4763372923235995152et_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1671_subrelI,axiom,
    ! [R2: set_Pr7600907837789447088it_nat,S2: set_Pr7600907837789447088it_nat] :
      ( ! [X3: b,Y3: produc6653097349344004940it_nat] :
          ( ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X3 @ Y3 ) @ R2 )
         => ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le4980869671742669840it_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1672_subrelI,axiom,
    ! [R2: set_Pr7098220151150636591it_nat,S2: set_Pr7098220151150636591it_nat] :
      ( ! [X3: a,Y3: produc6653097349344004940it_nat] :
          ( ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X3 @ Y3 ) @ R2 )
         => ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le4478181985103859343it_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1673_Heap__cases,axiom,
    ! [F: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
      ( ! [X3: product_unit,H4: produc6653097349344004940it_nat] :
          ( ( heap_T875086893843062177t_unit @ F @ H )
         != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X3 @ H4 ) ) )
     => ( ( heap_T875086893843062177t_unit @ F @ H )
        = none_P9117596204409417319it_nat ) ) ).

% Heap_cases
thf(fact_1674_Heap__cases,axiom,
    ! [F: heap_Time_Heap_b,H: heap_e7401611519738050253t_unit] :
      ( ! [X3: b,H4: produc6653097349344004940it_nat] :
          ( ( heap_Time_execute_b @ F @ H )
         != ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ X3 @ H4 ) ) )
     => ( ( heap_Time_execute_b @ F @ H )
        = none_P6779099274072355161it_nat ) ) ).

% Heap_cases
thf(fact_1675_Heap__cases,axiom,
    ! [F: heap_Time_Heap_a,H: heap_e7401611519738050253t_unit] :
      ( ! [X3: a,H4: produc6653097349344004940it_nat] :
          ( ( heap_Time_execute_a @ F @ H )
         != ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X3 @ H4 ) ) )
     => ( ( heap_Time_execute_a @ F @ H )
        = none_P2651198173097904984it_nat ) ) ).

% Heap_cases
thf(fact_1676_timeFrame_Ocases,axiom,
    ! [X: produc2207270350733924475it_nat] :
      ( ! [N5: nat,R4: b,H2: heap_e7401611519738050253t_unit,N4: nat] :
          ( X
         != ( produc4613957359643799859it_nat @ N5 @ ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) ) )
     => ~ ! [N5: nat] :
            ( X
           != ( produc4613957359643799859it_nat @ N5 @ none_P6779099274072355161it_nat ) ) ) ).

% timeFrame.cases
thf(fact_1677_timeFrame_Ocases,axiom,
    ! [X: produc4453839368661128058it_nat] :
      ( ! [N5: nat,R4: a,H2: heap_e7401611519738050253t_unit,N4: nat] :
          ( X
         != ( produc4111269673004989362it_nat @ N5 @ ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) ) )
     => ~ ! [N5: nat] :
            ( X
           != ( produc4111269673004989362it_nat @ N5 @ none_P2651198173097904984it_nat ) ) ) ).

% timeFrame.cases
thf(fact_1678_timeFrame_Ocases,axiom,
    ! [X: produc3911288613690379145it_nat] :
      ( ! [N5: nat,R4: product_unit,H2: heap_e7401611519738050253t_unit,N4: nat] :
          ( X
         != ( produc638857205735767105it_nat @ N5 @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) ) )
     => ~ ! [N5: nat] :
            ( X
           != ( produc638857205735767105it_nat @ N5 @ none_P9117596204409417319it_nat ) ) ) ).

% timeFrame.cases
thf(fact_1679_timeFrame_Opelims,axiom,
    ! [X: nat,Xa: option4065278094766928714it_nat,Y: option4065278094766928714it_nat] :
      ( ( ( heap_T7616092557645711336rame_b @ X @ Xa )
        = Y )
     => ( ( accp_P4016523247509365636it_nat @ heap_T8132184524487034139_rel_b @ ( produc4613957359643799859it_nat @ X @ Xa ) )
       => ( ! [R4: b,H2: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa
                = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) )
             => ( ( Y
                  = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N4 ) ) ) ) )
               => ~ ( accp_P4016523247509365636it_nat @ heap_T8132184524487034139_rel_b @ ( produc4613957359643799859it_nat @ X @ ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) ) ) ) )
         => ~ ( ( Xa = none_P6779099274072355161it_nat )
             => ( ( Y = none_P6779099274072355161it_nat )
               => ~ ( accp_P4016523247509365636it_nat @ heap_T8132184524487034139_rel_b @ ( produc4613957359643799859it_nat @ X @ none_P6779099274072355161it_nat ) ) ) ) ) ) ) ).

% timeFrame.pelims
thf(fact_1680_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 ) )
       => ( ! [R4: a,H2: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa
                = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) )
             => ( ( Y
                  = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N4 ) ) ) ) )
               => ~ ( accp_P6263092265436569219it_nat @ heap_T8132184524487034138_rel_a @ ( produc4111269673004989362it_nat @ X @ ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ 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_1681_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 ) )
       => ( ! [R4: product_unit,H2: heap_e7401611519738050253t_unit,N4: nat] :
              ( ( Xa
                = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ N4 ) ) ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N4 ) ) ) ) )
               => ~ ( accp_P414730952086964626it_nat @ heap_T996182799752388649t_unit @ ( produc638857205735767105it_nat @ X @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R4 @ ( produc584006145561248582it_nat @ H2 @ 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_1682_disjE__realizer2,axiom,
    ! [P2: $o,Q2: num > $o,X: option_num,R3: int > $o,F: int,G: num > int] :
      ( ( case_option_o_num @ P2 @ Q2 @ X )
     => ( ( P2
         => ( R3 @ F ) )
       => ( ! [Q5: num] :
              ( ( Q2 @ Q5 )
             => ( R3 @ ( G @ Q5 ) ) )
         => ( R3 @ ( case_option_int_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_1683_disjE__realizer2,axiom,
    ! [P2: $o,Q2: num > $o,X: option_num,R3: option_num > $o,F: option_num,G: num > option_num] :
      ( ( case_option_o_num @ P2 @ Q2 @ X )
     => ( ( P2
         => ( R3 @ F ) )
       => ( ! [Q5: num] :
              ( ( Q2 @ Q5 )
             => ( R3 @ ( G @ Q5 ) ) )
         => ( R3 @ ( case_o6005452278849405969um_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_1684_disjE__realizer2,axiom,
    ! [P2: $o,Q2: num > $o,X: option_num,R3: num > $o,F: num,G: num > num] :
      ( ( case_option_o_num @ P2 @ Q2 @ X )
     => ( ( P2
         => ( R3 @ F ) )
       => ( ! [Q5: num] :
              ( ( Q2 @ Q5 )
             => ( R3 @ ( G @ Q5 ) ) )
         => ( R3 @ ( case_option_num_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_1685_set__to__map__def,axiom,
    ( set_to_map_int_int
    = ( ^ [S4: set_Pr958786334691620121nt_int,K3: int] :
          ( eps_Opt_int
          @ ^ [V2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ K3 @ V2 ) @ S4 ) ) ) ) ).

% set_to_map_def
thf(fact_1686_set__to__map__def,axiom,
    ( set_to6040779677306527128et_nat
    = ( ^ [S4: set_Pr3286484037609594932et_nat,K3: produc3658429121746597890et_nat > $o] :
          ( eps_Op2013419657081471078et_nat
          @ ^ [V2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ K3 @ V2 ) @ S4 ) ) ) ) ).

% set_to_map_def
thf(fact_1687_set__to__map__def,axiom,
    ( set_to2047380710992656148et_nat
    = ( ^ [S4: set_Pr8536935166611901872et_nat,K3: produc3658429121746597890et_nat > $o] :
          ( eps_Op6423059015816587746et_nat
          @ ^ [V2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ K3 @ V2 ) @ S4 ) ) ) ) ).

% set_to_map_def
thf(fact_1688_set__to__map__def,axiom,
    ( set_to6136816377707935252it_nat
    = ( ^ [S4: set_Pr7600907837789447088it_nat,K3: b] :
          ( eps_Op8301357815426737072it_nat
          @ ^ [V2: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ K3 @ V2 ) @ S4 ) ) ) ) ).

% set_to_map_def
thf(fact_1689_set__to__map__def,axiom,
    ( set_to2008915276733485075it_nat
    = ( ^ [S4: set_Pr7098220151150636591it_nat,K3: a] :
          ( eps_Op8301357815426737072it_nat
          @ ^ [V2: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ K3 @ V2 ) @ S4 ) ) ) ) ).

% set_to_map_def
thf(fact_1690_guard__def,axiom,
    ( heap_Time_guard_b
    = ( ^ [P5: heap_e7401611519738050253t_unit > $o,F3: heap_e7401611519738050253t_unit > produc7388388658123137530it_nat] :
          ( heap_Time_Heap_b2
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( if_opt7386294288321364996it_nat @ ( P5 @ H6 ) @ ( some_P2818173045054083285it_nat @ ( F3 @ H6 ) ) @ none_P6779099274072355161it_nat ) ) ) ) ).

% guard_def
thf(fact_1691_guard__def,axiom,
    ( heap_Time_guard_a
    = ( ^ [P5: heap_e7401611519738050253t_unit > $o,F3: heap_e7401611519738050253t_unit > produc3260487557148687353it_nat] :
          ( heap_Time_Heap_a2
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( if_opt6883606601682554499it_nat @ ( P5 @ H6 ) @ ( some_P7913643980934408916it_nat @ ( F3 @ H6 ) ) @ none_P2651198173097904984it_nat ) ) ) ) ).

% guard_def
thf(fact_1692_guard__def,axiom,
    ( heap_T8440541562793052209t_unit
    = ( ^ [P5: heap_e7401611519738050253t_unit > $o,F3: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( if_opt1729522071442692626it_nat @ ( P5 @ H6 ) @ ( some_P1914260805536162275it_nat @ ( F3 @ H6 ) ) @ none_P9117596204409417319it_nat ) ) ) ) ).

% guard_def
thf(fact_1693_Heap_Oinject,axiom,
    ! [X: heap_e7401611519738050253t_unit > option8956607266484857688it_nat,Ya: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( ( heap_T6183433275982383450t_unit @ X )
        = ( heap_T6183433275982383450t_unit @ Ya ) )
      = ( X = Ya ) ) ).

% Heap.inject
thf(fact_1694_Heap__execute,axiom,
    ! [F: heap_Time_Heap_b] :
      ( ( heap_Time_Heap_b2 @ ( heap_Time_execute_b @ F ) )
      = F ) ).

% Heap_execute
thf(fact_1695_Heap__execute,axiom,
    ! [F: heap_Time_Heap_a] :
      ( ( heap_Time_Heap_a2 @ ( heap_Time_execute_a @ F ) )
      = F ) ).

% Heap_execute
thf(fact_1696_Heap__execute,axiom,
    ! [F: heap_T5738788834812785303t_unit] :
      ( ( heap_T6183433275982383450t_unit @ ( heap_T875086893843062177t_unit @ F ) )
      = F ) ).

% Heap_execute
thf(fact_1697_Heap_Oexhaust,axiom,
    ! [Y: heap_T5738788834812785303t_unit] :
      ~ ! [X3: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
          ( Y
         != ( heap_T6183433275982383450t_unit @ X3 ) ) ).

% Heap.exhaust
thf(fact_1698_execute_Osimps,axiom,
    ! [F: heap_e7401611519738050253t_unit > option4065278094766928714it_nat] :
      ( ( heap_Time_execute_b @ ( heap_Time_Heap_b2 @ F ) )
      = F ) ).

% execute.simps
thf(fact_1699_execute_Osimps,axiom,
    ! [F: heap_e7401611519738050253t_unit > option3562590408128118217it_nat] :
      ( ( heap_Time_execute_a @ ( heap_Time_Heap_a2 @ F ) )
      = F ) ).

% execute.simps
thf(fact_1700_execute_Osimps,axiom,
    ! [F: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T6183433275982383450t_unit @ F ) )
      = F ) ).

% execute.simps
thf(fact_1701_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_1702_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_1703_case__optionE,axiom,
    ! [P2: $o,Q2: produc7388388658123137530it_nat > $o,X: option4065278094766928714it_nat] :
      ( ( case_o7395119307266030407it_nat @ P2 @ Q2 @ X )
     => ( ( ( X = none_P6779099274072355161it_nat )
         => ~ P2 )
       => ~ ! [Y3: produc7388388658123137530it_nat] :
              ( ( X
                = ( some_P2818173045054083285it_nat @ Y3 ) )
             => ~ ( Q2 @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_1704_case__optionE,axiom,
    ! [P2: $o,Q2: produc3260487557148687353it_nat > $o,X: option3562590408128118217it_nat] :
      ( ( case_o3267218206291580230it_nat @ P2 @ Q2 @ X )
     => ( ( ( X = none_P2651198173097904984it_nat )
         => ~ P2 )
       => ~ ! [Y3: produc3260487557148687353it_nat] :
              ( ( X
                = ( some_P7913643980934408916it_nat @ Y3 ) )
             => ~ ( Q2 @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_1705_case__optionE,axiom,
    ! [P2: $o,Q2: produc8664842809031399944it_nat > $o,X: option8956607266484857688it_nat] :
      ( ( case_o2686588417244861013it_nat @ P2 @ Q2 @ X )
     => ( ( ( X = none_P9117596204409417319it_nat )
         => ~ P2 )
       => ~ ! [Y3: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ Y3 ) )
             => ~ ( Q2 @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_1706_case__optionE,axiom,
    ! [P2: $o,Q2: num > $o,X: option_num] :
      ( ( case_option_o_num @ P2 @ Q2 @ X )
     => ( ( ( X = none_num )
         => ~ P2 )
       => ~ ! [Y3: num] :
              ( ( X
                = ( some_num @ Y3 ) )
             => ~ ( Q2 @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_1707_boolean__algebra__cancel_Osup2,axiom,
    ! [B5: set_Pr8551490117392284871at_nat,K2: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat] :
      ( ( B5
        = ( sup_su3035147773818789531at_nat @ K2 @ B ) )
     => ( ( sup_su3035147773818789531at_nat @ A @ B5 )
        = ( sup_su3035147773818789531at_nat @ K2 @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1708_boolean__algebra__cancel_Osup2,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,K2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( B5
        = ( sup_su5525570899277871387at_nat @ K2 @ B ) )
     => ( ( sup_su5525570899277871387at_nat @ A @ B5 )
        = ( sup_su5525570899277871387at_nat @ K2 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1709_boolean__algebra__cancel_Osup2,axiom,
    ! [B5: set_nat,K2: set_nat,B: set_nat,A: set_nat] :
      ( ( B5
        = ( sup_sup_set_nat @ K2 @ B ) )
     => ( ( sup_sup_set_nat @ A @ B5 )
        = ( sup_sup_set_nat @ K2 @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1710_boolean__algebra__cancel_Osup2,axiom,
    ! [B5: nat,K2: nat,B: nat,A: nat] :
      ( ( B5
        = ( sup_sup_nat @ K2 @ B ) )
     => ( ( sup_sup_nat @ A @ B5 )
        = ( sup_sup_nat @ K2 @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1711_boolean__algebra__cancel_Osup2,axiom,
    ! [B5: int,K2: int,B: int,A: int] :
      ( ( B5
        = ( sup_sup_int @ K2 @ B ) )
     => ( ( sup_sup_int @ A @ B5 )
        = ( sup_sup_int @ K2 @ ( sup_sup_int @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1712_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,K2: set_Pr8551490117392284871at_nat,A: set_Pr8551490117392284871at_nat,B: set_Pr8551490117392284871at_nat] :
      ( ( A4
        = ( sup_su3035147773818789531at_nat @ K2 @ A ) )
     => ( ( sup_su3035147773818789531at_nat @ A4 @ B )
        = ( sup_su3035147773818789531at_nat @ K2 @ ( sup_su3035147773818789531at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1713_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,K2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A4
        = ( sup_su5525570899277871387at_nat @ K2 @ A ) )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ B )
        = ( sup_su5525570899277871387at_nat @ K2 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1714_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: set_nat,K2: set_nat,A: set_nat,B: set_nat] :
      ( ( A4
        = ( sup_sup_set_nat @ K2 @ A ) )
     => ( ( sup_sup_set_nat @ A4 @ B )
        = ( sup_sup_set_nat @ K2 @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1715_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4
        = ( sup_sup_nat @ K2 @ A ) )
     => ( ( sup_sup_nat @ A4 @ B )
        = ( sup_sup_nat @ K2 @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1716_boolean__algebra__cancel_Osup1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( sup_sup_int @ K2 @ A ) )
     => ( ( sup_sup_int @ A4 @ B )
        = ( sup_sup_int @ K2 @ ( sup_sup_int @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1717_less__eq__option__def,axiom,
    ( ord_le353528952715127954et_int
    = ( ^ [X4: option_set_int,Y4: option_set_int] :
          ( case_o223999843215110191et_int @ $true
          @ ^ [Z: set_int] : ( case_o223999843215110191et_int @ $false @ ( ord_less_eq_set_int @ Z ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_1718_less__eq__option__def,axiom,
    ( ord_le2406147912482264968on_rat
    = ( ^ [X4: option_rat,Y4: option_rat] :
          ( case_option_o_rat @ $true
          @ ^ [Z: rat] : ( case_option_o_rat @ $false @ ( ord_less_eq_rat @ Z ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_1719_less__eq__option__def,axiom,
    ( ord_le6622620407824499402on_num
    = ( ^ [X4: option_num,Y4: option_num] :
          ( case_option_o_num @ $true
          @ ^ [Z: num] : ( case_option_o_num @ $false @ ( ord_less_eq_num @ Z ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_1720_less__eq__option__def,axiom,
    ( ord_le5914376470875661696on_nat
    = ( ^ [X4: option_nat,Y4: option_nat] :
          ( case_option_o_nat @ $true
          @ ^ [Z: nat] : ( case_option_o_nat @ $false @ ( ord_less_eq_nat @ Z ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_1721_less__eq__option__def,axiom,
    ( ord_le1736525451366464988on_int
    = ( ^ [X4: option_int,Y4: option_int] :
          ( case_option_o_int @ $true
          @ ^ [Z: int] : ( case_option_o_int @ $false @ ( ord_less_eq_int @ Z ) @ Y4 )
          @ X4 ) ) ) ).

% less_eq_option_def
thf(fact_1722_less__option__def,axiom,
    ( ord_less_option_assn
    = ( ^ [X4: option_assn] :
          ( case_option_o_assn @ $false
          @ ^ [Y4: assn] :
              ( case_option_o_assn @ $true
              @ ^ [Z: assn] : ( ord_less_assn @ Z @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_1723_less__option__def,axiom,
    ( ord_less_option_rat
    = ( ^ [X4: option_rat] :
          ( case_option_o_rat @ $false
          @ ^ [Y4: rat] :
              ( case_option_o_rat @ $true
              @ ^ [Z: rat] : ( ord_less_rat @ Z @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_1724_less__option__def,axiom,
    ( ord_less_option_num
    = ( ^ [X4: option_num] :
          ( case_option_o_num @ $false
          @ ^ [Y4: num] :
              ( case_option_o_num @ $true
              @ ^ [Z: num] : ( ord_less_num @ Z @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_1725_less__option__def,axiom,
    ( ord_less_option_nat
    = ( ^ [X4: option_nat] :
          ( case_option_o_nat @ $false
          @ ^ [Y4: nat] :
              ( case_option_o_nat @ $true
              @ ^ [Z: nat] : ( ord_less_nat @ Z @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_1726_less__option__def,axiom,
    ( ord_less_option_int
    = ( ^ [X4: option_int] :
          ( case_option_o_int @ $false
          @ ^ [Y4: int] :
              ( case_option_o_int @ $true
              @ ^ [Z: int] : ( ord_less_int @ Z @ Y4 )
              @ X4 ) ) ) ) ).

% less_option_def
thf(fact_1727_raise__def,axiom,
    ( heap_T2927564422264180874t_unit
    = ( ^ [S5: list_char] :
          ( heap_T6183433275982383450t_unit
          @ ^ [Uu2: heap_e7401611519738050253t_unit] : none_P9117596204409417319it_nat ) ) ) ).

% raise_def
thf(fact_1728_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_1729_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_1730_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_1731_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_1732_accp__subset,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
     => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_1733_accp__subset__induct,axiom,
    ! [D4: product_prod_num_num > $o,R3: product_prod_num_num > product_prod_num_num > $o,X: product_prod_num_num,P2: product_prod_num_num > $o] :
      ( ( ord_le2239182809043710856_num_o @ D4 @ ( accp_P3113834385874906142um_num @ R3 ) )
     => ( ! [X3: product_prod_num_num,Z2: product_prod_num_num] :
            ( ( D4 @ X3 )
           => ( ( R3 @ Z2 @ X3 )
             => ( D4 @ Z2 ) ) )
       => ( ( D4 @ X )
         => ( ! [X3: product_prod_num_num] :
                ( ( D4 @ X3 )
               => ( ! [Z6: product_prod_num_num] :
                      ( ( R3 @ Z6 @ X3 )
                     => ( P2 @ Z6 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1734_accp__subset__induct,axiom,
    ! [D4: product_prod_nat_nat > $o,R3: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R3 ) )
     => ( ! [X3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
            ( ( D4 @ X3 )
           => ( ( R3 @ Z2 @ X3 )
             => ( D4 @ Z2 ) ) )
       => ( ( D4 @ X )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( D4 @ X3 )
               => ( ! [Z6: product_prod_nat_nat] :
                      ( ( R3 @ Z6 @ X3 )
                     => ( P2 @ Z6 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1735_accp__subset__induct,axiom,
    ! [D4: product_prod_int_int > $o,R3: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int,P2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R3 ) )
     => ( ! [X3: product_prod_int_int,Z2: product_prod_int_int] :
            ( ( D4 @ X3 )
           => ( ( R3 @ Z2 @ X3 )
             => ( D4 @ Z2 ) ) )
       => ( ( D4 @ X )
         => ( ! [X3: product_prod_int_int] :
                ( ( D4 @ X3 )
               => ( ! [Z6: product_prod_int_int] :
                      ( ( R3 @ Z6 @ X3 )
                     => ( P2 @ Z6 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1736_accp__subset__induct,axiom,
    ! [D4: produc2732055786443039994et_nat > $o,R3: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat,P2: produc2732055786443039994et_nat > $o] :
      ( ( ord_le1658592502415435381_nat_o @ D4 @ ( accp_P1862375125659990705et_nat @ R3 ) )
     => ( ! [X3: produc2732055786443039994et_nat,Z2: produc2732055786443039994et_nat] :
            ( ( D4 @ X3 )
           => ( ( R3 @ Z2 @ X3 )
             => ( D4 @ Z2 ) ) )
       => ( ( D4 @ X )
         => ( ! [X3: produc2732055786443039994et_nat] :
                ( ( D4 @ X3 )
               => ( ! [Z6: produc2732055786443039994et_nat] :
                      ( ( R3 @ Z6 @ X3 )
                     => ( P2 @ Z6 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1737_accp__subset__induct,axiom,
    ! [D4: nat > $o,R3: nat > nat > $o,X: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R3 ) )
     => ( ! [X3: nat,Z2: nat] :
            ( ( D4 @ X3 )
           => ( ( R3 @ Z2 @ X3 )
             => ( D4 @ Z2 ) ) )
       => ( ( D4 @ X )
         => ( ! [X3: nat] :
                ( ( D4 @ X3 )
               => ( ! [Z6: nat] :
                      ( ( R3 @ Z6 @ X3 )
                     => ( P2 @ Z6 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1738_execute__assert_I2_J,axiom,
    ! [P2: b > $o,X: b,H: heap_e7401611519738050253t_unit] :
      ( ~ ( P2 @ X )
     => ( ( heap_Time_execute_b @ ( heap_Time_assert_b @ P2 @ X ) @ H )
        = none_P6779099274072355161it_nat ) ) ).

% execute_assert(2)
thf(fact_1739_execute__assert_I2_J,axiom,
    ! [P2: a > $o,X: a,H: heap_e7401611519738050253t_unit] :
      ( ~ ( P2 @ X )
     => ( ( heap_Time_execute_a @ ( heap_Time_assert_a @ P2 @ X ) @ H )
        = none_P2651198173097904984it_nat ) ) ).

% execute_assert(2)
thf(fact_1740_mod__h__bot__iff_I7_J,axiom,
    ! [P2: assn,Q2: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( sup_sup_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        | ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(7)
thf(fact_1741_combine__options__def,axiom,
    ( combin724284867043296059it_nat
    = ( ^ [F3: produc7388388658123137530it_nat > produc7388388658123137530it_nat > produc7388388658123137530it_nat,X4: option4065278094766928714it_nat,Y4: option4065278094766928714it_nat] :
          ( case_o5901184372446974929it_nat @ Y4
          @ ^ [Z: produc7388388658123137530it_nat] :
              ( case_o5901184372446974929it_nat @ ( some_P2818173045054083285it_nat @ Z )
              @ ^ [Aa: produc7388388658123137530it_nat] : ( some_P2818173045054083285it_nat @ ( F3 @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% combine_options_def
thf(fact_1742_combine__options__def,axiom,
    ( combin5819755802923621690it_nat
    = ( ^ [F3: produc3260487557148687353it_nat > produc3260487557148687353it_nat > produc3260487557148687353it_nat,X4: option3562590408128118217it_nat,Y4: option3562590408128118217it_nat] :
          ( case_o6061097939050036047it_nat @ Y4
          @ ^ [Z: produc3260487557148687353it_nat] :
              ( case_o6061097939050036047it_nat @ ( some_P7913643980934408916it_nat @ Z )
              @ ^ [Aa: produc3260487557148687353it_nat] : ( some_P7913643980934408916it_nat @ ( F3 @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% combine_options_def
thf(fact_1743_combine__options__def,axiom,
    ( combin4318129983670048329it_nat
    = ( ^ [F3: produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat,X4: option8956607266484857688it_nat,Y4: option8956607266484857688it_nat] :
          ( case_o2963978774867076333it_nat @ Y4
          @ ^ [Z: produc8664842809031399944it_nat] :
              ( case_o2963978774867076333it_nat @ ( some_P1914260805536162275it_nat @ Z )
              @ ^ [Aa: produc8664842809031399944it_nat] : ( some_P1914260805536162275it_nat @ ( F3 @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% combine_options_def
thf(fact_1744_combine__options__def,axiom,
    ( combine_options_num
    = ( ^ [F3: num > num > num,X4: option_num,Y4: option_num] :
          ( case_o6005452278849405969um_num @ Y4
          @ ^ [Z: num] :
              ( case_o6005452278849405969um_num @ ( some_num @ Z )
              @ ^ [Aa: num] : ( some_num @ ( F3 @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% combine_options_def
thf(fact_1745_reflcl__set__eq,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su362511073950362882_nat_o
        @ ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R2 )
        @ ^ [Y6: product_prod_nat_nat,Z4: product_prod_nat_nat] : Y6 = Z4 )
      = ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ ( sup_su718114333110466843at_nat @ R2 @ id_Pro2258643101195443293at_nat ) ) ) ) ).

% reflcl_set_eq
thf(fact_1746_reflcl__set__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R2 )
        @ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
      = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ ( sup_su6327502436637775413at_nat @ R2 @ id_nat2 ) ) ) ) ).

% reflcl_set_eq
thf(fact_1747_reflcl__set__eq,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( ( sup_sup_int_int_o
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R2 )
        @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
      = ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ ( sup_su6024340866399070445nt_int @ R2 @ id_int2 ) ) ) ) ).

% reflcl_set_eq
thf(fact_1748_reflcl__set__eq,axiom,
    ! [R2: set_Pr8551490117392284871at_nat] :
      ( ( sup_su2200014604384089602_nat_o
        @ ^ [X4: multis2468970476368604999at_nat,Y4: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X4 @ Y4 ) @ R2 )
        @ ^ [Y6: multis2468970476368604999at_nat,Z4: multis2468970476368604999at_nat] : Y6 = Z4 )
      = ( ^ [X4: multis2468970476368604999at_nat,Y4: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X4 @ Y4 ) @ ( sup_su3035147773818789531at_nat @ R2 @ id_mul2649389997224486051at_nat ) ) ) ) ).

% reflcl_set_eq
thf(fact_1749_reflcl__set__eq,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su7519161239522478338_nat_o
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ R2 )
        @ ^ [Y6: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] : Y6 = Z4 )
      = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ ( sup_su5525570899277871387at_nat @ R2 @ id_set1796276052702428605at_nat ) ) ) ) ).

% reflcl_set_eq
thf(fact_1750_in__measure,axiom,
    ! [X: int,Y: int,F: int > nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_1751_wait__bind__decon,axiom,
    ! [P2: assn,M: heap_Time_Heap_a,Q2: a > assn,N2: nat] :
      ( ( hoare_hoare_triple_a @ P2 @ M @ Q2 )
     => ( hoare_hoare_triple_a @ P2
        @ ( heap_T757603679106148408unit_a @ ( heap_Time_wait @ N2 )
          @ ^ [Uu2: product_unit] : M )
        @ Q2 ) ) ).

% wait_bind_decon
thf(fact_1752_wait__bind__decon,axiom,
    ! [P2: assn,M: heap_Time_Heap_b,Q2: b > assn,N2: nat] :
      ( ( hoare_hoare_triple_b @ P2 @ M @ Q2 )
     => ( hoare_hoare_triple_b @ P2
        @ ( heap_T757603679106148409unit_b @ ( heap_Time_wait @ N2 )
          @ ^ [Uu2: product_unit] : M )
        @ Q2 ) ) ).

% wait_bind_decon
thf(fact_1753_wait__bind__decon,axiom,
    ! [P2: assn,M: heap_T5738788834812785303t_unit,Q2: product_unit > assn,N2: nat] :
      ( ( hoare_8945653483474564448t_unit @ P2 @ M @ Q2 )
     => ( hoare_8945653483474564448t_unit @ P2
        @ ( heap_T2633723481742716231t_unit @ ( heap_Time_wait @ N2 )
          @ ^ [Uu2: product_unit] : M )
        @ Q2 ) ) ).

% wait_bind_decon
thf(fact_1754_empty__Collect__eq,axiom,
    ! [P2: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P2 ) )
      = ( ! [X4: list_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1755_empty__Collect__eq,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o] :
      ( ( bot_bo1488462491386950373nt_int
        = ( collec5210948495886036740nt_int @ P2 ) )
      = ( ! [X4: set_Pr958786334691620121nt_int] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1756_empty__Collect__eq,axiom,
    ! [P2: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P2 ) )
      = ( ! [X4: set_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1757_empty__Collect__eq,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P2 ) )
      = ( ! [X4: product_prod_nat_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1758_empty__Collect__eq,axiom,
    ! [P2: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P2 ) )
      = ( ! [X4: $o] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1759_empty__Collect__eq,axiom,
    ! [P2: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P2 ) )
      = ( ! [X4: nat] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1760_empty__Collect__eq,axiom,
    ! [P2: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P2 ) )
      = ( ! [X4: int] :
            ~ ( P2 @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_1761_Collect__empty__eq,axiom,
    ! [P2: list_nat > $o] :
      ( ( ( collect_list_nat @ P2 )
        = bot_bot_set_list_nat )
      = ( ! [X4: list_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1762_Collect__empty__eq,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o] :
      ( ( ( collec5210948495886036740nt_int @ P2 )
        = bot_bo1488462491386950373nt_int )
      = ( ! [X4: set_Pr958786334691620121nt_int] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1763_Collect__empty__eq,axiom,
    ! [P2: set_nat > $o] :
      ( ( ( collect_set_nat @ P2 )
        = bot_bot_set_set_nat )
      = ( ! [X4: set_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1764_Collect__empty__eq,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P2 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1765_Collect__empty__eq,axiom,
    ! [P2: $o > $o] :
      ( ( ( collect_o @ P2 )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1766_Collect__empty__eq,axiom,
    ! [P2: nat > $o] :
      ( ( ( collect_nat @ P2 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1767_Collect__empty__eq,axiom,
    ! [P2: int > $o] :
      ( ( ( collect_int @ P2 )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ~ ( P2 @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_1768_all__not__in__conv,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ( ! [X4: set_Pr958786334691620121nt_int] :
            ~ ( member2340774599025711042nt_int @ X4 @ A4 ) )
      = ( A4 = bot_bo1488462491386950373nt_int ) ) ).

% all_not_in_conv
thf(fact_1769_all__not__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ( ! [X4: set_nat] :
            ~ ( member_set_nat @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_1770_all__not__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ! [X4: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
thf(fact_1771_all__not__in__conv,axiom,
    ! [A4: set_o] :
      ( ( ! [X4: $o] :
            ~ ( member_o @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_1772_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X4: nat] :
            ~ ( member_nat @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_1773_all__not__in__conv,axiom,
    ! [A4: set_int] :
      ( ( ! [X4: int] :
            ~ ( member_int @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_1774_empty__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int] :
      ~ ( member2340774599025711042nt_int @ C2 @ bot_bo1488462491386950373nt_int ) ).

% empty_iff
thf(fact_1775_empty__iff,axiom,
    ! [C2: set_nat] :
      ~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_1776_empty__iff,axiom,
    ! [C2: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ).

% empty_iff
thf(fact_1777_empty__iff,axiom,
    ! [C2: $o] :
      ~ ( member_o @ C2 @ bot_bot_set_o ) ).

% empty_iff
thf(fact_1778_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_1779_empty__iff,axiom,
    ! [C2: int] :
      ~ ( member_int @ C2 @ bot_bot_set_int ) ).

% empty_iff
thf(fact_1780_subset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
thf(fact_1781_subset__empty,axiom,
    ! [A4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ bot_bot_set_o )
      = ( A4 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1782_subset__empty,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_1783_subset__empty,axiom,
    ! [A4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ bot_bot_set_int )
      = ( A4 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_1784_empty__subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A4 ) ).

% empty_subsetI
thf(fact_1785_empty__subsetI,axiom,
    ! [A4: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A4 ) ).

% empty_subsetI
thf(fact_1786_empty__subsetI,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).

% empty_subsetI
thf(fact_1787_empty__subsetI,axiom,
    ! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).

% empty_subsetI
thf(fact_1788_sup__bot__left,axiom,
    ! [X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_1789_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_1790_sup__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_1791_sup__bot__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X )
      = X ) ).

% sup_bot_left
thf(fact_1792_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_1793_sup__bot__left,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X )
      = X ) ).

% sup_bot_left
thf(fact_1794_sup__bot__right,axiom,
    ! [X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ bot_bo8422036546324065075at_nat )
      = X ) ).

% sup_bot_right
thf(fact_1795_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
      = X ) ).

% sup_bot_right
thf(fact_1796_sup__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% sup_bot_right
thf(fact_1797_sup__bot__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% sup_bot_right
thf(fact_1798_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_1799_sup__bot__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% sup_bot_right
thf(fact_1800_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_1801_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_1802_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_1803_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_1804_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_1805_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_1806_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_1807_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_1808_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_1809_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_1810_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_1811_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_1812_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_1813_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_1814_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_1815_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_1816_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_1817_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_1818_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1819_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1820_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1821_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1822_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1823_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1824_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_1825_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_1826_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_1827_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_1828_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_1829_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_1830_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A @ bot_bo8422036546324065075at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1831_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ bot_bo228742789529271731at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1832_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A @ bot_bo2099793752762293965at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1833_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1834_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1835_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1836_Un__empty,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( ( sup_su3035147773818789531at_nat @ A4 @ B5 )
        = bot_bo8422036546324065075at_nat )
      = ( ( A4 = bot_bo8422036546324065075at_nat )
        & ( B5 = bot_bo8422036546324065075at_nat ) ) ) ).

% Un_empty
thf(fact_1837_Un__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A4 @ B5 )
        = bot_bo228742789529271731at_nat )
      = ( ( A4 = bot_bo228742789529271731at_nat )
        & ( B5 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
thf(fact_1838_Un__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A4 = bot_bo2099793752762293965at_nat )
        & ( B5 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
thf(fact_1839_Un__empty,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ( sup_sup_set_o @ A4 @ B5 )
        = bot_bot_set_o )
      = ( ( A4 = bot_bot_set_o )
        & ( B5 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_1840_Un__empty,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( sup_sup_set_nat @ A4 @ B5 )
        = bot_bot_set_nat )
      = ( ( A4 = bot_bot_set_nat )
        & ( B5 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_1841_Un__empty,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( sup_sup_set_int @ A4 @ B5 )
        = bot_bot_set_int )
      = ( ( A4 = bot_bot_set_int )
        & ( B5 = bot_bot_set_int ) ) ) ).

% Un_empty
thf(fact_1842_pair__in__Id__conv,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ id_Pro2258643101195443293at_nat )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_1843_pair__in__Id__conv,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ A @ B ) @ id_mul2649389997224486051at_nat )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_1844_pair__in__Id__conv,axiom,
    ! [A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ id_nat2 )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_1845_pair__in__Id__conv,axiom,
    ! [A: int,B: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ id_int2 )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_1846_IdI,axiom,
    ! [A: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ id_Pro2258643101195443293at_nat ) ).

% IdI
thf(fact_1847_IdI,axiom,
    ! [A: multis2468970476368604999at_nat] : ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ A @ A ) @ id_mul2649389997224486051at_nat ) ).

% IdI
thf(fact_1848_IdI,axiom,
    ! [A: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ id_nat2 ) ).

% IdI
thf(fact_1849_IdI,axiom,
    ! [A: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ id_int2 ) ).

% IdI
thf(fact_1850_combine__options__simps_I3_J,axiom,
    ! [F: produc7388388658123137530it_nat > produc7388388658123137530it_nat > produc7388388658123137530it_nat,A: produc7388388658123137530it_nat,B: produc7388388658123137530it_nat] :
      ( ( combin724284867043296059it_nat @ F @ ( some_P2818173045054083285it_nat @ A ) @ ( some_P2818173045054083285it_nat @ B ) )
      = ( some_P2818173045054083285it_nat @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_1851_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_1852_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_1853_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_1854_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_1855_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_1856_bijective__Id,axiom,
    biject3503750217840948301at_nat @ id_Pro2258643101195443293at_nat ).

% bijective_Id
thf(fact_1857_bijective__Id,axiom,
    biject5714339216877808333at_nat @ id_mul2649389997224486051at_nat ).

% bijective_Id
thf(fact_1858_bijective__Id,axiom,
    bijective_nat_nat @ id_nat2 ).

% bijective_Id
thf(fact_1859_memb__imp__not__empty,axiom,
    ! [X: set_Pr958786334691620121nt_int,S: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ X @ S )
     => ( S != bot_bo1488462491386950373nt_int ) ) ).

% memb_imp_not_empty
thf(fact_1860_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_1861_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_1862_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_1863_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_1864_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_1865_set__notEmptyE,axiom,
    ! [S: set_se6260736226359567993nt_int] :
      ( ( S != bot_bo1488462491386950373nt_int )
     => ~ ! [X3: set_Pr958786334691620121nt_int] :
            ~ ( member2340774599025711042nt_int @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_1866_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_1867_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_1868_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X3: $o] :
            ~ ( member_o @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_1869_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_1870_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_1871_ex__in__conv,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ( ? [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ A4 ) )
      = ( A4 != bot_bo1488462491386950373nt_int ) ) ).

% ex_in_conv
thf(fact_1872_ex__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ( ? [X4: set_nat] : ( member_set_nat @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_1873_ex__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ? [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = ( A4 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
thf(fact_1874_ex__in__conv,axiom,
    ! [A4: set_o] :
      ( ( ? [X4: $o] : ( member_o @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_1875_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_1876_ex__in__conv,axiom,
    ! [A4: set_int] :
      ( ( ? [X4: int] : ( member_int @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_1877_equals0I,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ! [Y3: set_Pr958786334691620121nt_int] :
          ~ ( member2340774599025711042nt_int @ Y3 @ A4 )
     => ( A4 = bot_bo1488462491386950373nt_int ) ) ).

% equals0I
thf(fact_1878_equals0I,axiom,
    ! [A4: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_1879_equals0I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ Y3 @ A4 )
     => ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
thf(fact_1880_equals0I,axiom,
    ! [A4: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A4 )
     => ( A4 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_1881_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_1882_equals0I,axiom,
    ! [A4: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int @ Y3 @ A4 )
     => ( A4 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_1883_equals0D,axiom,
    ! [A4: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( A4 = bot_bo1488462491386950373nt_int )
     => ~ ( member2340774599025711042nt_int @ A @ A4 ) ) ).

% equals0D
thf(fact_1884_equals0D,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( A4 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_1885_equals0D,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_1886_equals0D,axiom,
    ! [A4: set_o,A: $o] :
      ( ( A4 = bot_bot_set_o )
     => ~ ( member_o @ A @ A4 ) ) ).

% equals0D
thf(fact_1887_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_1888_equals0D,axiom,
    ! [A4: set_int,A: int] :
      ( ( A4 = bot_bot_set_int )
     => ~ ( member_int @ A @ A4 ) ) ).

% equals0D
thf(fact_1889_emptyE,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ~ ( member2340774599025711042nt_int @ A @ bot_bo1488462491386950373nt_int ) ).

% emptyE
thf(fact_1890_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_1891_emptyE,axiom,
    ! [A: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% emptyE
thf(fact_1892_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_1893_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_1894_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_1895_Set_Oempty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X4: list_nat] : $false ) ) ).

% Set.empty_def
thf(fact_1896_Set_Oempty__def,axiom,
    ( bot_bo1488462491386950373nt_int
    = ( collec5210948495886036740nt_int
      @ ^ [X4: set_Pr958786334691620121nt_int] : $false ) ) ).

% Set.empty_def
thf(fact_1897_Set_Oempty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X4: set_nat] : $false ) ) ).

% Set.empty_def
thf(fact_1898_Set_Oempty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X4: product_prod_nat_nat] : $false ) ) ).

% Set.empty_def
thf(fact_1899_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X4: $o] : $false ) ) ).

% Set.empty_def
thf(fact_1900_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $false ) ) ).

% Set.empty_def
thf(fact_1901_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X4: int] : $false ) ) ).

% Set.empty_def
thf(fact_1902_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% bot.extremum
thf(fact_1903_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_1904_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_1905_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_1906_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1907_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
thf(fact_1908_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_1909_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_1910_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_1911_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1912_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1913_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_1914_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_1915_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_1916_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1917_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_1918_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_1919_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1920_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1921_bot_Oextremum__strict,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ bot_bot_assn ) ).

% bot.extremum_strict
thf(fact_1922_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1923_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_1924_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_1925_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_1926_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_1927_bot_Onot__eq__extremum,axiom,
    ! [A: assn] :
      ( ( A != bot_bot_assn )
      = ( ord_less_assn @ bot_bot_assn @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1928_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1929_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_1930_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_1931_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_1932_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_1933_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_1934_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_1935_IdE,axiom,
    ! [P3: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ P3 @ id_Pro2258643101195443293at_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ( P3
           != ( produc6161850002892822231at_nat @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_1936_IdE,axiom,
    ! [P3: produc4166570645942440679at_nat] :
      ( ( member6689249552917799696at_nat @ P3 @ id_mul2649389997224486051at_nat )
     => ~ ! [X3: multis2468970476368604999at_nat] :
            ( P3
           != ( produc4348348721325984599at_nat @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_1937_IdE,axiom,
    ! [P3: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ P3 @ id_nat2 )
     => ~ ! [X3: nat] :
            ( P3
           != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_1938_IdE,axiom,
    ! [P3: product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ P3 @ id_int2 )
     => ~ ! [X3: int] :
            ( P3
           != ( product_Pair_int_int @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_1939_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ id_Pro2258643101195443293at_nat )
     => ( A = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
thf(fact_1940_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ A @ B ) @ id_mul2649389997224486051at_nat )
     => ( A = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
thf(fact_1941_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ id_nat2 )
     => ( A = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
thf(fact_1942_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: int,B: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ id_int2 )
     => ( A = B ) ) ).

% BNF_Greatest_Fixpoint.IdD
thf(fact_1943_subset__emptyI,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ~ ( member2340774599025711042nt_int @ X3 @ A4 )
     => ( ord_le483042692224249369nt_int @ A4 @ bot_bo1488462491386950373nt_int ) ) ).

% subset_emptyI
thf(fact_1944_subset__emptyI,axiom,
    ! [A4: set_set_nat] :
      ( ! [X3: set_nat] :
          ~ ( member_set_nat @ X3 @ A4 )
     => ( ord_le6893508408891458716et_nat @ A4 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_1945_subset__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ X3 @ A4 )
     => ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).

% subset_emptyI
thf(fact_1946_subset__emptyI,axiom,
    ! [A4: set_o] :
      ( ! [X3: $o] :
          ~ ( member_o @ X3 @ A4 )
     => ( ord_less_eq_set_o @ A4 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_1947_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A4 )
     => ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1948_subset__emptyI,axiom,
    ! [A4: set_int] :
      ( ! [X3: int] :
          ~ ( member_int @ X3 @ A4 )
     => ( ord_less_eq_set_int @ A4 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_1949_Un__empty__left,axiom,
    ! [B5: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ bot_bo8422036546324065075at_nat @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_1950_Un__empty__left,axiom,
    ! [B5: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_1951_Un__empty__left,axiom,
    ! [B5: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_1952_Un__empty__left,axiom,
    ! [B5: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_1953_Un__empty__left,axiom,
    ! [B5: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_1954_Un__empty__left,axiom,
    ! [B5: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_1955_Un__empty__right,axiom,
    ! [A4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A4 @ bot_bo8422036546324065075at_nat )
      = A4 ) ).

% Un_empty_right
thf(fact_1956_Un__empty__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ bot_bo228742789529271731at_nat )
      = A4 ) ).

% Un_empty_right
thf(fact_1957_Un__empty__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = A4 ) ).

% Un_empty_right
thf(fact_1958_Un__empty__right,axiom,
    ! [A4: set_o] :
      ( ( sup_sup_set_o @ A4 @ bot_bot_set_o )
      = A4 ) ).

% Un_empty_right
thf(fact_1959_Un__empty__right,axiom,
    ! [A4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Un_empty_right
thf(fact_1960_Un__empty__right,axiom,
    ! [A4: set_int] :
      ( ( sup_sup_set_int @ A4 @ bot_bot_set_int )
      = A4 ) ).

% Un_empty_right
thf(fact_1961_not__psubset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_1962_not__psubset__empty,axiom,
    ! [A4: set_o] :
      ~ ( ord_less_set_o @ A4 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_1963_not__psubset__empty,axiom,
    ! [A4: set_nat] :
      ~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_1964_not__psubset__empty,axiom,
    ! [A4: set_int] :
      ~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_1965_mod__h__bot__indep,axiom,
    ! [P2: assn,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H5 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_execute__assert_I1_J,axiom,
    ! [P2: product_unit > $o,X: product_unit,H: heap_e7401611519738050253t_unit] :
      ( ( P2 @ X )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T4208721593536448476t_unit @ P2 @ X ) @ H )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ).

% execute_assert(1)
thf(fact_1974_execute__assert_I1_J,axiom,
    ! [P2: b > $o,X: b,H: heap_e7401611519738050253t_unit] :
      ( ( P2 @ X )
     => ( ( heap_Time_execute_b @ ( heap_Time_assert_b @ P2 @ X ) @ H )
        = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ X @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ).

% execute_assert(1)
thf(fact_1975_execute__assert_I1_J,axiom,
    ! [P2: a > $o,X: a,H: heap_e7401611519738050253t_unit] :
      ( ( P2 @ X )
     => ( ( heap_Time_execute_a @ ( heap_Time_assert_a @ P2 @ X ) @ H )
        = ( some_P7913643980934408916it_nat @ ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ).

% execute_assert(1)
thf(fact_1976_the__dflt__None__empty,axiom,
    ( ( dflt_N6592383573632408824at_nat @ bot_bo2099793752762293965at_nat )
    = none_s625347054029921090at_nat ) ).

% the_dflt_None_empty
thf(fact_1977_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_o @ bot_bot_set_o )
    = none_set_o ) ).

% the_dflt_None_empty
thf(fact_1978_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_nat @ bot_bot_set_nat )
    = none_set_nat ) ).

% the_dflt_None_empty
thf(fact_1979_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_int @ bot_bot_set_int )
    = none_set_int ) ).

% the_dflt_None_empty
thf(fact_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_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_1988_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_1989_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_1990_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_1991_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_1992_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_1993_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_1994_false__rule,axiom,
    ! [C2: heap_Time_Heap_a,Q2: a > assn] : ( hoare_hoare_triple_a @ bot_bot_assn @ C2 @ Q2 ) ).

% false_rule
thf(fact_1995_false__rule,axiom,
    ! [C2: heap_Time_Heap_b,Q2: b > assn] : ( hoare_hoare_triple_b @ bot_bot_assn @ C2 @ Q2 ) ).

% false_rule
thf(fact_1996_false__rule,axiom,
    ! [C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn] : ( hoare_8945653483474564448t_unit @ bot_bot_assn @ C2 @ Q2 ) ).

% false_rule
thf(fact_1997_bijective__Empty,axiom,
    bijective_nat_nat @ bot_bo2099793752762293965at_nat ).

% bijective_Empty
thf(fact_1998_set__to__map__empty,axiom,
    ( ( set_to_map_nat_nat @ bot_bo2099793752762293965at_nat )
    = ( ^ [X4: nat] : none_nat ) ) ).

% set_to_map_empty
thf(fact_1999_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_2000_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_2001_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_2002_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_2003_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_2004_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_2005_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_2006_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_2007_bot__empty__eq2,axiom,
    ( bot_bo5216386038637673661_nat_o
    = ( ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ bot_bo8983147464310796868it_nat ) ) ) ).

% bot_empty_eq2
thf(fact_2008_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_2009_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_2010_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_2011_bot__set__def,axiom,
    ( bot_bo1488462491386950373nt_int
    = ( collec5210948495886036740nt_int @ bot_bo2686080419298087992_int_o ) ) ).

% bot_set_def
thf(fact_2012_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_2013_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_2014_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_2015_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_2016_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_2017_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
thf(fact_2018_mod__false,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ~ ( rep_assn @ bot_bot_assn @ H ) ).

% mod_false
thf(fact_2019_less__by__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).

% less_by_empty
thf(fact_2020_raise__rule,axiom,
    ! [S2: list_char,Q2: a > assn] : ( hoare_hoare_triple_a @ bot_bot_assn @ ( heap_Time_raise_a @ S2 ) @ Q2 ) ).

% raise_rule
thf(fact_2021_raise__rule,axiom,
    ! [S2: list_char,Q2: b > assn] : ( hoare_hoare_triple_b @ bot_bot_assn @ ( heap_Time_raise_b @ S2 ) @ Q2 ) ).

% raise_rule
thf(fact_2022_raise__rule,axiom,
    ! [S2: list_char,Q2: product_unit > assn] : ( hoare_8945653483474564448t_unit @ bot_bot_assn @ ( heap_T2927564422264180874t_unit @ S2 ) @ Q2 ) ).

% raise_rule
thf(fact_2023_raise__iff,axiom,
    ! [P2: assn,S2: list_char,Q2: a > assn] :
      ( ( hoare_hoare_triple_a @ P2 @ ( heap_Time_raise_a @ S2 ) @ Q2 )
      = ( P2 = bot_bot_assn ) ) ).

% raise_iff
thf(fact_2024_raise__iff,axiom,
    ! [P2: assn,S2: list_char,Q2: b > assn] :
      ( ( hoare_hoare_triple_b @ P2 @ ( heap_Time_raise_b @ S2 ) @ Q2 )
      = ( P2 = bot_bot_assn ) ) ).

% raise_iff
thf(fact_2025_raise__iff,axiom,
    ! [P2: assn,S2: list_char,Q2: product_unit > assn] :
      ( ( hoare_8945653483474564448t_unit @ P2 @ ( heap_T2927564422264180874t_unit @ S2 ) @ Q2 )
      = ( P2 = bot_bot_assn ) ) ).

% raise_iff
thf(fact_2026_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_2027_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_2028_dflt__None__set__def,axiom,
    ( dflt_N6592383573632408824at_nat
    = ( ^ [S4: set_Pr1261947904930325089at_nat] : ( if_opt7704869406773131885at_nat @ ( S4 = bot_bo2099793752762293965at_nat ) @ none_s625347054029921090at_nat @ ( some_s147305329494351046at_nat @ S4 ) ) ) ) ).

% dflt_None_set_def
thf(fact_2029_dflt__None__set__def,axiom,
    ( dflt_None_set_o
    = ( ^ [S4: set_o] : ( if_option_set_o @ ( S4 = bot_bot_set_o ) @ none_set_o @ ( some_set_o @ S4 ) ) ) ) ).

% dflt_None_set_def
thf(fact_2030_dflt__None__set__def,axiom,
    ( dflt_None_set_nat
    = ( ^ [S4: set_nat] : ( if_option_set_nat @ ( S4 = bot_bot_set_nat ) @ none_set_nat @ ( some_set_nat @ S4 ) ) ) ) ).

% dflt_None_set_def
thf(fact_2031_dflt__None__set__def,axiom,
    ( dflt_None_set_int
    = ( ^ [S4: set_int] : ( if_option_set_int @ ( S4 = bot_bot_set_int ) @ none_set_int @ ( some_set_int @ S4 ) ) ) ) ).

% dflt_None_set_def
thf(fact_2032_rel__of__empty,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( rel_of_nat_nat
        @ ^ [X4: nat] : none_nat
        @ P2 )
      = bot_bo2099793752762293965at_nat ) ).

% rel_of_empty
thf(fact_2033_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A5: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A5 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_2034_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2035_discrete,axiom,
    ( ord_less_int
    = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2036_tap__def,axiom,
    ( heap_Time_tap_b
    = ( ^ [F3: heap_e7401611519738050253t_unit > b] :
          ( heap_Time_Heap_b2
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ ( F3 @ H6 ) @ ( produc584006145561248582it_nat @ H6 @ one_one_nat ) ) ) ) ) ) ).

% tap_def
thf(fact_2037_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_2038_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_2039_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_2040_execute__tap,axiom,
    ! [F: heap_e7401611519738050253t_unit > b,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_execute_b @ ( heap_Time_tap_b @ F ) @ H )
      = ( some_P2818173045054083285it_nat @ ( produc4082563078715348724it_nat @ ( F @ H ) @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ).

% execute_tap
thf(fact_2041_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_2042_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_2043_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_2044_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_2045_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_2046_less__add__one,axiom,
    ! [A: code_integer] : ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_2047_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_2048_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2049_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2050_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_2051_return__wp__rule,axiom,
    ! [Q2: a > assn,X: a] : ( hoare_hoare_triple_a @ ( Q2 @ X ) @ ( heap_Time_return_a @ X ) @ Q2 ) ).

% return_wp_rule
thf(fact_2052_return__wp__rule,axiom,
    ! [Q2: b > assn,X: b] : ( hoare_hoare_triple_b @ ( Q2 @ X ) @ ( heap_Time_return_b @ X ) @ Q2 ) ).

% return_wp_rule
thf(fact_2053_return__wp__rule,axiom,
    ! [Q2: product_unit > assn,X: product_unit] : ( hoare_8945653483474564448t_unit @ ( Q2 @ X ) @ ( heap_T7507251653302230130t_unit @ X ) @ Q2 ) ).

% return_wp_rule
thf(fact_2054_wait__rule,axiom,
    ! [N2: nat] :
      ( hoare_8945653483474564448t_unit @ one_one_assn @ ( heap_Time_wait @ N2 )
      @ ^ [Uu2: product_unit] : one_one_assn ) ).

% wait_rule
thf(fact_2055_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_2056_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_2057_bounded__Max__nat,axiom,
    ! [P2: nat > $o,X: nat,M6: nat] :
      ( ( P2 @ X )
     => ( ! [X3: nat] :
            ( ( P2 @ X3 )
           => ( ord_less_eq_nat @ X3 @ M6 ) )
       => ~ ! [M5: nat] :
              ( ( P2 @ M5 )
             => ~ ! [X6: nat] :
                    ( ( P2 @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_2058_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_2059_return__def,axiom,
    ( heap_Time_return_b
    = ( ^ [X4: b] :
          ( heap_Time_heap_b
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc4082563078715348724it_nat @ X4 @ ( produc584006145561248582it_nat @ H6 @ one_one_nat ) ) ) ) ) ).

% return_def
thf(fact_2060_return__def,axiom,
    ( heap_Time_return_a
    = ( ^ [X4: a] :
          ( heap_Time_heap_a
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc9178034014595674355it_nat @ X4 @ ( produc584006145561248582it_nat @ H6 @ one_one_nat ) ) ) ) ) ).

% return_def
thf(fact_2061_execute__lookup,axiom,
    ! [R2: ref_Product_unit,H: heap_e7401611519738050253t_unit] :
      ( ( heap_T875086893843062177t_unit @ ( ref_lo7930775386976318366t_unit @ R2 ) @ H )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ ( ref_get_Product_unit @ H @ R2 ) @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ).

% execute_lookup
thf(fact_2062_execute__return,axiom,
    ! [X: product_unit] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T7507251653302230130t_unit @ X ) )
      = ( comp_P3118722334806803912t_unit @ some_P1914260805536162275it_nat
        @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H6 @ one_one_nat ) ) ) ) ).

% execute_return
thf(fact_2063_execute__return,axiom,
    ! [X: b] :
      ( ( heap_Time_execute_b @ ( heap_Time_return_b @ X ) )
      = ( comp_P6401406225962756324t_unit @ some_P2818173045054083285it_nat
        @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc4082563078715348724it_nat @ X @ ( produc584006145561248582it_nat @ H6 @ one_one_nat ) ) ) ) ).

% execute_return
thf(fact_2064_execute__return,axiom,
    ! [X: a] :
      ( ( heap_Time_execute_a @ ( heap_Time_return_a @ X ) )
      = ( comp_P8552900652980440422t_unit @ some_P7913643980934408916it_nat
        @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H6 @ one_one_nat ) ) ) ) ).

% execute_return
thf(fact_2065_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_2066_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_2067_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_2068_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_2069_le__numeral__extra_I4_J,axiom,
    ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).

% le_numeral_extra(4)
thf(fact_2070_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_2071_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2072_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2073_map__to__set__empty,axiom,
    ( ( map_to_set_nat_nat
      @ ^ [X4: nat] : none_nat )
    = bot_bo2099793752762293965at_nat ) ).

% map_to_set_empty
thf(fact_2074_type__copy__map__cong0,axiom,
    ! [M6: code_integer > code_integer,G: code_integer > code_integer,X: code_integer,N7: code_integer > code_integer,H: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M6 @ ( G @ X ) )
        = ( N7 @ ( H @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M6 ) @ G @ X )
        = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ N7 ) @ H @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_2075_type__copy__map__cong0,axiom,
    ! [M6: code_integer > code_integer,G: code_integer > code_integer,X: code_integer,N7: ( code_integer > code_integer ) > code_integer,H: code_integer > code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M6 @ ( G @ X ) )
        = ( N7 @ ( H @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M6 ) @ G @ X )
        = ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ N7 ) @ H @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_2076_type__copy__map__cong0,axiom,
    ! [M6: code_integer > code_integer > code_integer,G: code_integer > code_integer,X: code_integer,N7: code_integer > code_integer > code_integer,H: code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M6 @ ( G @ X ) )
        = ( N7 @ ( H @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M6 ) @ G @ X )
        = ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ N7 ) @ H @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_2077_type__copy__map__cong0,axiom,
    ! [M6: code_integer > code_integer > code_integer,G: code_integer > code_integer,X: code_integer,N7: ( code_integer > code_integer ) > code_integer > code_integer,H: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M6 @ ( G @ X ) )
        = ( N7 @ ( H @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M6 ) @ G @ X )
        = ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ F @ N7 ) @ H @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_2078_type__copy__map__cong0,axiom,
    ! [M6: ( code_integer > code_integer ) > code_integer,G: code_integer > code_integer > code_integer,X: code_integer,N7: code_integer > code_integer,H: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M6 @ ( G @ X ) )
        = ( N7 @ ( H @ X ) ) )
     => ( ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ M6 ) @ G @ X )
        = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ N7 ) @ H @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_2079_type__copy__map__cong0,axiom,
    ! [M6: ( code_integer > code_integer ) > code_integer > code_integer,G: code_integer > code_integer > code_integer,X: code_integer,N7: code_integer > code_integer > code_integer,H: code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M6 @ ( G @ X ) )
        = ( N7 @ ( H @ X ) ) )
     => ( ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ F @ M6 ) @ G @ X )
        = ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ N7 ) @ H @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_2080_type__copy__map__cong0,axiom,
    ! [M6: int > int,G: int > int,X: int,N7: int > int,H: int > int,F: int > nat] :
      ( ( ( M6 @ ( G @ X ) )
        = ( N7 @ ( H @ X ) ) )
     => ( ( comp_int_nat_int @ ( comp_int_nat_int @ F @ M6 ) @ G @ X )
        = ( comp_int_nat_int @ ( comp_int_nat_int @ F @ N7 ) @ H @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_2081_comp__cong__left,axiom,
    ! [X: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer] :
      ( ( X = Y )
     => ( ( comp_C1593894019821074884nteger @ X @ F )
        = ( comp_C1593894019821074884nteger @ Y @ F ) ) ) ).

% comp_cong_left
thf(fact_2082_comp__cong__left,axiom,
    ! [X: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer] :
      ( ( X = Y )
     => ( ( comp_C8797469213163452608nteger @ X @ F )
        = ( comp_C8797469213163452608nteger @ Y @ F ) ) ) ).

% comp_cong_left
thf(fact_2083_comp__cong__left,axiom,
    ! [X: int > nat,Y: int > nat,F: int > int] :
      ( ( X = Y )
     => ( ( comp_int_nat_int @ X @ F )
        = ( comp_int_nat_int @ Y @ F ) ) ) ).

% comp_cong_left
thf(fact_2084_comp__cong__right,axiom,
    ! [X: code_integer > code_integer,Y: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( X = Y )
     => ( ( comp_C1593894019821074884nteger @ F @ X )
        = ( comp_C1593894019821074884nteger @ F @ Y ) ) ) ).

% comp_cong_right
thf(fact_2085_comp__cong__right,axiom,
    ! [X: code_integer > code_integer > code_integer,Y: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( X = Y )
     => ( ( comp_C8797469213163452608nteger @ F @ X )
        = ( comp_C8797469213163452608nteger @ F @ Y ) ) ) ).

% comp_cong_right
thf(fact_2086_comp__cong__right,axiom,
    ! [X: int > int,Y: int > int,F: int > nat] :
      ( ( X = Y )
     => ( ( comp_int_nat_int @ F @ X )
        = ( comp_int_nat_int @ F @ Y ) ) ) ).

% comp_cong_right
thf(fact_2087_fun__comp__eq__conv,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C1593894019821074884nteger @ F @ G )
        = Fg )
      = ( ! [X4: code_integer] :
            ( ( F @ ( G @ X4 ) )
            = ( Fg @ X4 ) ) ) ) ).

% fun_comp_eq_conv
thf(fact_2088_fun__comp__eq__conv,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C8797469213163452608nteger @ F @ G )
        = Fg )
      = ( ! [X4: code_integer] :
            ( ( F @ ( G @ X4 ) )
            = ( Fg @ X4 ) ) ) ) ).

% fun_comp_eq_conv
thf(fact_2089_fun__comp__eq__conv,axiom,
    ! [F: int > nat,G: int > int,Fg: int > nat] :
      ( ( ( comp_int_nat_int @ F @ G )
        = Fg )
      = ( ! [X4: int] :
            ( ( F @ ( G @ X4 ) )
            = ( Fg @ X4 ) ) ) ) ).

% fun_comp_eq_conv
thf(fact_2090_comp__cong,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,F4: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer,X10: code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G2 @ X10 ) ) )
     => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
        = ( comp_C1593894019821074884nteger @ F4 @ G2 @ X10 ) ) ) ).

% comp_cong
thf(fact_2091_comp__cong,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,F4: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer,X10: code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G2 @ X10 ) ) )
     => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
        = ( comp_C8797469213163452608nteger @ F4 @ G2 @ X10 ) ) ) ).

% comp_cong
thf(fact_2092_comp__cong,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,F4: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer,X10: code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G2 @ X10 ) ) )
     => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
        = ( comp_C1593894019821074884nteger @ F4 @ G2 @ X10 ) ) ) ).

% comp_cong
thf(fact_2093_comp__cong,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,F4: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer,X10: code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G2 @ X10 ) ) )
     => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
        = ( comp_C8797469213163452608nteger @ F4 @ G2 @ X10 ) ) ) ).

% comp_cong
thf(fact_2094_comp__cong,axiom,
    ! [F: int > nat,G: int > int,X: int,F4: int > nat,G2: int > int,X10: int] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G2 @ X10 ) ) )
     => ( ( comp_int_nat_int @ F @ G @ X )
        = ( comp_int_nat_int @ F4 @ G2 @ X10 ) ) ) ).

% comp_cong
thf(fact_2095_execute__heap,axiom,
    ! [F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T6927113302350381334t_unit @ F ) )
      = ( comp_P3118722334806803912t_unit @ some_P1914260805536162275it_nat @ F ) ) ).

% execute_heap
thf(fact_2096_execute__heap,axiom,
    ! [F: heap_e7401611519738050253t_unit > produc7388388658123137530it_nat] :
      ( ( heap_Time_execute_b @ ( heap_Time_heap_b @ F ) )
      = ( comp_P6401406225962756324t_unit @ some_P2818173045054083285it_nat @ F ) ) ).

% execute_heap
thf(fact_2097_execute__heap,axiom,
    ! [F: heap_e7401611519738050253t_unit > produc3260487557148687353it_nat] :
      ( ( heap_Time_execute_a @ ( heap_Time_heap_a @ F ) )
      = ( comp_P8552900652980440422t_unit @ some_P7913643980934408916it_nat @ F ) ) ).

% execute_heap
thf(fact_2098_heap__def,axiom,
    ( heap_Time_heap_b
    = ( ^ [F3: heap_e7401611519738050253t_unit > produc7388388658123137530it_nat] : ( heap_Time_Heap_b2 @ ( comp_P6401406225962756324t_unit @ some_P2818173045054083285it_nat @ F3 ) ) ) ) ).

% heap_def
thf(fact_2099_heap__def,axiom,
    ( heap_Time_heap_a
    = ( ^ [F3: heap_e7401611519738050253t_unit > produc3260487557148687353it_nat] : ( heap_Time_Heap_a2 @ ( comp_P8552900652980440422t_unit @ some_P7913643980934408916it_nat @ F3 ) ) ) ) ).

% heap_def
thf(fact_2100_heap__def,axiom,
    ( heap_T6927113302350381334t_unit
    = ( ^ [F3: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] : ( heap_T6183433275982383450t_unit @ ( comp_P3118722334806803912t_unit @ some_P1914260805536162275it_nat @ F3 ) ) ) ) ).

% heap_def
thf(fact_2101_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_2102_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_2103_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_2104_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_2105_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat,Acc: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
            = Acc ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
            = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_2106_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_2107_Set_Ois__empty__def,axiom,
    ( is_emp1662574758705540307at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] : A6 = bot_bo2099793752762293965at_nat ) ) ).

% Set.is_empty_def
thf(fact_2108_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A6: set_o] : A6 = bot_bot_set_o ) ) ).

% Set.is_empty_def
thf(fact_2109_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A6: set_nat] : A6 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
thf(fact_2110_Set_Ois__empty__def,axiom,
    ( is_empty_int
    = ( ^ [A6: set_int] : A6 = bot_bot_set_int ) ) ).

% Set.is_empty_def
thf(fact_2111_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_2112_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F3: nat > nat > nat,A5: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F3 @ A5 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_2113_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_2114_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_2115_pure__assn__eq__conv,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ( pure_assn @ P2 )
        = ( pure_assn @ Q2 ) )
      = ( P2 = Q2 ) ) ).

% pure_assn_eq_conv
thf(fact_2116_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_2117_pure__assn__eq__emp__iff,axiom,
    ! [P2: $o] :
      ( ( ( pure_assn @ P2 )
        = one_one_assn )
      = P2 ) ).

% pure_assn_eq_emp_iff
thf(fact_2118_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_2119_pure__assn__eq__false__iff,axiom,
    ! [P2: $o] :
      ( ( ( pure_assn @ P2 )
        = bot_bot_assn )
      = ~ P2 ) ).

% pure_assn_eq_false_iff
thf(fact_2120_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_2121_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_a,Q2: a > assn] :
      ( ( hoare_hoare_triple_a @ ( pure_assn @ B ) @ F @ Q2 )
      = ( B
       => ( hoare_hoare_triple_a @ one_one_assn @ F @ Q2 ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_2122_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_b,Q2: b > assn] :
      ( ( hoare_hoare_triple_b @ ( pure_assn @ B ) @ F @ Q2 )
      = ( B
       => ( hoare_hoare_triple_b @ one_one_assn @ F @ Q2 ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_2123_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T5738788834812785303t_unit,Q2: product_unit > assn] :
      ( ( hoare_8945653483474564448t_unit @ ( pure_assn @ B ) @ F @ Q2 )
      = ( B
       => ( hoare_8945653483474564448t_unit @ one_one_assn @ F @ Q2 ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_2124_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_a,Q2: a > assn] :
      ( ( B
       => ( hoare_hoare_triple_a @ one_one_assn @ F @ Q2 ) )
     => ( hoare_hoare_triple_a @ ( pure_assn @ B ) @ F @ Q2 ) ) ).

% norm_pre_pure_rule2
thf(fact_2125_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_b,Q2: b > assn] :
      ( ( B
       => ( hoare_hoare_triple_b @ one_one_assn @ F @ Q2 ) )
     => ( hoare_hoare_triple_b @ ( pure_assn @ B ) @ F @ Q2 ) ) ).

% norm_pre_pure_rule2
thf(fact_2126_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T5738788834812785303t_unit,Q2: product_unit > assn] :
      ( ( B
       => ( hoare_8945653483474564448t_unit @ one_one_assn @ F @ Q2 ) )
     => ( hoare_8945653483474564448t_unit @ ( pure_assn @ B ) @ F @ Q2 ) ) ).

% norm_pre_pure_rule2
thf(fact_2127_lookup__rule,axiom,
    ! [P3: ref_Product_unit,X: product_unit] :
      ( hoare_8945653483474564448t_unit @ ( sngr_a5825115052027484668t_unit @ P3 @ X ) @ ( ref_lo7930775386976318366t_unit @ P3 )
      @ ^ [R5: product_unit] : ( times_times_assn @ ( sngr_a5825115052027484668t_unit @ P3 @ X ) @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% lookup_rule
thf(fact_2128_K__record__comp,axiom,
    ! [C2: produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer] :
      ( ( comp_C1593894019821074884nteger
        @ ^ [X4: code_integer] : C2
        @ F )
      = ( ^ [X4: code_integer] : C2 ) ) ).

% K_record_comp
thf(fact_2129_K__record__comp,axiom,
    ! [C2: produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer] :
      ( ( comp_C8797469213163452608nteger
        @ ^ [X4: code_integer > code_integer] : C2
        @ F )
      = ( ^ [X4: code_integer] : C2 ) ) ).

% K_record_comp
thf(fact_2130_K__record__comp,axiom,
    ! [C2: nat,F: int > int] :
      ( ( comp_int_nat_int
        @ ^ [X4: int] : C2
        @ F )
      = ( ^ [X4: int] : C2 ) ) ).

% K_record_comp
thf(fact_2131_execute__ureturn,axiom,
    ! [X: product_unit] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T4284346855313245393t_unit @ X ) )
      = ( comp_P3118722334806803912t_unit @ some_P1914260805536162275it_nat
        @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H6 @ zero_zero_nat ) ) ) ) ).

% execute_ureturn
thf(fact_2132_execute__ureturn,axiom,
    ! [X: b] :
      ( ( heap_Time_execute_b @ ( heap_Time_ureturn_b @ X ) )
      = ( comp_P6401406225962756324t_unit @ some_P2818173045054083285it_nat
        @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc4082563078715348724it_nat @ X @ ( produc584006145561248582it_nat @ H6 @ zero_zero_nat ) ) ) ) ).

% execute_ureturn
thf(fact_2133_execute__ureturn,axiom,
    ! [X: a] :
      ( ( heap_Time_execute_a @ ( heap_Time_ureturn_a @ X ) )
      = ( comp_P8552900652980440422t_unit @ some_P7913643980934408916it_nat
        @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc9178034014595674355it_nat @ X @ ( produc584006145561248582it_nat @ H6 @ zero_zero_nat ) ) ) ) ).

% execute_ureturn
thf(fact_2134_mlex__leq,axiom,
    ! [F: int > nat,X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R3 )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_2135_mlex__iff,axiom,
    ! [X: int,Y: int,F: int > nat,R3: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R3 ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_2136_mlex__less,axiom,
    ! [F: int > nat,X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R3 ) ) ) ).

% mlex_less
thf(fact_2137_ent__pure__post__iff__sng,axiom,
    ! [P2: assn,B: $o] :
      ( ( entails @ P2 @ ( pure_assn @ B ) )
      = ( ! [H6: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P2 @ H6 )
           => B )
        & ( entails @ P2 @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_2138_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_2139_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
        = Y )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( Y
              = ( ~ ( ( As2 = bot_bot_set_nat )
                    & X ) ) ) ) ) ).

% pure_assn_raw.elims(1)
thf(fact_2140_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ~ ( ( As2 = bot_bot_set_nat )
                & X ) ) ) ).

% pure_assn_raw.elims(2)
thf(fact_2141_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_2142_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_2143_add__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add_0
thf(fact_2144_add__0,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A )
      = A ) ).

% add_0
thf(fact_2145_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_2146_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_2147_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_2148_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_2149_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_2150_add__cancel__right__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
thf(fact_2151_add__cancel__right__right,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A
        = ( plus_p7104986032573967614at_nat @ A @ B ) )
      = ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_right
thf(fact_2152_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_2153_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_2154_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_2155_add__cancel__right__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
thf(fact_2156_add__cancel__right__left,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A
        = ( plus_p7104986032573967614at_nat @ B @ A ) )
      = ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_left
thf(fact_2157_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_2158_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_2159_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_2160_add__cancel__left__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = A )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
thf(fact_2161_add__cancel__left__right,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A @ B )
        = A )
      = ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_right
thf(fact_2162_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_2163_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_2164_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_2165_add__cancel__left__left,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = A )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
thf(fact_2166_add__cancel__left__left,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B @ A )
        = A )
      = ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_left
thf(fact_2167_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_2168_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_2169_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_2170_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
thf(fact_2171_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2172_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2173_add_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.right_neutral
thf(fact_2174_add_Oright__neutral,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A @ zero_z1048942125864253310at_nat )
      = A ) ).

% add.right_neutral
thf(fact_2175_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_2176_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_2177_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_2178_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_2179_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_2180_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_2181_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_2182_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_2183_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_2184_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_2185_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_2186_star__false__left,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P2 )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_2187_star__false__right,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ P2 @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_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_2200_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_2201_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_2202_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_2203_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_2204_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_2205_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_2206_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_2207_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_2208_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_2209_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_2210_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_2211_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_2212_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_2213_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_2214_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_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_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_2233_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_2234_mod__pure__star__dist,axiom,
    ! [P2: assn,B: $o,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ H )
      = ( ( rep_assn @ P2 @ H )
        & B ) ) ).

% mod_pure_star_dist
thf(fact_2235_ent__pure__pre__iff,axiom,
    ! [P2: assn,B: $o,Q2: assn] :
      ( ( entails @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ Q2 )
      = ( B
       => ( entails @ P2 @ Q2 ) ) ) ).

% ent_pure_pre_iff
thf(fact_2236_norm__pre__pure__iff,axiom,
    ! [P2: assn,B: $o,F: heap_Time_Heap_a,Q2: a > assn] :
      ( ( hoare_hoare_triple_a @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ F @ Q2 )
      = ( B
       => ( hoare_hoare_triple_a @ P2 @ F @ Q2 ) ) ) ).

% norm_pre_pure_iff
thf(fact_2237_norm__pre__pure__iff,axiom,
    ! [P2: assn,B: $o,F: heap_Time_Heap_b,Q2: b > assn] :
      ( ( hoare_hoare_triple_b @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ F @ Q2 )
      = ( B
       => ( hoare_hoare_triple_b @ P2 @ F @ Q2 ) ) ) ).

% norm_pre_pure_iff
thf(fact_2238_norm__pre__pure__iff,axiom,
    ! [P2: assn,B: $o,F: heap_T5738788834812785303t_unit,Q2: product_unit > assn] :
      ( ( hoare_8945653483474564448t_unit @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ F @ Q2 )
      = ( B
       => ( hoare_8945653483474564448t_unit @ P2 @ F @ Q2 ) ) ) ).

% norm_pre_pure_iff
thf(fact_2239_ent__false__iff,axiom,
    ! [P2: assn] :
      ( ( entails @ P2 @ bot_bot_assn )
      = ( ! [H6: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P2 @ H6 ) ) ) ).

% ent_false_iff
thf(fact_2240_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q2: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q2 )
      = ( B
       => ( entails @ one_one_assn @ Q2 ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_2241_mod__h__bot__iff_I5_J,axiom,
    ! [P2: assn,Q2: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        & ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_2242_ent__pure__post__iff,axiom,
    ! [P2: assn,Q2: assn,B: $o] :
      ( ( entails @ P2 @ ( times_times_assn @ Q2 @ ( pure_assn @ B ) ) )
      = ( ! [H6: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P2 @ H6 )
           => B )
        & ( entails @ P2 @ Q2 ) ) ) ).

% ent_pure_post_iff
thf(fact_2243_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_2244_lambda__zero,axiom,
    ( ( ^ [H6: code_integer] : zero_z3403309356797280102nteger )
    = ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger ) ) ).

% lambda_zero
thf(fact_2245_lambda__zero,axiom,
    ( ( ^ [H6: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_2246_lambda__zero,axiom,
    ( ( ^ [H6: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_2247_lambda__zero,axiom,
    ( ( ^ [H6: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_2248_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C2 @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2249_mult__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2250_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2251_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2252_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C2 @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2253_mult__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2254_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2255_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2256_zero__le__square,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ A ) ) ).

% zero_le_square
thf(fact_2257_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2258_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2259_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_2260_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_2261_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_2262_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2263_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2264_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2265_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_2266_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_2267_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_2268_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2269_mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2270_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2271_mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2272_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2273_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2274_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2275_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_2276_mult__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_2277_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_2278_mult__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_2279_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_2280_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_2281_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_2282_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_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_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_2291_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_2292_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_2293_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_2294_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_2295_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_2296_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_2297_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_2298_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_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2306_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2307_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2308_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2309_ent__iffI,axiom,
    ! [A4: assn,B5: assn] :
      ( ( entails @ A4 @ B5 )
     => ( ( entails @ B5 @ A4 )
       => ( A4 = B5 ) ) ) ).

% ent_iffI
thf(fact_2310_ent__refl,axiom,
    ! [P2: assn] : ( entails @ P2 @ P2 ) ).

% ent_refl
thf(fact_2311_ent__trans,axiom,
    ! [P2: assn,Q2: assn,R3: assn] :
      ( ( entails @ P2 @ Q2 )
     => ( ( entails @ Q2 @ R3 )
       => ( entails @ P2 @ R3 ) ) ) ).

% ent_trans
thf(fact_2312_ent__star__mono,axiom,
    ! [P2: assn,P6: assn,Q2: assn,Q4: assn] :
      ( ( entails @ P2 @ P6 )
     => ( ( entails @ Q2 @ Q4 )
       => ( entails @ ( times_times_assn @ P2 @ Q2 ) @ ( times_times_assn @ P6 @ Q4 ) ) ) ) ).

% ent_star_mono
thf(fact_2313_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P5: assn,Q3: assn] : ( times_times_assn @ Q3 @ P5 ) ) ) ).

% assn_times_comm
thf(fact_2314_assn__times__assoc,axiom,
    ! [P2: assn,Q2: assn,R3: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P2 @ Q2 ) @ R3 )
      = ( times_times_assn @ P2 @ ( times_times_assn @ Q2 @ R3 ) ) ) ).

% assn_times_assoc
thf(fact_2315_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_2316_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_2317_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_2318_not__square__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ A ) @ zero_z3403309356797280102nteger ) ).

% not_square_less_zero
thf(fact_2319_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2320_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2321_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_2322_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_2323_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_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_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_2332_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_2333_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_2334_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_2335_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_2336_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_2337_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_2338_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_2339_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_2340_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_2341_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_2342_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_2343_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_2344_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_2345_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_2346_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_2347_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_2348_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_2349_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_2350_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_2351_mult__less__cancel__left__neg,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
        = ( ord_le6747313008572928689nteger @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2352_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2353_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2354_mult__less__cancel__left__pos,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
        = ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2355_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2356_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2357_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2358_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2359_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2360_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2361_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2362_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2363_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2364_mult__less__cancel__left__disj,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2365_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2366_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2367_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2368_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2369_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2370_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2371_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2372_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2373_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2374_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2375_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2376_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2377_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2378_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2379_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2380_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2381_mult__le__cancel__left,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2382_mult__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2383_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2384_mult__le__cancel__right,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2385_mult__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2386_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2387_mult__left__less__imp__less,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2388_mult__left__less__imp__less,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2389_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2390_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2391_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2392_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2393_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2394_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2395_mult__less__cancel__left,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2396_mult__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2397_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2398_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2399_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2400_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2401_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2402_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2403_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2404_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2405_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2406_mult__less__cancel__right,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2407_mult__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2408_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2409_mult__le__cancel__left__neg,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
        = ( ord_le3102999989581377725nteger @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2410_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2411_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2412_mult__le__cancel__left__pos,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
        = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2413_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2414_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2415_mult__left__le__imp__le,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2416_mult__left__le__imp__le,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2417_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2418_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2419_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2420_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2421_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2422_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2423_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2424_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2425_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2426_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2427_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C2 @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2428_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2429_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2430_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2431_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_2432_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_2433_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_2434_mult__left__le,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_2435_mult__left__le,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ C2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_2436_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_2437_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_2438_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_2439_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_2440_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_2441_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_2442_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_2443_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_2444_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_2445_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_2446_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_2447_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_2448_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_2449_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_2450_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_2451_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_2452_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_2453_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_2454_distrib__right,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_2455_distrib__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_2456_distrib__right,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_2457_distrib__left,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_2458_distrib__left,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_2459_distrib__left,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_2460_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2461_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2462_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2463_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2464_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2465_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2466_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2467_le__numeral__extra_I3_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% le_numeral_extra(3)
thf(fact_2468_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2469_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2470_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2471_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_2472_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_numeral_extra(3)
thf(fact_2473_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_2474_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_2475_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_2476_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_2477_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_2478_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_2479_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_2480_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2481_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2482_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2483_add_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.comm_neutral
thf(fact_2484_add_Ocomm__neutral,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A @ zero_z1048942125864253310at_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2485_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2486_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2487_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2488_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2489_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2490_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2491_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2492_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2493_verit__sum__simplify,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% verit_sum_simplify
thf(fact_2494_verit__sum__simplify,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A @ zero_z1048942125864253310at_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_2495_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_2496_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_2497_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_2498_ent__fwd,axiom,
    ! [P2: assn,H: produc3658429121746597890et_nat,Q2: assn] :
      ( ( rep_assn @ P2 @ H )
     => ( ( entails @ P2 @ Q2 )
       => ( rep_assn @ Q2 @ H ) ) ) ).

% ent_fwd
thf(fact_2499_entailsD,axiom,
    ! [P2: assn,Q2: assn,H: produc3658429121746597890et_nat] :
      ( ( entails @ P2 @ Q2 )
     => ( ( rep_assn @ P2 @ H )
       => ( rep_assn @ Q2 @ H ) ) ) ).

% entailsD
thf(fact_2500_entailsI,axiom,
    ! [P2: assn,Q2: assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H2 )
         => ( rep_assn @ Q2 @ H2 ) )
     => ( entails @ P2 @ Q2 ) ) ).

% entailsI
thf(fact_2501_entails__def,axiom,
    ( entails
    = ( ^ [P5: assn,Q3: assn] :
        ! [H6: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P5 @ H6 )
         => ( rep_assn @ Q3 @ H6 ) ) ) ) ).

% entails_def
thf(fact_2502_mod__starD,axiom,
    ! [A4: assn,B5: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A4 @ B5 ) @ H )
     => ? [H12: produc3658429121746597890et_nat,H23: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A4 @ H12 )
          & ( rep_assn @ B5 @ H23 ) ) ) ).

% mod_starD
thf(fact_2503_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_2504_infinite__descent0,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( ~ ( P2 @ N5 )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N5 )
                  & ~ ( P2 @ M4 ) ) ) )
       => ( P2 @ N2 ) ) ) ).

% infinite_descent0
thf(fact_2505_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_2506_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_2507_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_2508_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_2509_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_2510_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_2511_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2512_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_2513_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_2514_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_2515_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_2516_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_2517_cons__post__rule,axiom,
    ! [P2: assn,C2: heap_Time_Heap_a,Q2: a > assn,Q4: a > assn] :
      ( ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 )
     => ( ! [X3: a] : ( entails @ ( Q2 @ X3 ) @ ( Q4 @ X3 ) )
       => ( hoare_hoare_triple_a @ P2 @ C2 @ Q4 ) ) ) ).

% cons_post_rule
thf(fact_2518_cons__post__rule,axiom,
    ! [P2: assn,C2: heap_Time_Heap_b,Q2: b > assn,Q4: b > assn] :
      ( ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 )
     => ( ! [X3: b] : ( entails @ ( Q2 @ X3 ) @ ( Q4 @ X3 ) )
       => ( hoare_hoare_triple_b @ P2 @ C2 @ Q4 ) ) ) ).

% cons_post_rule
thf(fact_2519_cons__post__rule,axiom,
    ! [P2: assn,C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn,Q4: product_unit > assn] :
      ( ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 )
     => ( ! [X3: product_unit] : ( entails @ ( Q2 @ X3 ) @ ( Q4 @ X3 ) )
       => ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q4 ) ) ) ).

% cons_post_rule
thf(fact_2520_cons__rule,axiom,
    ! [P2: assn,P6: assn,Q2: a > assn,Q4: a > assn,C2: heap_Time_Heap_a] :
      ( ( entails @ P2 @ P6 )
     => ( ! [X3: a] : ( entails @ ( Q2 @ X3 ) @ ( Q4 @ X3 ) )
       => ( ( hoare_hoare_triple_a @ P6 @ C2 @ Q2 )
         => ( hoare_hoare_triple_a @ P2 @ C2 @ Q4 ) ) ) ) ).

% cons_rule
thf(fact_2521_cons__rule,axiom,
    ! [P2: assn,P6: assn,Q2: b > assn,Q4: b > assn,C2: heap_Time_Heap_b] :
      ( ( entails @ P2 @ P6 )
     => ( ! [X3: b] : ( entails @ ( Q2 @ X3 ) @ ( Q4 @ X3 ) )
       => ( ( hoare_hoare_triple_b @ P6 @ C2 @ Q2 )
         => ( hoare_hoare_triple_b @ P2 @ C2 @ Q4 ) ) ) ) ).

% cons_rule
thf(fact_2522_cons__rule,axiom,
    ! [P2: assn,P6: assn,Q2: product_unit > assn,Q4: product_unit > assn,C2: heap_T5738788834812785303t_unit] :
      ( ( entails @ P2 @ P6 )
     => ( ! [X3: product_unit] : ( entails @ ( Q2 @ X3 ) @ ( Q4 @ X3 ) )
       => ( ( hoare_8945653483474564448t_unit @ P6 @ C2 @ Q2 )
         => ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q4 ) ) ) ) ).

% cons_rule
thf(fact_2523_ent__disjE,axiom,
    ! [A4: assn,C4: assn,B5: assn] :
      ( ( entails @ A4 @ C4 )
     => ( ( entails @ B5 @ C4 )
       => ( entails @ ( sup_sup_assn @ A4 @ B5 ) @ C4 ) ) ) ).

% ent_disjE
thf(fact_2524_ent__disjI1,axiom,
    ! [P2: assn,Q2: assn,R3: assn] :
      ( ( entails @ ( sup_sup_assn @ P2 @ Q2 ) @ R3 )
     => ( entails @ P2 @ R3 ) ) ).

% ent_disjI1
thf(fact_2525_ent__disjI2,axiom,
    ! [P2: assn,Q2: assn,R3: assn] :
      ( ( entails @ ( sup_sup_assn @ P2 @ Q2 ) @ R3 )
     => ( entails @ Q2 @ R3 ) ) ).

% ent_disjI2
thf(fact_2526_ent__disjI1_H,axiom,
    ! [A4: assn,B5: assn,C4: assn] :
      ( ( entails @ A4 @ B5 )
     => ( entails @ A4 @ ( sup_sup_assn @ B5 @ C4 ) ) ) ).

% ent_disjI1'
thf(fact_2527_ent__disjI2_H,axiom,
    ! [A4: assn,C4: assn,B5: assn] :
      ( ( entails @ A4 @ C4 )
     => ( entails @ A4 @ ( sup_sup_assn @ B5 @ C4 ) ) ) ).

% ent_disjI2'
thf(fact_2528_ent__disjI1__direct,axiom,
    ! [A4: assn,B5: assn] : ( entails @ A4 @ ( sup_sup_assn @ A4 @ B5 ) ) ).

% ent_disjI1_direct
thf(fact_2529_ent__disjI2__direct,axiom,
    ! [B5: assn,A4: assn] : ( entails @ B5 @ ( sup_sup_assn @ A4 @ B5 ) ) ).

% ent_disjI2_direct
thf(fact_2530_star__or__dist1,axiom,
    ! [A4: assn,B5: assn,C4: assn] :
      ( ( times_times_assn @ ( sup_sup_assn @ A4 @ B5 ) @ C4 )
      = ( sup_sup_assn @ ( times_times_assn @ A4 @ C4 ) @ ( times_times_assn @ B5 @ C4 ) ) ) ).

% star_or_dist1
thf(fact_2531_star__or__dist2,axiom,
    ! [C4: assn,A4: assn,B5: assn] :
      ( ( times_times_assn @ C4 @ ( sup_sup_assn @ A4 @ B5 ) )
      = ( sup_sup_assn @ ( times_times_assn @ C4 @ A4 ) @ ( times_times_assn @ C4 @ B5 ) ) ) ).

% star_or_dist2
thf(fact_2532_assn__one__left,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ one_one_assn @ P2 )
      = P2 ) ).

% assn_one_left
thf(fact_2533_ent__false,axiom,
    ! [P2: assn] : ( entails @ bot_bot_assn @ P2 ) ).

% ent_false
thf(fact_2534_mult__le__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ ( times_3573771949741848930nteger @ C2 @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2535_mult__le__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2536_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2537_mult__le__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ C2 )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2538_mult__le__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2539_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2540_mult__le__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ ( times_3573771949741848930nteger @ B @ C2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2541_mult__le__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2542_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2543_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ C2 )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2544_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2545_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2546_mult__less__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ ( times_3573771949741848930nteger @ C2 @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2547_mult__less__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2548_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2549_mult__less__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ C2 )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2550_mult__less__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2551_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2552_mult__less__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ ( times_3573771949741848930nteger @ B @ C2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2553_mult__less__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2554_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2555_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ C2 )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2556_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2557_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2558_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_2559_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_2560_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_2561_lambda__one,axiom,
    ( ( ^ [X4: code_integer] : X4 )
    = ( times_3573771949741848930nteger @ one_one_Code_integer ) ) ).

% lambda_one
thf(fact_2562_lambda__one,axiom,
    ( ( ^ [X4: assn] : X4 )
    = ( times_times_assn @ one_one_assn ) ) ).

% lambda_one
thf(fact_2563_lambda__one,axiom,
    ( ( ^ [X4: rat] : X4 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_2564_lambda__one,axiom,
    ( ( ^ [X4: nat] : X4 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_2565_lambda__one,axiom,
    ( ( ^ [X4: int] : X4 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_2566_cons__pre__rule,axiom,
    ! [P2: assn,P6: assn,C2: heap_Time_Heap_a,Q2: a > assn] :
      ( ( entails @ P2 @ P6 )
     => ( ( hoare_hoare_triple_a @ P6 @ C2 @ Q2 )
       => ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 ) ) ) ).

% cons_pre_rule
thf(fact_2567_cons__pre__rule,axiom,
    ! [P2: assn,P6: assn,C2: heap_Time_Heap_b,Q2: b > assn] :
      ( ( entails @ P2 @ P6 )
     => ( ( hoare_hoare_triple_b @ P6 @ C2 @ Q2 )
       => ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 ) ) ) ).

% cons_pre_rule
thf(fact_2568_cons__pre__rule,axiom,
    ! [P2: assn,P6: assn,C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn] :
      ( ( entails @ P2 @ P6 )
     => ( ( hoare_8945653483474564448t_unit @ P6 @ C2 @ Q2 )
       => ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 ) ) ) ).

% cons_pre_rule
thf(fact_2569_frame__rule,axiom,
    ! [P2: assn,C2: heap_Time_Heap_a,Q2: a > assn,R3: assn] :
      ( ( hoare_hoare_triple_a @ P2 @ C2 @ Q2 )
     => ( hoare_hoare_triple_a @ ( times_times_assn @ P2 @ R3 ) @ C2
        @ ^ [X4: a] : ( times_times_assn @ ( Q2 @ X4 ) @ R3 ) ) ) ).

% frame_rule
thf(fact_2570_frame__rule,axiom,
    ! [P2: assn,C2: heap_Time_Heap_b,Q2: b > assn,R3: assn] :
      ( ( hoare_hoare_triple_b @ P2 @ C2 @ Q2 )
     => ( hoare_hoare_triple_b @ ( times_times_assn @ P2 @ R3 ) @ C2
        @ ^ [X4: b] : ( times_times_assn @ ( Q2 @ X4 ) @ R3 ) ) ) ).

% frame_rule
thf(fact_2571_frame__rule,axiom,
    ! [P2: assn,C2: heap_T5738788834812785303t_unit,Q2: product_unit > assn,R3: assn] :
      ( ( hoare_8945653483474564448t_unit @ P2 @ C2 @ Q2 )
     => ( hoare_8945653483474564448t_unit @ ( times_times_assn @ P2 @ R3 ) @ C2
        @ ^ [X4: product_unit] : ( times_times_assn @ ( Q2 @ X4 ) @ R3 ) ) ) ).

% frame_rule
thf(fact_2572_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_2573_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_2574_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_2575_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_2576_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_2577_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_2578_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_2579_add__decreasing,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ C2 @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_2580_add__decreasing,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_2581_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_2582_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_2583_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C2 )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2584_add__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2585_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2586_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2587_add__decreasing2,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2588_add__decreasing2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2589_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2590_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2591_add__increasing2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2592_add__increasing2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2593_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2594_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2595_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_2596_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_2597_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_2598_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_2599_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_2600_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_2601_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_2602_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_2603_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_2604_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_2605_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_2606_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_2607_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_2608_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_2609_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_2610_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_2611_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_2612_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2613_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2614_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2615_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_2616_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_2617_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_2618_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_2619_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_2620_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_2621_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_2622_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_2623_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C2 )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2624_pos__add__strict,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2625_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2626_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2627_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_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_2635_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_2636_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_2637_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_2638_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_2639_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_2640_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2641_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2642_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2643_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_2644_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2645_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2646_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2647_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_2648_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2649_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2650_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2651_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ N2 )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_2652_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ? [K: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K )
          & ( ( plus_plus_nat @ I2 @ K )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_2653_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_2654_norm__pre__pure__rule1,axiom,
    ! [B: $o,P2: assn,F: heap_Time_Heap_a,Q2: a > assn] :
      ( ( B
       => ( hoare_hoare_triple_a @ P2 @ F @ Q2 ) )
     => ( hoare_hoare_triple_a @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ F @ Q2 ) ) ).

% norm_pre_pure_rule1
thf(fact_2655_norm__pre__pure__rule1,axiom,
    ! [B: $o,P2: assn,F: heap_Time_Heap_b,Q2: b > assn] :
      ( ( B
       => ( hoare_hoare_triple_b @ P2 @ F @ Q2 ) )
     => ( hoare_hoare_triple_b @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ F @ Q2 ) ) ).

% norm_pre_pure_rule1
thf(fact_2656_norm__pre__pure__rule1,axiom,
    ! [B: $o,P2: assn,F: heap_T5738788834812785303t_unit,Q2: product_unit > assn] :
      ( ( B
       => ( hoare_8945653483474564448t_unit @ P2 @ F @ Q2 ) )
     => ( hoare_8945653483474564448t_unit @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ F @ Q2 ) ) ).

% norm_pre_pure_rule1
thf(fact_2657_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_2658_ureturn__def,axiom,
    ( heap_Time_ureturn_b
    = ( ^ [X4: b] :
          ( heap_Time_heap_b
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc4082563078715348724it_nat @ X4 @ ( produc584006145561248582it_nat @ H6 @ zero_zero_nat ) ) ) ) ) ).

% ureturn_def
thf(fact_2659_ureturn__def,axiom,
    ( heap_Time_ureturn_a
    = ( ^ [X4: a] :
          ( heap_Time_heap_a
          @ ^ [H6: heap_e7401611519738050253t_unit] : ( produc9178034014595674355it_nat @ X4 @ ( produc584006145561248582it_nat @ H6 @ zero_zero_nat ) ) ) ) ) ).

% ureturn_def
thf(fact_2660_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_2661_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_2662_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_2663_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_2664_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_2665_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_2666_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_2667_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_2668_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_2669_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_2670_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_2671_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_2672_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_2673_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_2674_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_2675_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_2676_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C2 )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2677_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2678_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2679_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2680_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C2 )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2681_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2682_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2683_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2684_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_2685_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2686_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2687_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2688_return__sp__rule,axiom,
    ! [P2: assn,X: a] :
      ( hoare_hoare_triple_a @ P2 @ ( heap_Time_return_a @ X )
      @ ^ [R5: a] : ( times_times_assn @ P2 @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_2689_return__sp__rule,axiom,
    ! [P2: assn,X: b] :
      ( hoare_hoare_triple_b @ P2 @ ( heap_Time_return_b @ X )
      @ ^ [R5: b] : ( times_times_assn @ P2 @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_2690_return__sp__rule,axiom,
    ! [P2: assn,X: product_unit] :
      ( hoare_8945653483474564448t_unit @ P2 @ ( heap_T7507251653302230130t_unit @ X )
      @ ^ [R5: product_unit] : ( times_times_assn @ P2 @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_2691_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( ( As2 = bot_bot_set_nat )
              & X ) ) ) ).

% pure_assn_raw.elims(3)
thf(fact_2692_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_2693_double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_eq_0_iff
thf(fact_2694_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_2695_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_2696_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_2697_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_2698_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_2699_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_2700_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_2701_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_2702_mult__le__cancel__iff1,axiom,
    ! [Z3: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z3 )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z3 ) @ ( times_3573771949741848930nteger @ Y @ Z3 ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2703_mult__le__cancel__iff1,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ Y @ Z3 ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2704_mult__le__cancel__iff1,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2705_mult__le__cancel__iff2,axiom,
    ! [Z3: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z3 )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z3 @ X ) @ ( times_3573771949741848930nteger @ Z3 @ Y ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2706_mult__le__cancel__iff2,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ ( times_times_rat @ Z3 @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2707_mult__le__cancel__iff2,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X ) @ ( times_times_int @ Z3 @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2708_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_2709_divides__aux__eq,axiom,
    ! [Q6: code_integer,R2: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q6 @ R2 ) )
      = ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_2710_divides__aux__eq,axiom,
    ! [Q6: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q6 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_2711_divides__aux__eq,axiom,
    ! [Q6: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q6 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_2712_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 ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) )
               => ( ( As2 = bot_bot_set_nat )
                  & X ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
thf(fact_2713_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 ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) )
               => ~ ( ( As2 = bot_bot_set_nat )
                    & X ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
thf(fact_2714_mult__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K2 )
        = ( times_times_nat @ N2 @ K2 ) )
      = ( ( M = N2 )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2715_mult__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K2 @ M )
        = ( times_times_nat @ K2 @ N2 ) )
      = ( ( M = N2 )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2716_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2717_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_2718_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_2719_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_2720_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_2721_mult__less__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_2722_mult__le__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_2723_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2724_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_2725_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_2726_mult__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).

% mult_le_mono
thf(fact_2727_mult__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).

% mult_le_mono1
thf(fact_2728_mult__le__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).

% mult_le_mono2
thf(fact_2729_add__mult__distrib2,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K2 @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_2730_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K2 )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N2 @ K2 ) ) ) ).

% add_mult_distrib
thf(fact_2731_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_2732_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_2733_mult__less__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).

% mult_less_mono2
thf(fact_2734_mult__less__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).

% mult_less_mono1
thf(fact_2735_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_2736_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_2737_mlex__bound,axiom,
    ! [A: nat,A4: nat,B: nat,N7: nat] :
      ( ( ord_less_nat @ A @ A4 )
     => ( ( ord_less_nat @ B @ N7 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( times_times_nat @ A4 @ N7 ) ) ) ) ).

% mlex_bound
thf(fact_2738_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_2739_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_2740_linordered__field__no__lb,axiom,
    ! [X6: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X6 ) ).

% linordered_field_no_lb
thf(fact_2741_linordered__field__no__ub,axiom,
    ! [X6: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X6 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_2742_mult__less__iff1,axiom,
    ! [Z3: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z3 )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z3 ) @ ( times_3573771949741848930nteger @ Y @ Z3 ) )
        = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2743_mult__less__iff1,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z3 )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ Y @ Z3 ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2744_mult__less__iff1,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2745_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_2746_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_2747_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_2748_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 ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( Y
                  = ( ( As2 = bot_bot_set_nat )
                    & X ) )
               => ~ ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ) ) ) ).

% pure_assn_raw.pelims(1)
thf(fact_2749_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2750_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_2751_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2752_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ R2 @ C2 ) )
         != ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2753_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C2: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C2 ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2754_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C2: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C2 ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2755_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C2: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C2 ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2756_Euclid__induct,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 @ A3 @ B3 )
          = ( P2 @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P2 @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P2 @ A3 @ B3 )
             => ( P2 @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P2 @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_2757_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ( times_times_nat @ K2 @ M )
          = ( times_times_nat @ K2 @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2758_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J2: nat,K2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J2 ) @ U ) @ K2 ) ) ).

% left_add_mult_distrib
thf(fact_2759_add__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B
        = ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% add_0_iff
thf(fact_2760_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_2761_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_2762_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_2763_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D2 ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_2764_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_2765_crossproduct__noteq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_2766_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z3: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y ) @ ( times_times_rat @ X @ Z3 ) )
        = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z3 ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2767_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z3: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z3 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z3 ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2768_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z3 ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z3 ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2769_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2770_ent__mp,axiom,
    ! [P2: assn,Q2: assn] : ( entails @ ( times_times_assn @ P2 @ ( wand_assn @ P2 @ Q2 ) ) @ Q2 ) ).

% ent_mp
thf(fact_2771_ent__wandI,axiom,
    ! [Q2: assn,P2: assn,R3: assn] :
      ( ( entails @ ( times_times_assn @ Q2 @ P2 ) @ R3 )
     => ( entails @ P2 @ ( wand_assn @ Q2 @ R3 ) ) ) ).

% ent_wandI
thf(fact_2772_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_2773_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_2774_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_2775_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_2776_mod__star__conv,axiom,
    ! [A4: assn,B5: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A4 @ B5 ) @ H )
      = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
            ( ( H
              = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As22 ) ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( rep_assn @ A4 @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
            & ( rep_assn @ B5 @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).

% mod_star_conv
thf(fact_2777_star__assnI,axiom,
    ! [P2: assn,H: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,As5: set_nat] :
      ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ As ) )
     => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H @ As5 ) )
       => ( ( ( inf_inf_set_nat @ As @ As5 )
            = bot_bot_set_nat )
         => ( rep_assn @ ( times_times_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ).

% star_assnI
thf(fact_2778_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_2779_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_2780_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_2781_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_2782_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_2783_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_2784_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_2785_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_2786_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_2787_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_2788_Int__iff,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( inf_inf_set_o @ A4 @ B5 ) )
      = ( ( member_o @ C2 @ A4 )
        & ( member_o @ C2 @ B5 ) ) ) ).

% Int_iff
thf(fact_2789_Int__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( inf_in8396524679539076455nt_int @ A4 @ B5 ) )
      = ( ( member2340774599025711042nt_int @ C2 @ A4 )
        & ( member2340774599025711042nt_int @ C2 @ B5 ) ) ) ).

% Int_iff
thf(fact_2790_Int__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A4 @ B5 ) )
      = ( ( member_set_nat @ C2 @ A4 )
        & ( member_set_nat @ C2 @ B5 ) ) ) ).

% Int_iff
thf(fact_2791_Int__iff,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( inf_inf_set_int @ A4 @ B5 ) )
      = ( ( member_int @ C2 @ A4 )
        & ( member_int @ C2 @ B5 ) ) ) ).

% Int_iff
thf(fact_2792_Int__iff,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ C2 @ A4 )
        & ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% Int_iff
thf(fact_2793_Int__iff,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B5 ) )
      = ( ( member_nat @ C2 @ A4 )
        & ( member_nat @ C2 @ B5 ) ) ) ).

% Int_iff
thf(fact_2794_IntI,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ A4 )
     => ( ( member_o @ C2 @ B5 )
       => ( member_o @ C2 @ ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).

% IntI
thf(fact_2795_IntI,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ A4 )
     => ( ( member2340774599025711042nt_int @ C2 @ B5 )
       => ( member2340774599025711042nt_int @ C2 @ ( inf_in8396524679539076455nt_int @ A4 @ B5 ) ) ) ) ).

% IntI
thf(fact_2796_IntI,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ A4 )
     => ( ( member_set_nat @ C2 @ B5 )
       => ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A4 @ B5 ) ) ) ) ).

% IntI
thf(fact_2797_IntI,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ A4 )
     => ( ( member_int @ C2 @ B5 )
       => ( member_int @ C2 @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ).

% IntI
thf(fact_2798_IntI,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ A4 )
     => ( ( member8440522571783428010at_nat @ C2 @ B5 )
       => ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).

% IntI
thf(fact_2799_IntI,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ A4 )
     => ( ( member_nat @ C2 @ B5 )
       => ( member_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).

% IntI
thf(fact_2800_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z3 ) )
      = ( ( ord_less_eq_assn @ X @ Y )
        & ( ord_less_eq_assn @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_2801_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) )
      = ( ( ord_le3146513528884898305at_nat @ X @ Y )
        & ( ord_le3146513528884898305at_nat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_2802_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) )
      = ( ( ord_less_eq_set_nat @ X @ Y )
        & ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_2803_le__inf__iff,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z3 ) )
      = ( ( ord_le8369615600986905444_int_o @ X @ Y )
        & ( ord_le8369615600986905444_int_o @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_2804_le__inf__iff,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ X @ ( inf_inf_set_int @ Y @ Z3 ) )
      = ( ( ord_less_eq_set_int @ X @ Y )
        & ( ord_less_eq_set_int @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_2805_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z3 ) )
      = ( ( ord_less_eq_rat @ X @ Y )
        & ( ord_less_eq_rat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_2806_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_2807_le__inf__iff,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z3 ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( ord_less_eq_int @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_2808_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C2 ) )
      = ( ( ord_less_eq_assn @ A @ B )
        & ( ord_less_eq_assn @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_2809_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
      = ( ( ord_le3146513528884898305at_nat @ A @ B )
        & ( ord_le3146513528884898305at_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_2810_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
        & ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_2811_inf_Obounded__iff,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C2 ) )
      = ( ( ord_le8369615600986905444_int_o @ A @ B )
        & ( ord_le8369615600986905444_int_o @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_2812_inf_Obounded__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( inf_inf_set_int @ B @ C2 ) )
      = ( ( ord_less_eq_set_int @ A @ B )
        & ( ord_less_eq_set_int @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_2813_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C2 ) )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_2814_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_2815_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_2816_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_2817_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_2818_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_2819_inf__sup__absorb,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_2820_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_2821_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_2822_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_2823_inf__sup__absorb,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_2824_inf__sup__absorb,axiom,
    ! [X: int,Y: int] :
      ( ( inf_inf_int @ X @ ( sup_sup_int @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_2825_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_2826_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_2827_sup__inf__absorb,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_2828_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_2829_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_2830_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_2831_sup__inf__absorb,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_2832_sup__inf__absorb,axiom,
    ! [X: int,Y: int] :
      ( ( sup_sup_int @ X @ ( inf_inf_int @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_2833_Int__subset__iff,axiom,
    ! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C4 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
      = ( ( ord_le3146513528884898305at_nat @ C4 @ A4 )
        & ( ord_le3146513528884898305at_nat @ C4 @ B5 ) ) ) ).

% Int_subset_iff
thf(fact_2834_Int__subset__iff,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ C4 @ ( inf_inf_set_nat @ A4 @ B5 ) )
      = ( ( ord_less_eq_set_nat @ C4 @ A4 )
        & ( ord_less_eq_set_nat @ C4 @ B5 ) ) ) ).

% Int_subset_iff
thf(fact_2835_Int__subset__iff,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ C4 @ ( inf_inf_set_int @ A4 @ B5 ) )
      = ( ( ord_less_eq_set_int @ C4 @ A4 )
        & ( ord_less_eq_set_int @ C4 @ B5 ) ) ) ).

% Int_subset_iff
thf(fact_2836_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_2837_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_2838_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_2839_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_2840_inf__Some,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in7104746047340619750_int_o @ ( some_P180497116919641311_int_o @ X ) @ ( some_P180497116919641311_int_o @ Y ) )
      = ( some_P180497116919641311_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) ) ) ).

% inf_Some
thf(fact_2841_Int__Un__eq_I4_J,axiom,
    ! [T7: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ T7 @ ( inf_in2572325071724192079at_nat @ S @ T7 ) )
      = T7 ) ).

% Int_Un_eq(4)
thf(fact_2842_Int__Un__eq_I4_J,axiom,
    ! [T7: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ T7 @ ( inf_in1697001100524423349at_nat @ S @ T7 ) )
      = T7 ) ).

% Int_Un_eq(4)
thf(fact_2843_Int__Un__eq_I4_J,axiom,
    ! [T7: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ T7 @ ( inf_in7913087082777306421at_nat @ S @ T7 ) )
      = T7 ) ).

% Int_Un_eq(4)
thf(fact_2844_Int__Un__eq_I4_J,axiom,
    ! [T7: set_nat,S: set_nat] :
      ( ( sup_sup_set_nat @ T7 @ ( inf_inf_set_nat @ S @ T7 ) )
      = T7 ) ).

% Int_Un_eq(4)
thf(fact_2845_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T7 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_2846_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ S @ ( inf_in1697001100524423349at_nat @ S @ T7 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_2847_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ S @ ( inf_in7913087082777306421at_nat @ S @ T7 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_2848_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T7 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_2849_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T7 ) @ T7 )
      = T7 ) ).

% Int_Un_eq(2)
thf(fact_2850_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ S @ T7 ) @ T7 )
      = T7 ) ).

% Int_Un_eq(2)
thf(fact_2851_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T7 ) @ T7 )
      = T7 ) ).

% Int_Un_eq(2)
thf(fact_2852_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T7 ) @ T7 )
      = T7 ) ).

% Int_Un_eq(2)
thf(fact_2853_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T7 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_2854_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ S @ T7 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_2855_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T7 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_2856_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T7 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_2857_Un__Int__eq_I4_J,axiom,
    ! [T7: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ T7 @ ( sup_su6327502436637775413at_nat @ S @ T7 ) )
      = T7 ) ).

% Un_Int_eq(4)
thf(fact_2858_Un__Int__eq_I4_J,axiom,
    ! [T7: set_Pr8551490117392284871at_nat,S: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ T7 @ ( sup_su3035147773818789531at_nat @ S @ T7 ) )
      = T7 ) ).

% Un_Int_eq(4)
thf(fact_2859_Un__Int__eq_I4_J,axiom,
    ! [T7: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ T7 @ ( sup_su5525570899277871387at_nat @ S @ T7 ) )
      = T7 ) ).

% Un_Int_eq(4)
thf(fact_2860_Un__Int__eq_I4_J,axiom,
    ! [T7: set_nat,S: set_nat] :
      ( ( inf_inf_set_nat @ T7 @ ( sup_sup_set_nat @ S @ T7 ) )
      = T7 ) ).

% Un_Int_eq(4)
thf(fact_2861_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T7 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_2862_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ S @ ( sup_su3035147773818789531at_nat @ S @ T7 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_2863_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ S @ ( sup_su5525570899277871387at_nat @ S @ T7 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_2864_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T7 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_2865_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T7 ) @ T7 )
      = T7 ) ).

% Un_Int_eq(2)
thf(fact_2866_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ S @ T7 ) @ T7 )
      = T7 ) ).

% Un_Int_eq(2)
thf(fact_2867_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T7 ) @ T7 )
      = T7 ) ).

% Un_Int_eq(2)
thf(fact_2868_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T7 ) @ T7 )
      = T7 ) ).

% Un_Int_eq(2)
thf(fact_2869_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T7 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_2870_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ S @ T7 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_2871_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T7 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_2872_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T7 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_2873_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_2874_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_2875_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_2876_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_2877_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_2878_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_2879_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_2880_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_2881_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_2882_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_2883_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_2884_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_2885_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_2886_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_2887_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_2888_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_2889_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_2890_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_2891_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_2892_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_2893_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_2894_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_2895_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_2896_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_2897_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_2898_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_2899_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_2900_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_2901_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_2902_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_2903_Int__left__commute,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) )
      = ( inf_in2572325071724192079at_nat @ B5 @ ( inf_in2572325071724192079at_nat @ A4 @ C4 ) ) ) ).

% Int_left_commute
thf(fact_2904_Int__left__commute,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ B5 @ C4 ) )
      = ( inf_inf_set_nat @ B5 @ ( inf_inf_set_nat @ A4 @ C4 ) ) ) ).

% Int_left_commute
thf(fact_2905_Int__left__absorb,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
      = ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ).

% Int_left_absorb
thf(fact_2906_Int__left__absorb,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ A4 @ B5 ) )
      = ( inf_inf_set_nat @ A4 @ B5 ) ) ).

% Int_left_absorb
thf(fact_2907_Int__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B6 @ A6 ) ) ) ).

% Int_commute
thf(fact_2908_Int__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] : ( inf_inf_set_nat @ B6 @ A6 ) ) ) ).

% Int_commute
thf(fact_2909_Int__absorb,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ A4 )
      = A4 ) ).

% Int_absorb
thf(fact_2910_Int__absorb,axiom,
    ! [A4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ A4 )
      = A4 ) ).

% Int_absorb
thf(fact_2911_Int__assoc,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ C4 )
      = ( inf_in2572325071724192079at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) ) ) ).

% Int_assoc
thf(fact_2912_Int__assoc,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ C4 )
      = ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ B5 @ C4 ) ) ) ).

% Int_assoc
thf(fact_2913_IntD2,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( inf_inf_set_o @ A4 @ B5 ) )
     => ( member_o @ C2 @ B5 ) ) ).

% IntD2
thf(fact_2914_IntD2,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( inf_in8396524679539076455nt_int @ A4 @ B5 ) )
     => ( member2340774599025711042nt_int @ C2 @ B5 ) ) ).

% IntD2
thf(fact_2915_IntD2,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A4 @ B5 ) )
     => ( member_set_nat @ C2 @ B5 ) ) ).

% IntD2
thf(fact_2916_IntD2,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( inf_inf_set_int @ A4 @ B5 ) )
     => ( member_int @ C2 @ B5 ) ) ).

% IntD2
thf(fact_2917_IntD2,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
     => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ).

% IntD2
thf(fact_2918_IntD2,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B5 ) )
     => ( member_nat @ C2 @ B5 ) ) ).

% IntD2
thf(fact_2919_IntD1,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( inf_inf_set_o @ A4 @ B5 ) )
     => ( member_o @ C2 @ A4 ) ) ).

% IntD1
thf(fact_2920_IntD1,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( inf_in8396524679539076455nt_int @ A4 @ B5 ) )
     => ( member2340774599025711042nt_int @ C2 @ A4 ) ) ).

% IntD1
thf(fact_2921_IntD1,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A4 @ B5 ) )
     => ( member_set_nat @ C2 @ A4 ) ) ).

% IntD1
thf(fact_2922_IntD1,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( inf_inf_set_int @ A4 @ B5 ) )
     => ( member_int @ C2 @ A4 ) ) ).

% IntD1
thf(fact_2923_IntD1,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
     => ( member8440522571783428010at_nat @ C2 @ A4 ) ) ).

% IntD1
thf(fact_2924_IntD1,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B5 ) )
     => ( member_nat @ C2 @ A4 ) ) ).

% IntD1
thf(fact_2925_IntE,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( inf_inf_set_o @ A4 @ B5 ) )
     => ~ ( ( member_o @ C2 @ A4 )
         => ~ ( member_o @ C2 @ B5 ) ) ) ).

% IntE
thf(fact_2926_IntE,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( inf_in8396524679539076455nt_int @ A4 @ B5 ) )
     => ~ ( ( member2340774599025711042nt_int @ C2 @ A4 )
         => ~ ( member2340774599025711042nt_int @ C2 @ B5 ) ) ) ).

% IntE
thf(fact_2927_IntE,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( inf_inf_set_set_nat @ A4 @ B5 ) )
     => ~ ( ( member_set_nat @ C2 @ A4 )
         => ~ ( member_set_nat @ C2 @ B5 ) ) ) ).

% IntE
thf(fact_2928_IntE,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( inf_inf_set_int @ A4 @ B5 ) )
     => ~ ( ( member_int @ C2 @ A4 )
         => ~ ( member_int @ C2 @ B5 ) ) ) ).

% IntE
thf(fact_2929_IntE,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ C2 @ A4 )
         => ~ ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% IntE
thf(fact_2930_IntE,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B5 ) )
     => ~ ( ( member_nat @ C2 @ A4 )
         => ~ ( member_nat @ C2 @ B5 ) ) ) ).

% IntE
thf(fact_2931_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_2932_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_2933_Int__def,axiom,
    ( inf_in8396524679539076455nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] :
              ( ( member2340774599025711042nt_int @ X4 @ A6 )
              & ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_2934_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_2935_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_2936_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_2937_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_2938_Int__Collect,axiom,
    ! [X: $o,A4: set_o,P2: $o > $o] :
      ( ( member_o @ X @ ( inf_inf_set_o @ A4 @ ( collect_o @ P2 ) ) )
      = ( ( member_o @ X @ A4 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_2939_Int__Collect,axiom,
    ! [X: list_nat,A4: set_list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ X @ ( inf_inf_set_list_nat @ A4 @ ( collect_list_nat @ P2 ) ) )
      = ( ( member_list_nat @ X @ A4 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_2940_Int__Collect,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( member2340774599025711042nt_int @ X @ ( inf_in8396524679539076455nt_int @ A4 @ ( collec5210948495886036740nt_int @ P2 ) ) )
      = ( ( member2340774599025711042nt_int @ X @ A4 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_2941_Int__Collect,axiom,
    ! [X: set_nat,A4: set_set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ X @ ( inf_inf_set_set_nat @ A4 @ ( collect_set_nat @ P2 ) ) )
      = ( ( member_set_nat @ X @ A4 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_2942_Int__Collect,axiom,
    ! [X: int,A4: set_int,P2: int > $o] :
      ( ( member_int @ X @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) )
      = ( ( member_int @ X @ A4 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_2943_Int__Collect,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ ( inf_in2572325071724192079at_nat @ A4 @ ( collec3392354462482085612at_nat @ P2 ) ) )
      = ( ( member8440522571783428010at_nat @ X @ A4 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_2944_Int__Collect,axiom,
    ! [X: nat,A4: set_nat,P2: nat > $o] :
      ( ( member_nat @ X @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) )
      = ( ( member_nat @ X @ A4 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_2945_Collect__conj__eq,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P2 @ X4 )
            & ( Q2 @ X4 ) ) )
      = ( inf_inf_set_list_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q2 ) ) ) ).

% Collect_conj_eq
thf(fact_2946_Collect__conj__eq,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ( collec5210948495886036740nt_int
        @ ^ [X4: set_Pr958786334691620121nt_int] :
            ( ( P2 @ X4 )
            & ( Q2 @ X4 ) ) )
      = ( inf_in8396524679539076455nt_int @ ( collec5210948495886036740nt_int @ P2 ) @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ).

% Collect_conj_eq
thf(fact_2947_Collect__conj__eq,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P2 @ X4 )
            & ( Q2 @ X4 ) ) )
      = ( inf_inf_set_set_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q2 ) ) ) ).

% Collect_conj_eq
thf(fact_2948_Collect__conj__eq,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( P2 @ X4 )
            & ( Q2 @ X4 ) ) )
      = ( inf_inf_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q2 ) ) ) ).

% Collect_conj_eq
thf(fact_2949_Collect__conj__eq,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( P2 @ X4 )
            & ( Q2 @ X4 ) ) )
      = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P2 ) @ ( collec3392354462482085612at_nat @ Q2 ) ) ) ).

% Collect_conj_eq
thf(fact_2950_Collect__conj__eq,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P2 @ X4 )
            & ( Q2 @ X4 ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_conj_eq
thf(fact_2951_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_2952_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_2953_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_2954_inf__sup__ord_I2_J,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_2955_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_2956_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_2957_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_2958_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_2959_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_2960_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_2961_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_2962_inf__sup__ord_I1_J,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_2963_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_2964_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_2965_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_2966_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_2967_inf__le1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2968_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2969_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_2970_inf__le1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2971_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_2972_inf__le1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2973_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2974_inf__le1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2975_inf__le2,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2976_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2977_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_2978_inf__le2,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2979_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_2980_inf__le2,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2981_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2982_inf__le2,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2983_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_2984_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_2985_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_2986_le__infE,axiom,
    ! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ A @ B ) )
     => ~ ( ( ord_le8369615600986905444_int_o @ X @ A )
         => ~ ( ord_le8369615600986905444_int_o @ X @ B ) ) ) ).

% le_infE
thf(fact_2987_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_2988_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_2989_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_2990_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_2991_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_2992_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_2993_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_2994_le__infI,axiom,
    ! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ A )
     => ( ( ord_le8369615600986905444_int_o @ X @ B )
       => ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ) ).

% le_infI
thf(fact_2995_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_2996_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_2997_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_2998_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_2999_inf__mono,axiom,
    ! [A: assn,C2: assn,B: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A @ C2 )
     => ( ( ord_less_eq_assn @ B @ D2 )
       => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ ( inf_inf_assn @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_3000_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C2 )
     => ( ( ord_le3146513528884898305at_nat @ B @ D2 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_3001_inf__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_3002_inf__mono,axiom,
    ! [A: product_prod_int_int > $o,C2: product_prod_int_int > $o,B: product_prod_int_int > $o,D2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ C2 )
     => ( ( ord_le8369615600986905444_int_o @ B @ D2 )
       => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ ( inf_in3604695632404883862_int_o @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_3003_inf__mono,axiom,
    ! [A: set_int,C2: set_int,B: set_int,D2: set_int] :
      ( ( ord_less_eq_set_int @ A @ C2 )
     => ( ( ord_less_eq_set_int @ B @ D2 )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ ( inf_inf_set_int @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_3004_inf__mono,axiom,
    ! [A: rat,C2: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ C2 )
     => ( ( ord_less_eq_rat @ B @ D2 )
       => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ ( inf_inf_rat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_3005_inf__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_3006_inf__mono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_3007_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_3008_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_3009_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_3010_le__infI1,axiom,
    ! [A: product_prod_int_int > $o,X: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ X )
     => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_3011_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_3012_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_3013_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_3014_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_3015_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_3016_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_3017_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_3018_le__infI2,axiom,
    ! [B: product_prod_int_int > $o,X: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ B @ X )
     => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_3019_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_3020_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_3021_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_3022_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_3023_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( A
        = ( inf_inf_assn @ A @ B ) ) ) ).

% inf.orderE
thf(fact_3024_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_3025_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_3026_inf_OorderE,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ B )
     => ( A
        = ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ).

% inf.orderE
thf(fact_3027_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_3028_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( A
        = ( inf_inf_rat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_3029_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_3030_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( inf_inf_int @ A @ B ) ) ) ).

% inf.orderE
thf(fact_3031_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A
        = ( inf_inf_assn @ A @ B ) )
     => ( ord_less_eq_assn @ A @ B ) ) ).

% inf.orderI
thf(fact_3032_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_3033_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_3034_inf_OorderI,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( A
        = ( inf_in3604695632404883862_int_o @ A @ B ) )
     => ( ord_le8369615600986905444_int_o @ A @ B ) ) ).

% inf.orderI
thf(fact_3035_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_3036_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( inf_inf_rat @ A @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% inf.orderI
thf(fact_3037_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_3038_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( inf_inf_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% inf.orderI
thf(fact_3039_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_3040_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_3041_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_3042_inf__unique,axiom,
    ! [F: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o,X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
              ( ( ord_le8369615600986905444_int_o @ X3 @ Y3 )
             => ( ( ord_le8369615600986905444_int_o @ X3 @ Z2 )
               => ( ord_le8369615600986905444_int_o @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
         => ( ( inf_in3604695632404883862_int_o @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_3043_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_3044_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_3045_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_3046_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_3047_le__iff__inf,axiom,
    ( ord_less_eq_assn
    = ( ^ [X4: assn,Y4: assn] :
          ( ( inf_inf_assn @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_3048_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_3049_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_3050_le__iff__inf,axiom,
    ( ord_le8369615600986905444_int_o
    = ( ^ [X4: product_prod_int_int > $o,Y4: product_prod_int_int > $o] :
          ( ( inf_in3604695632404883862_int_o @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_3051_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_3052_le__iff__inf,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( inf_inf_rat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_3053_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( inf_inf_nat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_3054_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( inf_inf_int @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_3055_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( inf_inf_assn @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_3056_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_3057_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_3058_inf_Oabsorb1,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ B )
     => ( ( inf_in3604695632404883862_int_o @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_3059_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_3060_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_3061_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_3062_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_3063_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( inf_inf_assn @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_3064_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_3065_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_3066_inf_Oabsorb2,axiom,
    ! [B: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ B @ A )
     => ( ( inf_in3604695632404883862_int_o @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_3067_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_3068_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_3069_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_3070_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_3071_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( inf_inf_assn @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_3072_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_3073_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_3074_inf__absorb1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ Y )
     => ( ( inf_in3604695632404883862_int_o @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_3075_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_3076_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( inf_inf_rat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_3077_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_3078_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( inf_inf_int @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_3079_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( inf_inf_assn @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_3080_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_3081_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_3082_inf__absorb2,axiom,
    ! [Y: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ Y @ X )
     => ( ( inf_in3604695632404883862_int_o @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_3083_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_3084_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( inf_inf_rat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_3085_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_3086_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( inf_inf_int @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_3087_inf_OboundedE,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C2 ) )
     => ~ ( ( ord_less_eq_assn @ A @ B )
         => ~ ( ord_less_eq_assn @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_3088_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ A @ B )
         => ~ ( ord_le3146513528884898305at_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_3089_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ~ ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_3090_inf_OboundedE,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C2 ) )
     => ~ ( ( ord_le8369615600986905444_int_o @ A @ B )
         => ~ ( ord_le8369615600986905444_int_o @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_3091_inf_OboundedE,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( inf_inf_set_int @ B @ C2 ) )
     => ~ ( ( ord_less_eq_set_int @ A @ B )
         => ~ ( ord_less_eq_set_int @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_3092_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_rat @ A @ B )
         => ~ ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_3093_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_3094_inf_OboundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_3095_inf_OboundedI,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_assn @ A @ C2 )
       => ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_3096_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( ord_le3146513528884898305at_nat @ A @ C2 )
       => ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_3097_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ A @ C2 )
       => ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_3098_inf_OboundedI,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ B )
     => ( ( ord_le8369615600986905444_int_o @ A @ C2 )
       => ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_3099_inf_OboundedI,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ A @ C2 )
       => ( ord_less_eq_set_int @ A @ ( inf_inf_set_int @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_3100_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ A @ C2 )
       => ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_3101_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C2 )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_3102_inf_OboundedI,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C2 )
       => ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_3103_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Z3 )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_3104_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( ord_le3146513528884898305at_nat @ X @ Z3 )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_3105_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Z3 )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_3106_inf__greatest,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ Y )
     => ( ( ord_le8369615600986905444_int_o @ X @ Z3 )
       => ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_3107_inf__greatest,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ X @ Z3 )
       => ( ord_less_eq_set_int @ X @ ( inf_inf_set_int @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_3108_inf__greatest,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Z3 )
       => ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_3109_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z3 )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_3110_inf__greatest,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Z3 )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_3111_inf_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [A5: assn,B4: assn] :
          ( A5
          = ( inf_inf_assn @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_3112_inf_Oorder__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( A5
          = ( inf_in2572325071724192079at_nat @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_3113_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( A5
          = ( inf_inf_set_nat @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_3114_inf_Oorder__iff,axiom,
    ( ord_le8369615600986905444_int_o
    = ( ^ [A5: product_prod_int_int > $o,B4: product_prod_int_int > $o] :
          ( A5
          = ( inf_in3604695632404883862_int_o @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_3115_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( A5
          = ( inf_inf_set_int @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_3116_inf_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( A5
          = ( inf_inf_rat @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_3117_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( A5
          = ( inf_inf_nat @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_3118_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( A5
          = ( inf_inf_int @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_3119_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_3120_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_3121_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_3122_inf_Ocobounded1,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_3123_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_3124_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_3125_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_3126_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_3127_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_3128_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_3129_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_3130_inf_Ocobounded2,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_3131_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_3132_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_3133_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_3134_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_3135_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [A5: assn,B4: assn] :
          ( ( inf_inf_assn @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_3136_inf_Oabsorb__iff1,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_3137_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( inf_inf_set_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_3138_inf_Oabsorb__iff1,axiom,
    ( ord_le8369615600986905444_int_o
    = ( ^ [A5: product_prod_int_int > $o,B4: product_prod_int_int > $o] :
          ( ( inf_in3604695632404883862_int_o @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_3139_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( inf_inf_set_int @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_3140_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( inf_inf_rat @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_3141_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( inf_inf_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_3142_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( inf_inf_int @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_3143_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [B4: assn,A5: assn] :
          ( ( inf_inf_assn @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_3144_inf_Oabsorb__iff2,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [B4: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_3145_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( ( inf_inf_set_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_3146_inf_Oabsorb__iff2,axiom,
    ( ord_le8369615600986905444_int_o
    = ( ^ [B4: product_prod_int_int > $o,A5: product_prod_int_int > $o] :
          ( ( inf_in3604695632404883862_int_o @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_3147_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( ( inf_inf_set_int @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_3148_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( inf_inf_rat @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_3149_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( inf_inf_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_3150_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( inf_inf_int @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_3151_inf_OcoboundedI1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ C2 )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_3152_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C2 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_3153_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_3154_inf_OcoboundedI1,axiom,
    ! [A: product_prod_int_int > $o,C2: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ C2 )
     => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_3155_inf_OcoboundedI1,axiom,
    ! [A: set_int,C2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ C2 )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_3156_inf_OcoboundedI1,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ C2 )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_3157_inf_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_3158_inf_OcoboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_3159_inf_OcoboundedI2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ C2 )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_3160_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ C2 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_3161_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_3162_inf_OcoboundedI2,axiom,
    ! [B: product_prod_int_int > $o,C2: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ B @ C2 )
     => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_3163_inf_OcoboundedI2,axiom,
    ! [B: set_int,C2: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ C2 )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_3164_inf_OcoboundedI2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ C2 )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_3165_inf_OcoboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_3166_inf_OcoboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ B @ C2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_3167_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_3168_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_3169_less__infI1,axiom,
    ! [A: product_prod_int_int > $o,X: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ A @ X )
     => ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_3170_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_3171_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_3172_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_3173_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_3174_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_3175_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_3176_less__infI2,axiom,
    ! [B: product_prod_int_int > $o,X: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ B @ X )
     => ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_3177_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_3178_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_3179_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_3180_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_3181_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_3182_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_3183_inf_Oabsorb3,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ A @ B )
     => ( ( inf_in3604695632404883862_int_o @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_3184_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( inf_inf_assn @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_3185_inf_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_3186_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_3187_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_3188_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_3189_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_3190_inf_Oabsorb4,axiom,
    ! [B: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ B @ A )
     => ( ( inf_in3604695632404883862_int_o @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_3191_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( inf_inf_assn @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_3192_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_3193_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_3194_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_3195_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
     => ~ ( ( ord_le7866589430770878221at_nat @ A @ B )
         => ~ ( ord_le7866589430770878221at_nat @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_3196_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
     => ~ ( ( ord_less_set_nat @ A @ B )
         => ~ ( ord_less_set_nat @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_3197_inf_Ostrict__boundedE,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C2: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C2 ) )
     => ~ ( ( ord_le8213806771718485336_int_o @ A @ B )
         => ~ ( ord_le8213806771718485336_int_o @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_3198_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ ( inf_inf_assn @ B @ C2 ) )
     => ~ ( ( ord_less_assn @ A @ B )
         => ~ ( ord_less_assn @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_3199_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( inf_inf_rat @ B @ C2 ) )
     => ~ ( ( ord_less_rat @ A @ B )
         => ~ ( ord_less_rat @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_3200_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_3201_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( inf_inf_int @ B @ C2 ) )
     => ~ ( ( ord_less_int @ A @ B )
         => ~ ( ord_less_int @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_3202_inf_Ostrict__order__iff,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( ( A5
            = ( inf_in2572325071724192079at_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_3203_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( A5
            = ( inf_inf_set_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_3204_inf_Ostrict__order__iff,axiom,
    ( ord_le8213806771718485336_int_o
    = ( ^ [A5: product_prod_int_int > $o,B4: product_prod_int_int > $o] :
          ( ( A5
            = ( inf_in3604695632404883862_int_o @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_3205_inf_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [A5: assn,B4: assn] :
          ( ( A5
            = ( inf_inf_assn @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_3206_inf_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( A5
            = ( inf_inf_rat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_3207_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( A5
            = ( inf_inf_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_3208_inf_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( A5
            = ( inf_inf_int @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_3209_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ C2 )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_3210_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ C2 )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_3211_inf_Ostrict__coboundedI1,axiom,
    ! [A: product_prod_int_int > $o,C2: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ A @ C2 )
     => ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_3212_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( ( ord_less_assn @ A @ C2 )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_3213_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ A @ C2 )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_3214_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ A @ C2 )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_3215_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ A @ C2 )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_3216_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ C2 )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_3217_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ C2 )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_3218_inf_Ostrict__coboundedI2,axiom,
    ! [B: product_prod_int_int > $o,C2: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ B @ C2 )
     => ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_3219_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( ord_less_assn @ B @ C2 )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_3220_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_rat @ B @ C2 )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_3221_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ B @ C2 )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_3222_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_int @ B @ C2 )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_3223_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z3: 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 @ Z3 ) )
        = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z3 ) ) ) ) ).

% distrib_imp1
thf(fact_3224_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: 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 @ Z3 ) )
        = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp1
thf(fact_3225_distrib__imp1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
          ( ( inf_in3604695632404883862_int_o @ X3 @ ( sup_su8463660629351352368_int_o @ Y3 @ Z2 ) )
          = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X3 @ Y3 ) @ ( inf_in3604695632404883862_int_o @ X3 @ Z2 ) ) )
     => ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z3 ) )
        = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z3 ) ) ) ) ).

% distrib_imp1
thf(fact_3226_distrib__imp1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: 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 @ Z3 ) )
        = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp1
thf(fact_3227_distrib__imp1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: 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 @ Z3 ) )
        = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp1
thf(fact_3228_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: 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 @ Z3 ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp1
thf(fact_3229_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z3: 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 @ Z3 ) )
        = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp1
thf(fact_3230_distrib__imp1,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ! [X3: int,Y3: int,Z2: int] :
          ( ( inf_inf_int @ X3 @ ( sup_sup_int @ Y3 @ Z2 ) )
          = ( sup_sup_int @ ( inf_inf_int @ X3 @ Y3 ) @ ( inf_inf_int @ X3 @ Z2 ) ) )
     => ( ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z3 ) )
        = ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z3 ) ) ) ) ).

% distrib_imp1
thf(fact_3231_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z3: 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 @ Z3 ) )
        = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z3 ) ) ) ) ).

% distrib_imp2
thf(fact_3232_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: 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 @ Z3 ) )
        = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp2
thf(fact_3233_distrib__imp2,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
          ( ( sup_su8463660629351352368_int_o @ X3 @ ( inf_in3604695632404883862_int_o @ Y3 @ Z2 ) )
          = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X3 @ Y3 ) @ ( sup_su8463660629351352368_int_o @ X3 @ Z2 ) ) )
     => ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z3 ) )
        = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z3 ) ) ) ) ).

% distrib_imp2
thf(fact_3234_distrib__imp2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: 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 @ Z3 ) )
        = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp2
thf(fact_3235_distrib__imp2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: 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 @ Z3 ) )
        = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp2
thf(fact_3236_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z3: 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 @ Z3 ) )
        = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp2
thf(fact_3237_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z3: 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 @ Z3 ) )
        = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z3 ) ) ) ) ).

% distrib_imp2
thf(fact_3238_distrib__imp2,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ! [X3: int,Y3: int,Z2: int] :
          ( ( sup_sup_int @ X3 @ ( inf_inf_int @ Y3 @ Z2 ) )
          = ( inf_inf_int @ ( sup_sup_int @ X3 @ Y3 ) @ ( sup_sup_int @ X3 @ Z2 ) ) )
     => ( ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z3 ) )
        = ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z3 ) ) ) ) ).

% distrib_imp2
thf(fact_3239_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z3 ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z3 ) ) ) ).

% inf_sup_distrib1
thf(fact_3240_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z3 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z3 ) ) ) ).

% inf_sup_distrib1
thf(fact_3241_inf__sup__distrib1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z3 ) )
      = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z3 ) ) ) ).

% inf_sup_distrib1
thf(fact_3242_inf__sup__distrib1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z3 ) ) ) ).

% inf_sup_distrib1
thf(fact_3243_inf__sup__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z3 ) ) ) ).

% inf_sup_distrib1
thf(fact_3244_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z3 ) ) ) ).

% inf_sup_distrib1
thf(fact_3245_inf__sup__distrib1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) )
      = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z3 ) ) ) ).

% inf_sup_distrib1
thf(fact_3246_inf__sup__distrib1,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z3 ) )
      = ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z3 ) ) ) ).

% inf_sup_distrib1
thf(fact_3247_inf__sup__distrib2,axiom,
    ! [Y: assn,Z3: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z3 ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z3 @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3248_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z3 ) @ X )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z3 @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3249_inf__sup__distrib2,axiom,
    ! [Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ Z3 ) @ X )
      = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ X ) @ ( inf_in3604695632404883862_int_o @ Z3 @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3250_inf__sup__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) @ X )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ X ) @ ( inf_in1697001100524423349at_nat @ Z3 @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3251_inf__sup__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) @ X )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ X ) @ ( inf_in7913087082777306421at_nat @ Z3 @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3252_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z3: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z3 ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z3 @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3253_inf__sup__distrib2,axiom,
    ! [Y: nat,Z3: nat,X: nat] :
      ( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X )
      = ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z3 @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3254_inf__sup__distrib2,axiom,
    ! [Y: int,Z3: int,X: int] :
      ( ( inf_inf_int @ ( sup_sup_int @ Y @ Z3 ) @ X )
      = ( sup_sup_int @ ( inf_inf_int @ Y @ X ) @ ( inf_inf_int @ Z3 @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_3255_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z3 ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z3 ) ) ) ).

% sup_inf_distrib1
thf(fact_3256_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z3 ) ) ) ).

% sup_inf_distrib1
thf(fact_3257_sup__inf__distrib1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z3 ) )
      = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z3 ) ) ) ).

% sup_inf_distrib1
thf(fact_3258_sup__inf__distrib1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z3 ) )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z3 ) ) ) ).

% sup_inf_distrib1
thf(fact_3259_sup__inf__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z3 ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z3 ) ) ) ).

% sup_inf_distrib1
thf(fact_3260_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ).

% sup_inf_distrib1
thf(fact_3261_sup__inf__distrib1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) )
      = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z3 ) ) ) ).

% sup_inf_distrib1
thf(fact_3262_sup__inf__distrib1,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z3 ) )
      = ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z3 ) ) ) ).

% sup_inf_distrib1
thf(fact_3263_sup__inf__distrib2,axiom,
    ! [Y: assn,Z3: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z3 ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z3 @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3264_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) @ X )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z3 @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3265_sup__inf__distrib2,axiom,
    ! [Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ Z3 ) @ X )
      = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ X ) @ ( sup_su8463660629351352368_int_o @ Z3 @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3266_sup__inf__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ Z3 ) @ X )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ X ) @ ( sup_su3035147773818789531at_nat @ Z3 @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3267_sup__inf__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ Z3 ) @ X )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ X ) @ ( sup_su5525570899277871387at_nat @ Z3 @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3268_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z3: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z3 ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z3 @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3269_sup__inf__distrib2,axiom,
    ! [Y: nat,Z3: nat,X: nat] :
      ( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z3 ) @ X )
      = ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z3 @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3270_sup__inf__distrib2,axiom,
    ! [Y: int,Z3: int,X: int] :
      ( ( sup_sup_int @ ( inf_inf_int @ Y @ Z3 ) @ X )
      = ( inf_inf_int @ ( sup_sup_int @ Y @ X ) @ ( sup_sup_int @ Z3 @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_3271_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z3 ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z3 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_3272_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z3 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z3 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_3273_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z3 ) )
      = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z3 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_3274_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z3 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_3275_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z3 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_3276_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z3 ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_3277_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z3 ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z3 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_3278_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z3 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_3279_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z3 ) )
      = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z3 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_3280_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z3 ) )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z3 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_3281_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z3 ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z3 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_3282_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_3283_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z3: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z3 ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z3 @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_3284_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z3 ) @ X )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z3 @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_3285_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ Z3 ) @ X )
      = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ X ) @ ( inf_in3604695632404883862_int_o @ Z3 @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_3286_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) @ X )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ X ) @ ( inf_in1697001100524423349at_nat @ Z3 @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_3287_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) @ X )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ X ) @ ( inf_in7913087082777306421at_nat @ Z3 @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_3288_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z3: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z3 ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z3 @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_3289_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z3: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z3 ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z3 @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_3290_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) @ X )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z3 @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_3291_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ Z3 ) @ X )
      = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ X ) @ ( sup_su8463660629351352368_int_o @ Z3 @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_3292_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat,X: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ Y @ Z3 ) @ X )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ Y @ X ) @ ( sup_su3035147773818789531at_nat @ Z3 @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_3293_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ Z3 ) @ X )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ X ) @ ( sup_su5525570899277871387at_nat @ Z3 @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_3294_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z3: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z3 ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z3 @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_3295_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ).

% add_divide_distrib
thf(fact_3296_disjointI,axiom,
    ! [A: set_se6260736226359567993nt_int,B: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ X3 @ A )
         => ~ ( member2340774599025711042nt_int @ X3 @ B ) )
     => ( ( inf_in8396524679539076455nt_int @ A @ B )
        = bot_bo1488462491386950373nt_int ) ) ).

% disjointI
thf(fact_3297_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_3298_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_3299_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_3300_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_3301_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_3302_disjoint__iff__not__equal,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A4 )
           => ! [Y4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y4 @ B5 )
               => ( X4 != Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_3303_disjoint__iff__not__equal,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ A4 @ B5 )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ A4 )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ B5 )
               => ( X4 = ~ Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_3304_disjoint__iff__not__equal,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( inf_inf_set_nat @ A4 @ B5 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ B5 )
               => ( X4 != Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_3305_disjoint__iff__not__equal,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( inf_inf_set_int @ A4 @ B5 )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A4 )
           => ! [Y4: int] :
                ( ( member_int @ Y4 @ B5 )
               => ( X4 != Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_3306_Int__empty__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_3307_Int__empty__right,axiom,
    ! [A4: set_o] :
      ( ( inf_inf_set_o @ A4 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_3308_Int__empty__right,axiom,
    ! [A4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_3309_Int__empty__right,axiom,
    ! [A4: set_int] :
      ( ( inf_inf_set_int @ A4 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% Int_empty_right
thf(fact_3310_Int__empty__left,axiom,
    ! [B5: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B5 )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_3311_Int__empty__left,axiom,
    ! [B5: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B5 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_3312_Int__empty__left,axiom,
    ! [B5: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B5 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_3313_Int__empty__left,axiom,
    ! [B5: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ B5 )
      = bot_bot_set_int ) ).

% Int_empty_left
thf(fact_3314_disjoint__iff,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( ( inf_in8396524679539076455nt_int @ A4 @ B5 )
        = bot_bo1488462491386950373nt_int )
      = ( ! [X4: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X4 @ A4 )
           => ~ ( member2340774599025711042nt_int @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_3315_disjoint__iff,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ A4 @ B5 )
        = bot_bot_set_set_nat )
      = ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
           => ~ ( member_set_nat @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_3316_disjoint__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A4 )
           => ~ ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_3317_disjoint__iff,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ A4 @ B5 )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ A4 )
           => ~ ( member_o @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_3318_disjoint__iff,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( inf_inf_set_nat @ A4 @ B5 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ~ ( member_nat @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_3319_disjoint__iff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( inf_inf_set_int @ A4 @ B5 )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A4 )
           => ~ ( member_int @ X4 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_3320_Int__emptyI,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ X3 @ A4 )
         => ~ ( member2340774599025711042nt_int @ X3 @ B5 ) )
     => ( ( inf_in8396524679539076455nt_int @ A4 @ B5 )
        = bot_bo1488462491386950373nt_int ) ) ).

% Int_emptyI
thf(fact_3321_Int__emptyI,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A4 )
         => ~ ( member_set_nat @ X3 @ B5 ) )
     => ( ( inf_inf_set_set_nat @ A4 @ B5 )
        = bot_bot_set_set_nat ) ) ).

% Int_emptyI
thf(fact_3322_Int__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ~ ( member8440522571783428010at_nat @ X3 @ B5 ) )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_3323_Int__emptyI,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ~ ( member_o @ X3 @ B5 ) )
     => ( ( inf_inf_set_o @ A4 @ B5 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_3324_Int__emptyI,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ~ ( member_nat @ X3 @ B5 ) )
     => ( ( inf_inf_set_nat @ A4 @ B5 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_3325_Int__emptyI,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ~ ( member_int @ X3 @ B5 ) )
     => ( ( inf_inf_set_int @ A4 @ B5 )
        = bot_bot_set_int ) ) ).

% Int_emptyI
thf(fact_3326_Int__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ C4 )
     => ( ( ord_le3146513528884898305at_nat @ B5 @ D4 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ ( inf_in2572325071724192079at_nat @ C4 @ D4 ) ) ) ) ).

% Int_mono
thf(fact_3327_Int__mono,axiom,
    ! [A4: set_nat,C4: set_nat,B5: set_nat,D4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C4 )
     => ( ( ord_less_eq_set_nat @ B5 @ D4 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ ( inf_inf_set_nat @ C4 @ D4 ) ) ) ) ).

% Int_mono
thf(fact_3328_Int__mono,axiom,
    ! [A4: set_int,C4: set_int,B5: set_int,D4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ C4 )
     => ( ( ord_less_eq_set_int @ B5 @ D4 )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A4 @ B5 ) @ ( inf_inf_set_int @ C4 @ D4 ) ) ) ) ).

% Int_mono
thf(fact_3329_Int__lower1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ A4 ) ).

% Int_lower1
thf(fact_3330_Int__lower1,axiom,
    ! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ A4 ) ).

% Int_lower1
thf(fact_3331_Int__lower1,axiom,
    ! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A4 @ B5 ) @ A4 ) ).

% Int_lower1
thf(fact_3332_Int__lower2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ B5 ) ).

% Int_lower2
thf(fact_3333_Int__lower2,axiom,
    ! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ B5 ) ).

% Int_lower2
thf(fact_3334_Int__lower2,axiom,
    ! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A4 @ B5 ) @ B5 ) ).

% Int_lower2
thf(fact_3335_Int__absorb1,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
        = B5 ) ) ).

% Int_absorb1
thf(fact_3336_Int__absorb1,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( inf_inf_set_nat @ A4 @ B5 )
        = B5 ) ) ).

% Int_absorb1
thf(fact_3337_Int__absorb1,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( inf_inf_set_int @ A4 @ B5 )
        = B5 ) ) ).

% Int_absorb1
thf(fact_3338_Int__absorb2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
        = A4 ) ) ).

% Int_absorb2
thf(fact_3339_Int__absorb2,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( inf_inf_set_nat @ A4 @ B5 )
        = A4 ) ) ).

% Int_absorb2
thf(fact_3340_Int__absorb2,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( inf_inf_set_int @ A4 @ B5 )
        = A4 ) ) ).

% Int_absorb2
thf(fact_3341_Int__greatest,axiom,
    ! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C4 @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ C4 @ B5 )
       => ( ord_le3146513528884898305at_nat @ C4 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).

% Int_greatest
thf(fact_3342_Int__greatest,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ C4 @ A4 )
     => ( ( ord_less_eq_set_nat @ C4 @ B5 )
       => ( ord_less_eq_set_nat @ C4 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).

% Int_greatest
thf(fact_3343_Int__greatest,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ C4 @ A4 )
     => ( ( ord_less_eq_set_int @ C4 @ B5 )
       => ( ord_less_eq_set_int @ C4 @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ).

% Int_greatest
thf(fact_3344_Int__Collect__mono,axiom,
    ! [A4: set_o,B5: set_o,P2: $o > $o,Q2: $o > $o] :
      ( ( ord_less_eq_set_o @ A4 @ B5 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A4 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A4 @ ( collect_o @ P2 ) ) @ ( inf_inf_set_o @ B5 @ ( collect_o @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_3345_Int__Collect__mono,axiom,
    ! [A4: set_list_nat,B5: set_list_nat,P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ A4 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A4 @ ( collect_list_nat @ P2 ) ) @ ( inf_inf_set_list_nat @ B5 @ ( collect_list_nat @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_3346_Int__Collect__mono,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int,P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ( ord_le483042692224249369nt_int @ A4 @ B5 )
     => ( ! [X3: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X3 @ A4 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_le483042692224249369nt_int @ ( inf_in8396524679539076455nt_int @ A4 @ ( collec5210948495886036740nt_int @ P2 ) ) @ ( inf_in8396524679539076455nt_int @ B5 @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_3347_Int__Collect__mono,axiom,
    ! [A4: set_set_nat,B5: set_set_nat,P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A4 @ ( collect_set_nat @ P2 ) ) @ ( inf_inf_set_set_nat @ B5 @ ( collect_set_nat @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_3348_Int__Collect__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ ( collec3392354462482085612at_nat @ P2 ) ) @ ( inf_in2572325071724192079at_nat @ B5 @ ( collec3392354462482085612at_nat @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_3349_Int__Collect__mono,axiom,
    ! [A4: set_nat,B5: set_nat,P2: nat > $o,Q2: nat > $o] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B5 @ ( collect_nat @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_3350_Int__Collect__mono,axiom,
    ! [A4: set_int,B5: set_int,P2: int > $o,Q2: int > $o] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) @ ( inf_inf_set_int @ B5 @ ( collect_int @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_3351_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S6: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ S6 )
     => ( ( ( inf_in2572325071724192079at_nat @ A4 @ S6 )
          = ( inf_in2572325071724192079at_nat @ B5 @ S6 ) )
       => ( ( inf_in2572325071724192079at_nat @ A4 @ S )
          = ( inf_in2572325071724192079at_nat @ B5 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_3352_inter__eq__subsetI,axiom,
    ! [S: set_nat,S6: set_nat,A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ S6 )
     => ( ( ( inf_inf_set_nat @ A4 @ S6 )
          = ( inf_inf_set_nat @ B5 @ S6 ) )
       => ( ( inf_inf_set_nat @ A4 @ S )
          = ( inf_inf_set_nat @ B5 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_3353_inter__eq__subsetI,axiom,
    ! [S: set_int,S6: set_int,A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ S @ S6 )
     => ( ( ( inf_inf_set_int @ A4 @ S6 )
          = ( inf_inf_set_int @ B5 @ S6 ) )
       => ( ( inf_inf_set_int @ A4 @ S )
          = ( inf_inf_set_int @ B5 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_3354_Un__Int__crazy,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) ) @ ( inf_in2572325071724192079at_nat @ C4 @ A4 ) )
      = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) @ ( sup_su6327502436637775413at_nat @ B5 @ C4 ) ) @ ( sup_su6327502436637775413at_nat @ C4 @ A4 ) ) ) ).

% Un_Int_crazy
thf(fact_3355_Un__Int__crazy,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) @ ( inf_in1697001100524423349at_nat @ B5 @ C4 ) ) @ ( inf_in1697001100524423349at_nat @ C4 @ A4 ) )
      = ( inf_in1697001100524423349at_nat @ ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) ) @ ( sup_su3035147773818789531at_nat @ C4 @ A4 ) ) ) ).

% Un_Int_crazy
thf(fact_3356_Un__Int__crazy,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) @ ( inf_in7913087082777306421at_nat @ B5 @ C4 ) ) @ ( inf_in7913087082777306421at_nat @ C4 @ A4 ) )
      = ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) ) @ ( sup_su5525570899277871387at_nat @ C4 @ A4 ) ) ) ).

% Un_Int_crazy
thf(fact_3357_Un__Int__crazy,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ ( inf_inf_set_nat @ B5 @ C4 ) ) @ ( inf_inf_set_nat @ C4 @ A4 ) )
      = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ ( sup_sup_set_nat @ B5 @ C4 ) ) @ ( sup_sup_set_nat @ C4 @ A4 ) ) ) ).

% Un_Int_crazy
thf(fact_3358_Int__Un__distrib,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( sup_su6327502436637775413at_nat @ B5 @ C4 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ ( inf_in2572325071724192079at_nat @ A4 @ C4 ) ) ) ).

% Int_Un_distrib
thf(fact_3359_Int__Un__distrib,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) @ ( inf_in1697001100524423349at_nat @ A4 @ C4 ) ) ) ).

% Int_Un_distrib
thf(fact_3360_Int__Un__distrib,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) @ ( inf_in7913087082777306421at_nat @ A4 @ C4 ) ) ) ).

% Int_Un_distrib
thf(fact_3361_Int__Un__distrib,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C4 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ ( inf_inf_set_nat @ A4 @ C4 ) ) ) ).

% Int_Un_distrib
thf(fact_3362_Un__Int__distrib,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) @ ( sup_su6327502436637775413at_nat @ A4 @ C4 ) ) ) ).

% Un_Int_distrib
thf(fact_3363_Un__Int__distrib,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A4 @ ( inf_in1697001100524423349at_nat @ B5 @ C4 ) )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) @ ( sup_su3035147773818789531at_nat @ A4 @ C4 ) ) ) ).

% Un_Int_distrib
thf(fact_3364_Un__Int__distrib,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( inf_in7913087082777306421at_nat @ B5 @ C4 ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) @ ( sup_su5525570899277871387at_nat @ A4 @ C4 ) ) ) ).

% Un_Int_distrib
thf(fact_3365_Un__Int__distrib,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( inf_inf_set_nat @ B5 @ C4 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ ( sup_sup_set_nat @ A4 @ C4 ) ) ) ).

% Un_Int_distrib
thf(fact_3366_Int__Un__distrib2,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B5 @ C4 ) @ A4 )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B5 @ A4 ) @ ( inf_in2572325071724192079at_nat @ C4 @ A4 ) ) ) ).

% Int_Un_distrib2
thf(fact_3367_Int__Un__distrib2,axiom,
    ! [B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) @ A4 )
      = ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ B5 @ A4 ) @ ( inf_in1697001100524423349at_nat @ C4 @ A4 ) ) ) ).

% Int_Un_distrib2
thf(fact_3368_Int__Un__distrib2,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) @ A4 )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B5 @ A4 ) @ ( inf_in7913087082777306421at_nat @ C4 @ A4 ) ) ) ).

% Int_Un_distrib2
thf(fact_3369_Int__Un__distrib2,axiom,
    ! [B5: set_nat,C4: set_nat,A4: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B5 @ C4 ) @ A4 )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B5 @ A4 ) @ ( inf_inf_set_nat @ C4 @ A4 ) ) ) ).

% Int_Un_distrib2
thf(fact_3370_Un__Int__distrib2,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) @ A4 )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B5 @ A4 ) @ ( sup_su6327502436637775413at_nat @ C4 @ A4 ) ) ) ).

% Un_Int_distrib2
thf(fact_3371_Un__Int__distrib2,axiom,
    ! [B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ B5 @ C4 ) @ A4 )
      = ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ B5 @ A4 ) @ ( sup_su3035147773818789531at_nat @ C4 @ A4 ) ) ) ).

% Un_Int_distrib2
thf(fact_3372_Un__Int__distrib2,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B5 @ C4 ) @ A4 )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B5 @ A4 ) @ ( sup_su5525570899277871387at_nat @ C4 @ A4 ) ) ) ).

% Un_Int_distrib2
thf(fact_3373_Un__Int__distrib2,axiom,
    ! [B5: set_nat,C4: set_nat,A4: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B5 @ C4 ) @ A4 )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B5 @ A4 ) @ ( sup_sup_set_nat @ C4 @ A4 ) ) ) ).

% Un_Int_distrib2
thf(fact_3374_inf__option__def,axiom,
    ( inf_inf_option_assn
    = ( ^ [X4: option_assn,Y4: option_assn] :
          ( case_o4484465799723439917n_assn @ none_assn
          @ ^ [Z: assn] :
              ( case_o4484465799723439917n_assn @ none_assn
              @ ^ [Aa: assn] : ( some_assn @ ( inf_inf_assn @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_3375_inf__option__def,axiom,
    ( inf_in777885744494410645at_nat
    = ( ^ [X4: option8963830502488799655at_nat,Y4: option8963830502488799655at_nat] :
          ( case_o311359030874850053at_nat @ none_s625347054029921090at_nat
          @ ^ [Z: set_Pr1261947904930325089at_nat] :
              ( case_o311359030874850053at_nat @ none_s625347054029921090at_nat
              @ ^ [Aa: set_Pr1261947904930325089at_nat] : ( some_s147305329494351046at_nat @ ( inf_in2572325071724192079at_nat @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_3376_inf__option__def,axiom,
    ( inf_in7812914138253463912et_nat
    = ( ^ [X4: option_set_nat,Y4: option_set_nat] :
          ( case_o4054078431260844265et_nat @ none_set_nat
          @ ^ [Z: set_nat] :
              ( case_o4054078431260844265et_nat @ none_set_nat
              @ ^ [Aa: set_nat] : ( some_set_nat @ ( inf_inf_set_nat @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_3377_inf__option__def,axiom,
    ( inf_inf_option_nat
    = ( ^ [X4: option_nat,Y4: option_nat] :
          ( case_o7429725398727453821at_nat @ none_nat
          @ ^ [Z: nat] :
              ( case_o7429725398727453821at_nat @ none_nat
              @ ^ [Aa: nat] : ( some_nat @ ( inf_inf_nat @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_3378_inf__option__def,axiom,
    ( inf_in7104746047340619750_int_o
    = ( ^ [X4: option1893999432384633940_int_o,Y4: option1893999432384633940_int_o] :
          ( case_o670596351732880613_int_o @ none_P2320557853698873699_int_o
          @ ^ [Z: product_prod_int_int > $o] :
              ( case_o670596351732880613_int_o @ none_P2320557853698873699_int_o
              @ ^ [Aa: product_prod_int_int > $o] : ( some_P180497116919641311_int_o @ ( inf_in3604695632404883862_int_o @ Z @ Aa ) )
              @ Y4 )
          @ X4 ) ) ) ).

% inf_option_def
thf(fact_3379_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z3: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z3 ) ) @ ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3380_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3381_distrib__sup__le,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z3 ) ) @ ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3382_distrib__sup__le,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ X @ ( inf_in1697001100524423349at_nat @ Y @ Z3 ) ) @ ( inf_in1697001100524423349at_nat @ ( sup_su3035147773818789531at_nat @ X @ Y ) @ ( sup_su3035147773818789531at_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3383_distrib__sup__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z3 ) ) @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3384_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3385_distrib__sup__le,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ X @ ( inf_inf_set_int @ Y @ Z3 ) ) @ ( inf_inf_set_int @ ( sup_sup_set_int @ X @ Y ) @ ( sup_sup_set_int @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3386_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z3: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X @ ( inf_inf_rat @ Y @ Z3 ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X @ Y ) @ ( sup_sup_rat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3387_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3388_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z3: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z3 ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_3389_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z3: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z3 ) ) @ ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3390_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z3 ) ) @ ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3391_distrib__inf__le,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z3 ) ) @ ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3392_distrib__inf__le,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] : ( ord_le8081472938463900775at_nat @ ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ ( inf_in1697001100524423349at_nat @ X @ Z3 ) ) @ ( inf_in1697001100524423349at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3393_distrib__inf__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z3 ) ) @ ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3394_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z3 ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3395_distrib__inf__le,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ ( inf_inf_set_int @ X @ Y ) @ ( inf_inf_set_int @ X @ Z3 ) ) @ ( inf_inf_set_int @ X @ ( sup_sup_set_int @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3396_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z3: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X @ Y ) @ ( inf_inf_rat @ X @ Z3 ) ) @ ( inf_inf_rat @ X @ ( sup_sup_rat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3397_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z3 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3398_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z3: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z3 ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_3399_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_3400_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_3401_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_3402_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_3403_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_3404_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_3405_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_3406_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_3407_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_3408_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_3409_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_3410_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_3411_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_3412_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_3413_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_3414_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_3415_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_3416_divide__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).

% divide_right_mono
thf(fact_3417_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_3418_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_3419_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_3420_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_3421_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_3422_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( divide_divide_rat @ A @ C2 ) ) ) ) ).

% divide_right_mono_neg
thf(fact_3423_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_3424_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_3425_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_3426_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_3427_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_3428_divide__less__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_3429_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_3430_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).

% divide_strict_right_mono
thf(fact_3431_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_3432_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_3433_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_3434_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_3435_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_3436_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_3437_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_3438_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_3439_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_3440_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_3441_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ C4 )
        = ( inf_in2572325071724192079at_nat @ A4 @ ( sup_su6327502436637775413at_nat @ B5 @ C4 ) ) )
      = ( ord_le3146513528884898305at_nat @ C4 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_3442_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) @ C4 )
        = ( inf_in1697001100524423349at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) ) )
      = ( ord_le8081472938463900775at_nat @ C4 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_3443_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) @ C4 )
        = ( inf_in7913087082777306421at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) ) )
      = ( ord_le1268244103169919719at_nat @ C4 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_3444_Un__Int__assoc__eq,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ C4 )
        = ( inf_inf_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C4 ) ) )
      = ( ord_less_eq_set_nat @ C4 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_3445_Un__Int__assoc__eq,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ( sup_sup_set_int @ ( inf_inf_set_int @ A4 @ B5 ) @ C4 )
        = ( inf_inf_set_int @ A4 @ ( sup_sup_set_int @ B5 @ C4 ) ) )
      = ( ord_less_eq_set_int @ C4 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_3446_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_3447_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_3448_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_3449_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_3450_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_3451_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_3452_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_3453_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_3454_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_3455_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_3456_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_3457_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_3458_divide__le__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_3459_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z3: 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 @ Z3 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z3 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_3460_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z3: 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 @ Z3 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z3 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_3461_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z3: 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 @ Z3 )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z3 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_3462_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_3463_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_3464_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_3465_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_3466_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_3467_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_3468_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_3469_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_3470_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_3471_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_3472_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_3473_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_3474_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_3475_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_3476_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C2 ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3477_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3478_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3479_divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3480_less__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3481_neg__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3482_neg__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3483_pos__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3484_pos__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3485_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z3 @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z3 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3486_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z3 @ Y ) @ X )
       => ( ord_less_rat @ Z3 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3487_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3488_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3489_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_3490_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_3491_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z3 ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z3 ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3492_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z3 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3493_add__frac__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z3 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z3 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3494_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3495_add__num__frac,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z3 @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3496_add__divide__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z3 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% add_divide_eq_iff
thf(fact_3497_divide__add__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z3 ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_add_eq_iff
thf(fact_3498_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_3499_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_3500_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_3501_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_3502_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_3503_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_3504_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_3505_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_3506_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_3507_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_3508_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_3509_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_3510_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
        = zero_z3403309356797280102nteger ) ) ).

% zero_power
thf(fact_3511_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_3512_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_3513_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_3514_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_3515_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_3516_nat__mult__div__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
        = ( divide_divide_nat @ M @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_3517_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_3518_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_3519_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic8660852769118194346_int_o @ inf_in3604695632404883862_int_o @ ord_le8369615600986905444_int_o @ ord_le8213806771718485336_int_o ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_3520_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_3521_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_3522_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_3523_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_3524_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_3525_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_3526_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_3527_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_3528_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_3529_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3530_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ Y ) @ X )
       => ( ord_less_eq_rat @ Z3 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3531_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z3 @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z3 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3532_pos__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3533_pos__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3534_neg__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3535_neg__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3536_divide__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3537_le__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3538_divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3539_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_3540_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_3541_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_3542_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_3543_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_3544_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_3545_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_3546_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_3547_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_3548_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_3549_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_3550_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_3551_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_3552_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_3553_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_3554_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_3555_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_3556_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_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_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_3564_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_3565_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_3566_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_3567_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_3568_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_3569_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C2 @ B ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C2 @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3570_div__mult__self1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3571_div__mult__self1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3572_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,C2: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C2 @ B ) ) @ B )
        = ( plus_p4538020629002901425atural @ C2 @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_3573_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C2 ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C2 @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3574_div__mult__self2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3575_div__mult__self2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3576_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,C2: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C2 ) ) @ B )
        = ( plus_p4538020629002901425atural @ C2 @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_3577_div__mult__self3,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C2 @ B ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C2 @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3578_div__mult__self3,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3579_div__mult__self3,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3580_div__mult__self3,axiom,
    ! [B: code_natural,C2: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C2 @ B ) @ A ) @ B )
        = ( plus_p4538020629002901425atural @ C2 @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_3581_div__mult__self4,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C2 ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C2 @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3582_div__mult__self4,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3583_div__mult__self4,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3584_div__mult__self4,axiom,
    ! [B: code_natural,C2: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C2 ) @ A ) @ B )
        = ( plus_p4538020629002901425atural @ C2 @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_3585_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_3586_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_3587_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_3588_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_3589_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_3590_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_3591_mod__h__bot__iff_I6_J,axiom,
    ! [P2: assn,Q2: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( inf_inf_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        & ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(6)
thf(fact_3592_inf__Int__eq2,axiom,
    ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( inf_inf_int_int_o
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: int,Y4: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ ( inf_in2269163501485487111nt_int @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_3593_inf__Int__eq2,axiom,
    ! [R3: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( inf_in3295504058751909687_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_3594_inf__Int__eq2,axiom,
    ! [R3: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( inf_in2641120393918057659_nat_o
        @ ^ [X4: produc3658429121746597890et_nat > $o,Y4: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_3595_inf__Int__eq2,axiom,
    ! [R3: set_Pr7600907837789447088it_nat,S: set_Pr7600907837789447088it_nat] :
      ( ( inf_in844129575377303227_nat_o
        @ ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: b,Y4: produc6653097349344004940it_nat] : ( member8682313912305727889it_nat @ ( produc4082563078715348724it_nat @ X4 @ Y4 ) @ ( inf_in3995795194662975554it_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_3596_inf__Int__eq2,axiom,
    ! [R3: set_Pr7098220151150636591it_nat,S: set_Pr7098220151150636591it_nat] :
      ( ( inf_in8051596381320575356_nat_o
        @ ^ [X4: a,Y4: produc6653097349344004940it_nat] : ( member4554412811331277712it_nat @ ( produc9178034014595674355it_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_3597_inf__Int__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_inf_nat_nat_o
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R3 )
        @ ^ [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 @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_3598_mod__and__dist,axiom,
    ! [P2: assn,Q2: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( inf_inf_assn @ P2 @ Q2 ) @ H )
      = ( ( rep_assn @ P2 @ H )
        & ( rep_assn @ Q2 @ H ) ) ) ).

% mod_and_dist
thf(fact_3599_ent__conjI,axiom,
    ! [A4: assn,B5: assn,C4: assn] :
      ( ( entails @ A4 @ B5 )
     => ( ( entails @ A4 @ C4 )
       => ( entails @ A4 @ ( inf_inf_assn @ B5 @ C4 ) ) ) ) ).

% ent_conjI
thf(fact_3600_ent__conjE1,axiom,
    ! [A4: assn,C4: assn,B5: assn] :
      ( ( entails @ A4 @ C4 )
     => ( entails @ ( inf_inf_assn @ A4 @ B5 ) @ C4 ) ) ).

% ent_conjE1
thf(fact_3601_ent__conjE2,axiom,
    ! [B5: assn,C4: assn,A4: assn] :
      ( ( entails @ B5 @ C4 )
     => ( entails @ ( inf_inf_assn @ A4 @ B5 ) @ C4 ) ) ).

% ent_conjE2
thf(fact_3602_inf__Int__eq,axiom,
    ! [R3: set_o,S: set_o] :
      ( ( inf_inf_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R3 )
        @ ^ [X4: $o] : ( member_o @ X4 @ S ) )
      = ( ^ [X4: $o] : ( member_o @ X4 @ ( inf_inf_set_o @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_3603_inf__Int__eq,axiom,
    ! [R3: set_se6260736226359567993nt_int,S: set_se6260736226359567993nt_int] :
      ( ( inf_in5102985939729578038_int_o
        @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ R3 )
        @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ S ) )
      = ( ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ ( inf_in8396524679539076455nt_int @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_3604_inf__Int__eq,axiom,
    ! [R3: set_set_nat,S: set_set_nat] :
      ( ( inf_inf_set_nat_o
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ R3 )
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ S ) )
      = ( ^ [X4: set_nat] : ( member_set_nat @ X4 @ ( inf_inf_set_set_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_3605_inf__Int__eq,axiom,
    ! [R3: set_int,S: set_int] :
      ( ( inf_inf_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R3 )
        @ ^ [X4: int] : ( member_int @ X4 @ S ) )
      = ( ^ [X4: int] : ( member_int @ X4 @ ( inf_inf_set_int @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_3606_inf__Int__eq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in5163264567034779214_nat_o
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R3 )
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S ) )
      = ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( inf_in2572325071724192079at_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_3607_inf__Int__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( inf_inf_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R3 )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
      = ( ^ [X4: nat] : ( member_nat @ X4 @ ( inf_inf_set_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_3608_inf__Int__eq,axiom,
    ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( inf_in3604695632404883862_int_o
        @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ R3 )
        @ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ S ) )
      = ( ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ ( inf_in2269163501485487111nt_int @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_3609_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_3610_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_3611_inf__set__def,axiom,
    ( inf_in8396524679539076455nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ( inf_in5102985939729578038_int_o
            @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ A6 )
            @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_3612_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_3613_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_3614_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_3615_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_3616_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_3617_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_3618_div__le__mono,axiom,
    ! [M: nat,N2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K2 ) @ ( divide_divide_nat @ N2 @ K2 ) ) ) ).

% div_le_mono
thf(fact_3619_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_3620_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_3621_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_3622_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_3623_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N2 ) @ ( divide_divide_nat @ K2 @ M ) ) ) ) ).

% div_le_mono2
thf(fact_3624_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_3625_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_3626_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_3627_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_3628_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self2
thf(fact_3629_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_3630_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_3631_div__add__self2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% div_add_self2
thf(fact_3632_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self1
thf(fact_3633_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_3634_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_3635_div__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% div_add_self1
thf(fact_3636_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_3637_split__div,axiom,
    ! [P2: nat > $o,M: nat,N2: nat] :
      ( ( P2 @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P2 @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I: nat,J: nat] :
              ( ( ord_less_nat @ J @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I ) @ J ) )
               => ( P2 @ I ) ) ) ) ) ) ).

% split_div
thf(fact_3638_wand__assnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,R3: assn] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
     => ( ! [H4: heap_e7401611519738050253t_unit,As6: set_nat] :
            ( ( ( inf_inf_set_nat @ As @ As6 )
              = bot_bot_set_nat )
           => ( ( relH @ As @ H @ H4 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
               => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H4 @ As6 ) )
                 => ( rep_assn @ R3 @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) )
       => ( rep_assn @ ( wand_assn @ Q2 @ R3 ) @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ).

% wand_assnI
thf(fact_3639_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ? [As12: set_nat,As23: set_nat] :
                ( ( As2
                  = ( sup_sup_set_nat @ As12 @ As23 ) )
                & ( ( inf_inf_set_nat @ As12 @ As23 )
                  = bot_bot_set_nat )
                & ( X @ ( produc7507926704131184380et_nat @ H2 @ As12 ) )
                & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As23 ) ) ) ) ) ).

% times_assn_raw.elims(3)
thf(fact_3640_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ~ ? [As13: set_nat,As24: set_nat] :
                  ( ( As2
                    = ( sup_sup_set_nat @ As13 @ As24 ) )
                  & ( ( inf_inf_set_nat @ As13 @ As24 )
                    = bot_bot_set_nat )
                  & ( X @ ( produc7507926704131184380et_nat @ H2 @ As13 ) )
                  & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As24 ) ) ) ) ) ).

% times_assn_raw.elims(2)
thf(fact_3641_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 )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( Y
              = ( ~ ? [As1: set_nat,As22: set_nat] :
                      ( ( As2
                        = ( sup_sup_set_nat @ As1 @ As22 ) )
                      & ( ( inf_inf_set_nat @ As1 @ As22 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ) ) ).

% times_assn_raw.elims(1)
thf(fact_3642_times__assn__raw_Osimps,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( times_assn_raw @ P2 @ Q2 @ ( produc7507926704131184380et_nat @ H @ As ) )
      = ( ? [As1: set_nat,As22: set_nat] :
            ( ( As
              = ( sup_sup_set_nat @ As1 @ As22 ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( P2 @ ( produc7507926704131184380et_nat @ H @ As1 ) )
            & ( Q2 @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ).

% times_assn_raw.simps
thf(fact_3643_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 ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) )
               => ? [As12: set_nat,As23: set_nat] :
                    ( ( As2
                      = ( sup_sup_set_nat @ As12 @ As23 ) )
                    & ( ( inf_inf_set_nat @ As12 @ As23 )
                      = bot_bot_set_nat )
                    & ( X @ ( produc7507926704131184380et_nat @ H2 @ As12 ) )
                    & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As23 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
thf(fact_3644_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 ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) )
               => ~ ? [As13: set_nat,As24: set_nat] :
                      ( ( As2
                        = ( sup_sup_set_nat @ As13 @ As24 ) )
                      & ( ( inf_inf_set_nat @ As13 @ As24 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H2 @ As13 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As24 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
thf(fact_3645_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 ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( Y
                  = ( ? [As1: set_nat,As22: set_nat] :
                        ( ( As2
                          = ( sup_sup_set_nat @ As1 @ As22 ) )
                        & ( ( inf_inf_set_nat @ As1 @ As22 )
                          = bot_bot_set_nat )
                        & ( X @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
                        & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(1)
thf(fact_3646_in__range__empty,axiom,
    ! [H: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ).

% in_range_empty
thf(fact_3647_in__range__dist__union,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat,As5: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ ( sup_sup_set_nat @ As @ As5 ) ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
        & ( in_range @ ( produc7507926704131184380et_nat @ H @ As5 ) ) ) ) ).

% in_range_dist_union
thf(fact_3648_inf__unit__def,axiom,
    ( inf_inf_Product_unit
    = ( ^ [Uu3: product_unit,Uv: product_unit] : product_Unity ) ) ).

% inf_unit_def
thf(fact_3649_less__eq__assn__def,axiom,
    ( ord_less_eq_assn
    = ( ^ [A5: assn,B4: assn] :
          ( A5
          = ( inf_inf_assn @ A5 @ B4 ) ) ) ) ).

% less_eq_assn_def
thf(fact_3650_old_Ounit_Oexhaust,axiom,
    ! [Y: product_unit] : Y = product_Unity ).

% old.unit.exhaust
thf(fact_3651_sup__unit__def,axiom,
    ( sup_sup_Product_unit
    = ( ^ [Uu3: product_unit,Uv: product_unit] : product_Unity ) ) ).

% sup_unit_def
thf(fact_3652_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
thf(fact_3653_models__in__range,axiom,
    ! [P2: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P2 @ H )
     => ( in_range @ H ) ) ).

% models_in_range
thf(fact_3654_relH__in__rangeI_I2_J,axiom,
    ! [As: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H @ H5 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) ) ) ).

% relH_in_rangeI(2)
thf(fact_3655_relH__in__rangeI_I1_J,axiom,
    ! [As: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H @ H5 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ).

% relH_in_rangeI(1)
thf(fact_3656_relH__refl,axiom,
    ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
     => ( relH @ As @ H @ H ) ) ).

% relH_refl
thf(fact_3657_in__range__subset,axiom,
    ! [As: set_nat,As5: set_nat,H: heap_e7401611519738050253t_unit] :
      ( ( ord_less_eq_set_nat @ As @ As5 )
     => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As5 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) ) ) ) ).

% in_range_subset
thf(fact_3658_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_3659_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_3660_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( Y
              = ( ~ ! [X4: nat] :
                      ( ( member_nat @ X4 @ As2 )
                     => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H2 ) ) ) ) ) ) ) ).

% in_range.elims(1)
thf(fact_3661_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ~ ! [X6: nat] :
                  ( ( member_nat @ X6 @ As2 )
                 => ( ord_less_nat @ X6 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).

% in_range.elims(2)
thf(fact_3662_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ! [X3: nat] :
                ( ( member_nat @ X3 @ As2 )
               => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).

% in_range.elims(3)
thf(fact_3663_wand__raw_Osimps,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( wand_raw @ P2 @ Q2 @ ( produc7507926704131184380et_nat @ H @ As ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As ) )
        & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
            ( ( ( ( inf_inf_set_nat @ As @ As7 )
                = bot_bot_set_nat )
              & ( relH @ As @ H @ H7 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
              & ( P2 @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
           => ( Q2 @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As7 ) ) ) ) ) ) ).

% wand_raw.simps
thf(fact_3664_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 )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( Y
              = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As2 @ As7 )
                            = bot_bot_set_nat )
                          & ( relH @ As2 @ H2 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) ) ) ) ) ).

% wand_raw.elims(1)
thf(fact_3665_wand__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
                & ! [H8: heap_e7401611519738050253t_unit,As8: set_nat] :
                    ( ( ( ( inf_inf_set_nat @ As2 @ As8 )
                        = bot_bot_set_nat )
                      & ( relH @ As2 @ H2 @ H8 )
                      & ( in_range @ ( produc7507926704131184380et_nat @ H8 @ As2 ) )
                      & ( X @ ( produc7507926704131184380et_nat @ H8 @ As8 ) ) )
                   => ( Xa @ ( produc7507926704131184380et_nat @ H8 @ ( sup_sup_set_nat @ As2 @ As8 ) ) ) ) ) ) ) ).

% wand_raw.elims(2)
thf(fact_3666_wand__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
              & ! [H4: heap_e7401611519738050253t_unit,As6: set_nat] :
                  ( ( ( ( inf_inf_set_nat @ As2 @ As6 )
                      = bot_bot_set_nat )
                    & ( relH @ As2 @ H2 @ H4 )
                    & ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As2 ) )
                    & ( X @ ( produc7507926704131184380et_nat @ H4 @ As6 ) ) )
                 => ( Xa @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) ) ) ).

% wand_raw.elims(3)
thf(fact_3667_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 ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( Y
                  = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As2 @ As7 )
                            = bot_bot_set_nat )
                          & ( relH @ As2 @ H2 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(1)
thf(fact_3668_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 ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) )
               => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
                    & ! [H8: heap_e7401611519738050253t_unit,As8: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As2 @ As8 )
                            = bot_bot_set_nat )
                          & ( relH @ As2 @ H2 @ H8 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H8 @ As2 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H8 @ As8 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H8 @ ( sup_sup_set_nat @ As2 @ As8 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
thf(fact_3669_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 ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) )
               => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
                  & ! [H4: heap_e7401611519738050253t_unit,As6: set_nat] :
                      ( ( ( ( inf_inf_set_nat @ As2 @ As6 )
                          = bot_bot_set_nat )
                        & ( relH @ As2 @ H2 @ H4 )
                        & ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As2 ) )
                        & ( X @ ( produc7507926704131184380et_nat @ H4 @ As6 ) ) )
                     => ( Xa @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
thf(fact_3670_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( Y
                  = ( ! [X4: nat] :
                        ( ( member_nat @ X4 @ As2 )
                       => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H2 ) ) ) ) )
               => ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ) ) ).

% in_range.pelims(1)
thf(fact_3671_less__assn__def,axiom,
    ( ord_less_assn
    = ( ^ [A5: assn,B4: assn] :
          ( ( ord_less_eq_assn @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% less_assn_def
thf(fact_3672_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
               => ! [X3: nat] :
                    ( ( member_nat @ X3 @ As2 )
                   => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H2 ) ) ) ) ) ) ) ).

% in_range.pelims(3)
thf(fact_3673_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
               => ~ ! [X6: nat] :
                      ( ( member_nat @ X6 @ As2 )
                     => ( ord_less_nat @ X6 @ ( lim_Product_unit @ H2 ) ) ) ) ) ) ) ).

% in_range.pelims(2)
thf(fact_3674_default__unit__def,axiom,
    defaul566961228789861419t_unit = product_Unity ).

% default_unit_def
thf(fact_3675_verit__le__mono__div,axiom,
    ! [A4: nat,B5: nat,N2: nat] :
      ( ( ord_less_nat @ A4 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B5 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_3676_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_3677_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_3678_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_3679_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_3680_wand__assn__def,axiom,
    ( wand_assn
    = ( ^ [P5: assn,Q3: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P5 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).

% wand_assn_def
thf(fact_3681_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C2: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3682_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,C2: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3683_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C2: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3684_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C2: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3685_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,C2: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3686_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C2: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3687_distrib__left__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: rat,B: rat,C2: rat] :
      ( ( nO_MAT3836730678284324717al_rat @ ( divide5121882707175180666atural @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3688_distrib__left__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: nat,B: nat,C2: nat] :
      ( ( nO_MAT4471860738370820453al_nat @ ( divide5121882707175180666atural @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3689_distrib__left__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: int,B: int,C2: int] :
      ( ( nO_MAT4469370267861770177al_int @ ( divide5121882707175180666atural @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3690_distrib__left__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat,C2: rat] :
      ( ( nO_MATCH_rat_rat @ ( divide_divide_rat @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3691_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3692_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3693_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3694_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_3695_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_3696_mod__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_add_self1
thf(fact_3697_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_3698_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_3699_mod__add__self2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_add_self2
thf(fact_3700_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_3701_Rep__assn__inverse,axiom,
    ! [X: assn] :
      ( ( abs_assn @ ( rep_assn @ X ) )
      = X ) ).

% Rep_assn_inverse
thf(fact_3702_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_3703_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_3704_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_3705_of__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% of_nat_0
thf(fact_3706_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_3707_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_3708_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_3709_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3710_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_3711_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_3712_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_3713_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3714_mod__mult__self1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3715_mod__mult__self1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3716_mod__mult__self1,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C2 @ B ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3717_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3718_mod__mult__self2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3719_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C2 ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3720_mod__mult__self3,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3721_mod__mult__self3,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3722_mod__mult__self3,axiom,
    ! [C2: code_natural,B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C2 @ B ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3723_mod__mult__self4,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3724_mod__mult__self4,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3725_mod__mult__self4,axiom,
    ! [B: code_natural,C2: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C2 ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3726_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_3727_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_3728_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_3729_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_3730_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_3731_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_3732_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_3733_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_3734_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_3735_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_3736_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_3737_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_3738_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_3739_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_3740_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_3741_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_3742_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_3743_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_3744_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_3745_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_3746_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_3747_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_3748_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_3749_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_3750_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_3751_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_3752_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_3753_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_3754_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_3755_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_3756_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_3757_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_3758_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_3759_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_3760_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_3761_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_3762_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_3763_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_3764_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_3765_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_3766_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_3767_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_3768_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_3769_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_3770_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_3771_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_3772_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_3773_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_3774_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_3775_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_3776_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_3777_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_3778_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).

% mod_add_right_eq
thf(fact_3779_mod__add__right__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% mod_add_right_eq
thf(fact_3780_mod__add__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ B @ C2 ) ) @ C2 )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C2 ) ) ).

% mod_add_right_eq
thf(fact_3781_mod__add__left__eq,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ B ) @ C2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).

% mod_add_left_eq
thf(fact_3782_mod__add__left__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ B ) @ C2 )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% mod_add_left_eq
thf(fact_3783_mod__add__left__eq,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C2 ) @ B ) @ C2 )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C2 ) ) ).

% mod_add_left_eq
thf(fact_3784_mod__add__cong,axiom,
    ! [A: nat,C2: nat,A2: nat,B: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ C2 )
        = ( modulo_modulo_nat @ A2 @ C2 ) )
     => ( ( ( modulo_modulo_nat @ B @ C2 )
          = ( modulo_modulo_nat @ B2 @ C2 ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).

% mod_add_cong
thf(fact_3785_mod__add__cong,axiom,
    ! [A: int,C2: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C2 )
        = ( modulo_modulo_int @ A2 @ C2 ) )
     => ( ( ( modulo_modulo_int @ B @ C2 )
          = ( modulo_modulo_int @ B2 @ C2 ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 )
          = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ) ) ).

% mod_add_cong
thf(fact_3786_mod__add__cong,axiom,
    ! [A: code_natural,C2: code_natural,A2: code_natural,B: code_natural,B2: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ C2 )
        = ( modulo8411746178871703098atural @ A2 @ C2 ) )
     => ( ( ( modulo8411746178871703098atural @ B @ C2 )
          = ( modulo8411746178871703098atural @ B2 @ C2 ) )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C2 )
          = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) @ C2 ) ) ) ) ).

% mod_add_cong
thf(fact_3787_mod__add__eq,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).

% mod_add_eq
thf(fact_3788_mod__add__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% mod_add_eq
thf(fact_3789_mod__add__eq,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C2 ) @ ( modulo8411746178871703098atural @ B @ C2 ) ) @ C2 )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C2 ) ) ).

% mod_add_eq
thf(fact_3790_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_3791_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_3792_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_3793_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_3794_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_3795_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_3796_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_3797_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_3798_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_3799_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z3: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z3 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z3 ) ) ).

% zadd_int_left
thf(fact_3800_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_3801_int__plus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_3802_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_3803_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_3804_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_3805_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_3806_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_3807_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_3808_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_less_as_int
thf(fact_3809_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_3810_Abs__assn__eqI_I1_J,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( ( P2 @ H2 )
          = ( rep_assn @ Pr @ H2 ) )
     => ( ( abs_assn @ P2 )
        = Pr ) ) ).

% Abs_assn_eqI(1)
thf(fact_3811_Abs__assn__eqI_I2_J,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H2: produc3658429121746597890et_nat] :
          ( ( P2 @ H2 )
          = ( rep_assn @ Pr @ H2 ) )
     => ( Pr
        = ( abs_assn @ P2 ) ) ) ).

% Abs_assn_eqI(2)
thf(fact_3812_pure__assn__def,axiom,
    ( pure_assn
    = ( ^ [B4: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B4 ) ) ) ) ).

% pure_assn_def
thf(fact_3813_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_3814_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_3815_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_3816_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_3817_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_3818_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_3819_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3820_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_3821_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_3822_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_3823_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_3824_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_3825_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_3826_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_3827_mod__eqE,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C2 )
        = ( modulo_modulo_int @ B @ C2 ) )
     => ~ ! [D: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C2 @ D ) ) ) ) ).

% mod_eqE
thf(fact_3828_div__add1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 ) ) ) ).

% div_add1_eq
thf(fact_3829_div__add1__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 ) ) ) ).

% div_add1_eq
thf(fact_3830_div__add1__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C2 )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C2 ) @ ( divide5121882707175180666atural @ B @ C2 ) ) @ ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C2 ) @ ( modulo8411746178871703098atural @ B @ C2 ) ) @ C2 ) ) ) ).

% div_add1_eq
thf(fact_3831_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_3832_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_3833_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_3834_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_3835_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_3836_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_3837_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_3838_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_3839_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I2 ) @ ( semiri4939895301339042750nteger @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3840_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3841_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3842_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3843_gcd__nat__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M5: nat] : ( P2 @ M5 @ zero_zero_nat )
     => ( ! [M5: nat,N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( ( P2 @ N5 @ ( modulo_modulo_nat @ M5 @ N5 ) )
             => ( P2 @ M5 @ N5 ) ) )
       => ( P2 @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_3844_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_3845_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J2: int,K2: nat] :
      ( ( ord_less_int @ I2 @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J2 ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_3846_pos__int__cases,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ~ ! [N5: nat] :
            ( ( K2
              = ( semiri1314217659103216013at_int @ N5 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).

% pos_int_cases
thf(fact_3847_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ? [N5: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N5 )
          & ( K2
            = ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_3848_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_3849_bot__assn__def,axiom,
    ( bot_bot_assn
    = ( abs_assn
      @ ^ [Uu2: produc3658429121746597890et_nat] : $false ) ) ).

% bot_assn_def
thf(fact_3850_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_3851_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_3852_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_3853_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_3854_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_3855_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_3856_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_3857_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_3858_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_3859_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_3860_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_3861_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_3862_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_3863_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_3864_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_3865_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_3866_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_3867_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_3868_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_3869_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_3870_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_3871_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C2 )
      = ( plus_plus_nat @ A @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3872_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C2 )
      = ( plus_plus_int @ A @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3873_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C2 )
      = ( plus_p4538020629002901425atural @ A @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3874_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C2 )
      = ( plus_plus_nat @ A @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3875_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C2 )
      = ( plus_plus_int @ A @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3876_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C2 )
      = ( plus_p4538020629002901425atural @ A @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3877_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C2 ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_3878_div__mult1__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C2 ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_3879_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C2 )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C2 ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_3880_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_3881_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_3882_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_3883_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_3884_div__less__mono,axiom,
    ! [A4: nat,B5: nat,N2: nat] :
      ( ( ord_less_nat @ A4 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A4 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B5 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A4 @ N2 ) @ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_3885_div__mod__decomp,axiom,
    ! [A4: nat,N2: nat] :
      ( A4
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A4 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_3886_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q6: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q6 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q6 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_3887_sup__assn__def,axiom,
    ( sup_sup_assn
    = ( ^ [P5: assn,Q3: assn] :
          ( abs_assn
          @ ^ [H6: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P5 @ H6 )
              | ( rep_assn @ Q3 @ H6 ) ) ) ) ) ).

% sup_assn_def
thf(fact_3888_inf__assn__def,axiom,
    ( inf_inf_assn
    = ( ^ [P5: assn,Q3: assn] :
          ( abs_assn
          @ ^ [H6: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P5 @ H6 )
              & ( rep_assn @ Q3 @ H6 ) ) ) ) ) ).

% inf_assn_def
thf(fact_3889_split__mod,axiom,
    ! [P2: nat > $o,M: nat,N2: nat] :
      ( ( P2 @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P2 @ M ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I: nat,J: nat] :
              ( ( ord_less_nat @ J @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I ) @ J ) )
               => ( P2 @ J ) ) ) ) ) ) ).

% split_mod
thf(fact_3890_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C2 ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C2 ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3891_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3892_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3893_times__assn__def,axiom,
    ( times_times_assn
    = ( ^ [P5: assn,Q3: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P5 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).

% times_assn_def
thf(fact_3894_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: rat,A: rat,B: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3895_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: nat,A: nat,B: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3896_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: int,A: int,B: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3897_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: rat,A: rat,B: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3898_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: nat,A: nat,B: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3899_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: int,A: int,B: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3900_distrib__right__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C2: rat,A: rat,B: rat] :
      ( ( nO_MAT3836730678284324717al_rat @ ( divide5121882707175180666atural @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3901_distrib__right__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C2: nat,A: nat,B: nat] :
      ( ( nO_MAT4471860738370820453al_nat @ ( divide5121882707175180666atural @ X @ Y ) @ C2 )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3902_distrib__right__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C2: int,A: int,B: int] :
      ( ( nO_MAT4469370267861770177al_int @ ( divide5121882707175180666atural @ X @ Y ) @ C2 )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3903_distrib__right__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C2: rat,A: rat,B: rat] :
      ( ( nO_MATCH_rat_rat @ ( divide_divide_rat @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3904_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_3905_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I: rat] : ( plus_plus_rat @ I @ one_one_rat )
          @ N
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_3906_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I: int] : ( plus_plus_int @ I @ one_one_int )
          @ N
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_3907_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I: nat] : ( plus_plus_nat @ I @ one_one_nat )
          @ N
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_3908_of__nat__code,axiom,
    ( semiri4939895301339042750nteger
    = ( ^ [N: nat] :
          ( semiri4055485073559036834nteger
          @ ^ [I: code_integer] : ( plus_p5714425477246183910nteger @ I @ one_one_Code_integer )
          @ N
          @ zero_z3403309356797280102nteger ) ) ) ).

% of_nat_code
thf(fact_3909_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N5: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N5 ) ) ).

% reals_Archimedean2
thf(fact_3910_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N5: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N5 ) ) ).

% real_arch_simple
thf(fact_3911_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_3912_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_3913_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_3914_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_3915_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_3916_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_3917_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_3918_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_3919_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_3920_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_3921_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_3922_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_3923_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_3924_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ K2 @ zero_zero_int )
     => ( ( ord_less_int @ L @ K2 )
       => ( ( modulo_modulo_int @ K2 @ L )
          = K2 ) ) ) ).

% mod_neg_neg_trivial
thf(fact_3925_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ K2 @ L )
       => ( ( modulo_modulo_int @ K2 @ L )
          = K2 ) ) ) ).

% mod_pos_pos_trivial
thf(fact_3926_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ K2 @ L )
       => ( ( divide_divide_int @ K2 @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_3927_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ K2 @ zero_zero_int )
     => ( ( ord_less_int @ L @ K2 )
       => ( ( divide_divide_int @ K2 @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_3928_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3929_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_3930_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3931_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_3932_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_3933_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K2 ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_3934_Suc__mod__mult__self3,axiom,
    ! [K2: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K2 @ N2 ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_3935_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_3936_Suc__mod__mult__self1,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K2 @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_3937_zle__add1__eq__le,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z3 ) ) ).

% zle_add1_eq_le
thf(fact_3938_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_3939_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_3940_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( ( modulo_modulo_int @ I2 @ K2 )
        = I2 )
      = ( ( K2 = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K2 ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K2 @ I2 ) ) ) ) ).

% zmod_trivial_iff
thf(fact_3941_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K2 ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_3942_zero__le__imp__eq__int,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ? [N5: nat] :
          ( K2
          = ( semiri1314217659103216013at_int @ N5 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_3943_nonneg__int__cases,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ~ ! [N5: nat] :
            ( K2
           != ( semiri1314217659103216013at_int @ N5 ) ) ) ).

% nonneg_int_cases
thf(fact_3944_le__imp__0__less,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).

% le_imp_0_less
thf(fact_3945_int__one__le__iff__zero__less,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z3 )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% int_one_le_iff_zero_less
thf(fact_3946_zless__imp__add1__zle,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_int @ W2 @ Z3 )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z3 ) ) ).

% zless_imp_add1_zle
thf(fact_3947_add1__zle__eq,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z3 )
      = ( ord_less_int @ W2 @ Z3 ) ) ).

% add1_zle_eq
thf(fact_3948_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z: int] :
        ? [N: nat] :
          ( Z
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_3949_incr__mult__lemma,axiom,
    ! [D2: int,P2: int > $o,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int] :
            ( ( P2 @ X3 )
           => ( P2 @ ( plus_plus_int @ X3 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K2 )
         => ! [X6: int] :
              ( ( P2 @ X6 )
             => ( P2 @ ( plus_plus_int @ X6 @ ( times_times_int @ K2 @ D2 ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3950_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q7: int,R7: int,Q6: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q7 ) @ R7 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R7 )
           => ( ord_less_eq_int @ Q6 @ Q7 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3951_unique__quotient__lemma,axiom,
    ! [B: int,Q7: int,R7: int,Q6: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q7 ) @ R7 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R7 )
       => ( ( ord_less_int @ R7 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q7 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3952_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q6: int,R2: int,B2: int,Q7: int,R7: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R7 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R7 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ 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 @ Q7 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3953_zdiv__mono2__lemma,axiom,
    ! [B: int,Q6: int,R2: int,B2: int,Q7: int,R7: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R7 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R7 ) )
       => ( ( ord_less_int @ R7 @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q6 @ Q7 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3954_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q6: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_3955_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q6: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_3956_q__pos__lemma,axiom,
    ! [B2: int,Q7: int,R7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R7 ) )
     => ( ( ord_less_int @ R7 @ B2 )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ord_less_eq_int @ zero_zero_int @ Q7 ) ) ) ) ).

% q_pos_lemma
thf(fact_3957_split__zmod,axiom,
    ! [P2: int > $o,N2: int,K2: int] :
      ( ( P2 @ ( modulo_modulo_int @ N2 @ K2 ) )
      = ( ( ( K2 = zero_zero_int )
         => ( P2 @ N2 ) )
        & ( ( ord_less_int @ zero_zero_int @ K2 )
         => ! [I: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K2 )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ J ) ) )
        & ( ( ord_less_int @ K2 @ zero_zero_int )
         => ! [I: int,J: int] :
              ( ( ( ord_less_int @ K2 @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ J ) ) ) ) ) ).

% split_zmod
thf(fact_3958_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K2 @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_3959_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_3960_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K2 @ L )
          = ( plus_plus_int @ K2 @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_3961_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_3962_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_3963_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( ( divide_divide_int @ I2 @ K2 )
        = zero_zero_int )
      = ( ( K2 = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K2 ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K2 @ I2 ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_3964_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_3965_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_3966_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) )
      = ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K2 )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K2 @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_3967_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_eq_int @ L @ K2 )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) ) ) ) ).

% div_positive_int
thf(fact_3968_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_3969_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_3970_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K2 ) )
        = ( ord_less_eq_int @ K2 @ I2 ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_3971_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_3972_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_3973_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_3974_verit__le__mono__div__int,axiom,
    ! [A4: int,B5: int,N2: int] :
      ( ( ord_less_int @ A4 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A4 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B5 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B5 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_3975_zmod__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_3976_split__pos__lemma,axiom,
    ! [K2: int,P2: int > int > $o,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( P2 @ ( divide_divide_int @ N2 @ K2 ) @ ( modulo_modulo_int @ N2 @ K2 ) )
        = ( ! [I: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K2 )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ I @ J ) ) ) ) ) ).

% split_pos_lemma
thf(fact_3977_split__neg__lemma,axiom,
    ! [K2: int,P2: int > int > $o,N2: int] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ( ( P2 @ ( divide_divide_int @ N2 @ K2 ) @ ( modulo_modulo_int @ N2 @ K2 ) )
        = ( ! [I: int,J: int] :
              ( ( ( ord_less_int @ K2 @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ I @ J ) ) ) ) ) ).

% split_neg_lemma
thf(fact_3978_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q6: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q6 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3979_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q6: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q6 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3980_split__zdiv,axiom,
    ! [P2: int > $o,N2: int,K2: int] :
      ( ( P2 @ ( divide_divide_int @ N2 @ K2 ) )
      = ( ( ( K2 = zero_zero_int )
         => ( P2 @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K2 )
         => ! [I: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K2 )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ I ) ) )
        & ( ( ord_less_int @ K2 @ zero_zero_int )
         => ! [I: int,J: int] :
              ( ( ( ord_less_int @ K2 @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ I ) ) ) ) ) ).

% split_zdiv
thf(fact_3981_div__mod__decomp__int,axiom,
    ! [A4: int,N2: int] :
      ( A4
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A4 @ N2 ) ) ) ).

% div_mod_decomp_int
thf(fact_3982_int__div__less__self,axiom,
    ! [X: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K2 )
       => ( ord_less_int @ ( divide_divide_int @ X @ K2 ) @ X ) ) ) ).

% int_div_less_self
thf(fact_3983_zdiv__mono__strict,axiom,
    ! [A4: int,B5: int,N2: int] :
      ( ( ord_less_int @ A4 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ( ( modulo_modulo_int @ A4 @ N2 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B5 @ N2 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A4 @ N2 ) @ ( divide_divide_int @ B5 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_3984_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_3985_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_3986_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_3987_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K2 @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_3988_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K2 @ L ) ) ) ).

% neg_mod_bound
thf(fact_3989_zmult__zless__mono2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( ( ord_less_int @ I2 @ J2 )
     => ( ( ord_less_int @ zero_zero_int @ K2 )
       => ( ord_less_int @ ( times_times_int @ K2 @ I2 ) @ ( times_times_int @ K2 @ J2 ) ) ) ) ).

% zmult_zless_mono2
thf(fact_3990_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_3991_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_3992_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_3993_int__if,axiom,
    ! [P2: $o,A: nat,B: nat] :
      ( ( P2
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P2
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_3994_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( semiri1314217659103216013at_int @ A5 )
          = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_3995_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z: int] :
        ? [N: nat] :
          ( Z
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_3996_int__gr__induct,axiom,
    ! [K2: int,I2: int,P2: int > $o] :
      ( ( ord_less_int @ K2 @ I2 )
     => ( ( P2 @ ( plus_plus_int @ K2 @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K2 @ I3 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_gr_induct
thf(fact_3997_zless__add1__eq,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z3 @ one_one_int ) )
      = ( ( ord_less_int @ W2 @ Z3 )
        | ( W2 = Z3 ) ) ) ).

% zless_add1_eq
thf(fact_3998_odd__less__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
      = ( ord_less_int @ Z3 @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3999_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_4000_plus__int__code_I1_J,axiom,
    ! [K2: int] :
      ( ( plus_plus_int @ K2 @ zero_zero_int )
      = K2 ) ).

% plus_int_code(1)
thf(fact_4001_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_4002_odd__nonzero,axiom,
    ! [Z3: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_4003_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_4004_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_4005_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_4006_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(1)
thf(fact_4007_zdiv__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).

% zdiv_zmult2_eq
thf(fact_4008_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_4009_int__ge__induct,axiom,
    ! [K2: int,I2: int,P2: int > $o] :
      ( ( ord_less_eq_int @ K2 @ I2 )
     => ( ( P2 @ K2 )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K2 @ I3 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_ge_induct
thf(fact_4010_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_4011_imp__le__cong,axiom,
    ! [X: int,X10: int,P2: $o,P6: $o] :
      ( ( X = X10 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
         => ( P2 = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P2 )
          = ( ( ord_less_eq_int @ zero_zero_int @ X10 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_4012_conj__le__cong,axiom,
    ! [X: int,X10: int,P2: $o,P6: $o] :
      ( ( X = X10 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
         => ( P2 = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P2 )
          = ( ( ord_less_eq_int @ zero_zero_int @ X10 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_4013_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_4014_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_4015_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_4016_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_4017_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_4018_nat__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ( P2 @ N5 )
           => ( P2 @ ( suc @ N5 ) ) )
       => ( P2 @ N2 ) ) ) ).

% nat_induct
thf(fact_4019_diff__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P2 @ X3 @ Y3 )
             => ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P2 @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_4020_zero__induct,axiom,
    ! [P2: nat > $o,K2: nat] :
      ( ( P2 @ K2 )
     => ( ! [N5: nat] :
            ( ( P2 @ ( suc @ N5 ) )
           => ( P2 @ N5 ) )
       => ( P2 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_4021_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_4022_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_4023_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_4024_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% not0_implies_Suc
thf(fact_4025_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_4026_strict__inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ! [I3: nat] :
            ( ( J2
              = ( suc @ I3 ) )
           => ( P2 @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( P2 @ ( suc @ I3 ) )
               => ( P2 @ I3 ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_4027_less__Suc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J3: nat,K: nat] :
              ( ( ord_less_nat @ I3 @ J3 )
             => ( ( ord_less_nat @ J3 @ K )
               => ( ( P2 @ I3 @ J3 )
                 => ( ( P2 @ J3 @ K )
                   => ( P2 @ I3 @ K ) ) ) ) )
         => ( P2 @ I2 @ J2 ) ) ) ) ).

% less_Suc_induct
thf(fact_4028_less__trans__Suc,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ J2 @ K2 )
       => ( ord_less_nat @ ( suc @ I2 ) @ K2 ) ) ) ).

% less_trans_Suc
thf(fact_4029_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_4030_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_4031_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_4032_All__less__Suc,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N2 ) )
           => ( P2 @ I ) ) )
      = ( ( P2 @ N2 )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N2 )
           => ( P2 @ I ) ) ) ) ).

% All_less_Suc
thf(fact_4033_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_4034_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_4035_Ex__less__Suc,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N2 ) )
            & ( P2 @ I ) ) )
      = ( ( P2 @ N2 )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N2 )
            & ( P2 @ I ) ) ) ) ).

% Ex_less_Suc
thf(fact_4036_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_4037_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_4038_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_4039_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K2 )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I2 @ J3 )
           => ( K2
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_4040_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_4041_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ K2 )
     => ( ( K2
         != ( suc @ I2 ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I2 @ J3 )
             => ( K2
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_4042_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R3: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X3: nat] : ( R3 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( R3 @ X3 @ Y3 )
             => ( ( R3 @ Y3 @ Z2 )
               => ( R3 @ X3 @ Z2 ) ) )
         => ( ! [N5: nat] : ( R3 @ N5 @ ( suc @ N5 ) )
           => ( R3 @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_4043_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( P2 @ M )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ M @ N5 )
             => ( ( P2 @ N5 )
               => ( P2 @ ( suc @ N5 ) ) ) )
         => ( P2 @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_4044_full__nat__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N5 )
             => ( P2 @ M4 ) )
         => ( P2 @ N5 ) )
     => ( P2 @ N2 ) ) ).

% full_nat_induct
thf(fact_4045_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_4046_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_4047_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_4048_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_4049_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_4050_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_4051_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_4052_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_4053_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_4054_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_4055_Suc__mult__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ ( suc @ K2 ) @ M )
        = ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
      = ( M = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_4056_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_4057_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_4058_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_4059_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_4060_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_4061_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_4062_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_4063_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_4064_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_4065_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_4066_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_4067_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_4068_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_4069_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_4070_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_4071_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_4072_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_4073_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_4074_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_4075_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_4076_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_4077_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_4078_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_4079_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ N2 ) )
     != zero_z3403309356797280102nteger ) ).

% of_nat_neq_0
thf(fact_4080_Ex__less__Suc2,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N2 ) )
            & ( P2 @ I ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N2 )
            & ( P2 @ ( suc @ I ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_4081_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M3: nat] :
            ( N2
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_4082_All__less__Suc2,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N2 ) )
           => ( P2 @ I ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N2 )
           => ( P2 @ ( suc @ I ) ) ) ) ) ).

% All_less_Suc2
thf(fact_4083_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_4084_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( M = zero_zero_nat )
        | ? [J: nat] :
            ( ( M
              = ( suc @ J ) )
            & ( ord_less_nat @ J @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_4085_nat__compl__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N5 )
               => ( P2 @ Nn ) )
           => ( P2 @ ( suc @ N5 ) ) )
       => ( P2 @ N2 ) ) ) ).

% nat_compl_induct
thf(fact_4086_nat__compl__induct_H,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N5 )
               => ( P2 @ Nn ) )
           => ( P2 @ ( suc @ N5 ) ) )
       => ( P2 @ N2 ) ) ) ).

% nat_compl_induct'
thf(fact_4087_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_4088_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_4089_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_4090_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_4091_dec__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P2 @ I2 )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ I2 @ N5 )
             => ( ( ord_less_nat @ N5 @ J2 )
               => ( ( P2 @ N5 )
                 => ( P2 @ ( suc @ N5 ) ) ) ) )
         => ( P2 @ J2 ) ) ) ) ).

% dec_induct
thf(fact_4092_inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P2 @ J2 )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ I2 @ N5 )
             => ( ( ord_less_nat @ N5 @ J2 )
               => ( ( P2 @ ( suc @ N5 ) )
                 => ( P2 @ N5 ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% inc_induct
thf(fact_4093_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_4094_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_4095_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_4096_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_4097_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_4098_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_4099_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_4100_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_4101_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).

% less_add_Suc1
thf(fact_4102_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).

% less_add_Suc2
thf(fact_4103_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_4104_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% less_imp_Suc_add
thf(fact_4105_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_4106_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_4107_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_4108_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_4109_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_4110_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_4111_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_4112_mod__induct,axiom,
    ! [P2: nat > $o,N2: nat,P3: nat,M: nat] :
      ( ( P2 @ N2 )
     => ( ( ord_less_nat @ N2 @ P3 )
       => ( ( ord_less_nat @ M @ P3 )
         => ( ! [N5: nat] :
                ( ( ord_less_nat @ N5 @ P3 )
               => ( ( P2 @ N5 )
                 => ( P2 @ ( modulo_modulo_nat @ ( suc @ N5 ) @ P3 ) ) ) )
           => ( P2 @ M ) ) ) ) ) ).

% mod_induct
thf(fact_4113_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_4114_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_4115_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_4116_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_4117_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_4118_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_4119_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_4120_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_4121_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_4122_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_4123_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_4124_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_4125_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_4126_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_4127_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ N2 )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_nat @ K @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ ( suc @ K ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_4128_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_4129_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_4130_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_4131_nat__induct__non__zero,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P2 @ one_one_nat )
       => ( ! [N5: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N5 )
             => ( ( P2 @ N5 )
               => ( P2 @ ( suc @ N5 ) ) ) )
         => ( P2 @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_4132_power__gt__expt,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ K2 @ ( power_power_nat @ N2 @ K2 ) ) ) ).

% power_gt_expt
thf(fact_4133_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_4134_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_4135_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_4136_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_4137_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_4138_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_4139_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_4140_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_4141_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_4142_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_4143_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_4144_split__div_H,axiom,
    ! [P2: nat > $o,M: nat,N2: nat] :
      ( ( P2 @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
          & ( P2 @ zero_zero_nat ) )
        | ? [Q8: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q8 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q8 ) ) )
            & ( P2 @ Q8 ) ) ) ) ).

% split_div'
thf(fact_4145_option_Osize__gen_I2_J,axiom,
    ! [X: produc7388388658123137530it_nat > nat,X2: produc7388388658123137530it_nat] :
      ( ( size_o5858930521590659991it_nat @ X @ ( some_P2818173045054083285it_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4146_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_4147_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_4148_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_4149_Heap_Osize__gen,axiom,
    ! [Xa: product_unit > nat,X: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( heap_T4142866422068808505t_unit @ Xa @ ( heap_T6183433275982383450t_unit @ X ) )
      = ( suc @ zero_zero_nat ) ) ).

% Heap.size_gen
thf(fact_4150_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_4151_Heap_Osize_I2_J,axiom,
    ! [X: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( size_s1564113455978345259t_unit @ ( heap_T6183433275982383450t_unit @ X ) )
      = ( suc @ zero_zero_nat ) ) ).

% Heap.size(2)
thf(fact_4152_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q6: int,R2: int] :
      ( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q6 @ R2 ) )
      = ( ( K2
          = ( plus_plus_int @ ( times_times_int @ L @ Q6 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q6 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_4153_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_4154_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_4155_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_4156_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_4157_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P2: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I3: int,J3: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J3 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J3 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) )
             => ( P2 @ I3 @ J3 ) ) )
       => ( P2 @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_4158_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_4159_option_Osize_I4_J,axiom,
    ! [X2: produc7388388658123137530it_nat] :
      ( ( size_s2866289034186623454it_nat @ ( some_P2818173045054083285it_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_4160_option_Osize_I4_J,axiom,
    ! [X2: produc3260487557148687353it_nat] :
      ( ( size_s2363601347547812957it_nat @ ( some_P7913643980934408916it_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_4161_option_Osize_I4_J,axiom,
    ! [X2: produc8664842809031399944it_nat] :
      ( ( size_s8766407808098229740it_nat @ ( some_P1914260805536162275it_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_4162_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_4163_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K2 @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_4164_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K2 )
       => ( ( divide_divide_int @ K2 @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K2 @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_4165_int__power__div__base,axiom,
    ! [M: nat,K2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K2 )
       => ( ( divide_divide_int @ ( power_power_int @ K2 @ M ) @ K2 )
          = ( power_power_int @ K2 @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_4166_one__less__nat__eq,axiom,
    ! [Z3: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
      = ( ord_less_int @ one_one_int @ Z3 ) ) ).

% one_less_nat_eq
thf(fact_4167_verit__minus__simplify_I4_J,axiom,
    ! [B: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_4168_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_4169_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_4170_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_4171_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_4172_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_4173_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_4174_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_4175_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_4176_add__diff__cancel__right_H,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_4177_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_4178_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_4179_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_4180_add__diff__cancel__right,axiom,
    ! [A: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A @ C2 ) @ ( plus_p7104986032573967614at_nat @ B @ C2 ) )
      = ( minus_4286766774447292334at_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_4181_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_4182_add__diff__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_4183_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_4184_add__diff__cancel__left_H,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_4185_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_4186_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_4187_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_4188_add__diff__cancel__left,axiom,
    ! [C2: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ C2 @ A ) @ ( plus_p7104986032573967614at_nat @ C2 @ B ) )
      = ( minus_4286766774447292334at_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_4189_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_4190_add__diff__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_4191_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_4192_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_4193_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_4194_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_4195_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_4196_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_4197_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_4198_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_4199_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_4200_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_4201_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_4202_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_add_distrib
thf(fact_4203_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_4204_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4205_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4206_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4207_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4208_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4209_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4210_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_4211_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_4212_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_4213_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K2 ) ) ).

% Suc_diff_diff
thf(fact_4214_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_4215_diff__diff__left,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K2 )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).

% diff_diff_left
thf(fact_4216_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_4217_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_4218_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_4219_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_4220_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_4221_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_4222_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_4223_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_4224_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_4225_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_4226_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_4227_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_4228_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4229_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_4230_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_4231_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_4232_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_4233_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_4234_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_4235_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_4236_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_4237_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_4238_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_4239_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_4240_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_4241_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_4242_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_4243_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_4244_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_4245_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_4246_diff__add__zero,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A @ ( plus_p7104986032573967614at_nat @ A @ B ) )
      = zero_z1048942125864253310at_nat ) ).

% diff_add_zero
thf(fact_4247_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_4248_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_4249_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_4250_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_4251_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_4252_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_4253_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_4254_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_4255_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_4256_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_4257_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_4258_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_4259_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_4260_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_4261_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_4262_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_4263_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_4264_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_4265_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_4266_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4267_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4268_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4269_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4270_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( plus_p5714425477246183910nteger @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4271_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4272_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4273_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4274_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4275_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_4276_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( uminus7117520113953359693_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) )
      = ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_4277_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_4278_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_4279_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_4280_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_4281_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_4282_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_4283_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( uminus7117520113953359693_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
      = ( sup_su8463660629351352368_int_o @ ( uminus7117520113953359693_int_o @ X ) @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_4284_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_4285_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_4286_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_4287_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_4288_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_4289_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_4290_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_4291_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_4292_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J2 ) ) ) ).

% Nat.diff_diff_right
thf(fact_4293_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K2 ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_4294_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K2 ) ) ) ).

% Nat.add_diff_assoc
thf(fact_4295_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_4296_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_4297_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% add_neg_numeral_special(7)
thf(fact_4298_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(7)
thf(fact_4299_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% add_neg_numeral_special(7)
thf(fact_4300_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = zero_zero_int ) ).

% add_neg_numeral_special(8)
thf(fact_4301_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(8)
thf(fact_4302_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = zero_zero_rat ) ).

% add_neg_numeral_special(8)
thf(fact_4303_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_4304_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K2 ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( suc @ J2 ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_4305_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K2 ) ) @ I2 )
        = ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_4306_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_4307_nat__le__0,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ Z3 @ zero_zero_int )
     => ( ( nat2 @ Z3 )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_4308_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( ( nat2 @ I2 )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_4309_zless__nat__conj,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
      = ( ( ord_less_int @ zero_zero_int @ Z3 )
        & ( ord_less_int @ W2 @ Z3 ) ) ) ).

% zless_nat_conj
thf(fact_4310_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_4311_int__nat__eq,axiom,
    ! [Z3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
          = Z3 ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_4312_zle__diff1__eq,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z3 @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z3 ) ) ).

% zle_diff1_eq
thf(fact_4313_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_4314_zero__less__nat__eq,axiom,
    ! [Z3: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% zero_less_nat_eq
thf(fact_4315_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_4316_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_4317_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_4318_diff__commute,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J2 ) ) ).

% diff_commute
thf(fact_4319_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_4320_nat__diff__distrib,axiom,
    ! [Z7: int,Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z3 )
       => ( ( nat2 @ ( minus_minus_int @ Z3 @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_4321_group__cancel_Osub2,axiom,
    ! [B5: int,K2: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( minus_minus_int @ A @ B5 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4322_group__cancel_Osub2,axiom,
    ! [B5: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B5
        = ( plus_p5714425477246183910nteger @ K2 @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B5 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4323_group__cancel_Osub2,axiom,
    ! [B5: rat,K2: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K2 @ B ) )
     => ( ( minus_minus_rat @ A @ B5 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4324_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4325_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4326_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4327_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4328_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4329_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4330_int__minus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_4331_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_4332_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_4333_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_4334_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_4335_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_4336_diff__nat__eq__if,axiom,
    ! [Z7: int,Z3: int] :
      ( ( ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z3 ) ) )
      & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z3 @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z3 @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_4337_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C2 @ D2 ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4338_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4339_diff__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).

% diff_right_mono
thf(fact_4340_diff__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).

% diff_right_mono
thf(fact_4341_diff__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).

% diff_left_mono
thf(fact_4342_diff__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).

% diff_left_mono
thf(fact_4343_diff__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D2 @ C2 )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_4344_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D2 @ C2 )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_4345_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_4346_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_4347_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_4348_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_4349_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C2 @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_4350_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_4351_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D2 @ C2 )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_4352_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D2 @ C2 )
       => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_4353_diff__diff__eq,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( minus_4286766774447292334at_nat @ A @ B ) @ C2 )
      = ( minus_4286766774447292334at_nat @ A @ ( plus_p7104986032573967614at_nat @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_4354_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_4355_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_4356_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_4357_add__implies__diff,axiom,
    ! [C2: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ C2 @ B )
        = A )
     => ( C2
        = ( minus_4286766774447292334at_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_4358_add__implies__diff,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_4359_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_4360_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_4361_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_4362_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_4363_diff__add__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_4364_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_4365_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_4366_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_4367_add__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_4368_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_4369_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C2 @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_4370_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C2 @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_4371_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_4372_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_4373_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( minus_minus_rat @ A4 @ B )
        = ( plus_plus_rat @ K2 @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_4374_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( minus_minus_int @ A4 @ B )
        = ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_4375_inf__period_I2_J,axiom,
    ! [P2: rat > $o,D4: rat,Q2: rat > $o] :
      ( ! [X3: rat,K: rat] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D4 ) ) ) )
     => ( ! [X3: rat,K: rat] :
            ( ( Q2 @ X3 )
            = ( Q2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D4 ) ) ) )
       => ! [X6: rat,K4: rat] :
            ( ( ( P2 @ X6 )
              | ( Q2 @ X6 ) )
            = ( ( P2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4376_inf__period_I2_J,axiom,
    ! [P2: int > $o,D4: int,Q2: int > $o] :
      ( ! [X3: int,K: int] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D4 ) ) ) )
     => ( ! [X3: int,K: int] :
            ( ( Q2 @ X3 )
            = ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D4 ) ) ) )
       => ! [X6: int,K4: int] :
            ( ( ( P2 @ X6 )
              | ( Q2 @ X6 ) )
            = ( ( P2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4377_inf__period_I1_J,axiom,
    ! [P2: rat > $o,D4: rat,Q2: rat > $o] :
      ( ! [X3: rat,K: rat] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D4 ) ) ) )
     => ( ! [X3: rat,K: rat] :
            ( ( Q2 @ X3 )
            = ( Q2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D4 ) ) ) )
       => ! [X6: rat,K4: rat] :
            ( ( ( P2 @ X6 )
              & ( Q2 @ X6 ) )
            = ( ( P2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4378_inf__period_I1_J,axiom,
    ! [P2: int > $o,D4: int,Q2: int > $o] :
      ( ! [X3: int,K: int] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D4 ) ) ) )
     => ( ! [X3: int,K: int] :
            ( ( Q2 @ X3 )
            = ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D4 ) ) ) )
       => ! [X6: int,K4: int] :
            ( ( ( P2 @ X6 )
              & ( Q2 @ X6 ) )
            = ( ( P2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4379_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_4380_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_4381_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_4382_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_4383_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_4384_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_4385_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_4386_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_4387_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_4388_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_4389_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_4390_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_4391_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_4392_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_4393_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_4394_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_4395_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_4396_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_4397_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_4398_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_4399_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_4400_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_4401_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_4402_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_4403_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_4404_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_4405_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_4406_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_4407_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_4408_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_4409_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_4410_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4411_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4412_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4413_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4414_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4415_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4416_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( uminus_uminus_int @ A4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4417_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K2: code_integer,A: code_integer] :
      ( ( A4
        = ( plus_p5714425477246183910nteger @ K2 @ A ) )
     => ( ( uminus1351360451143612070nteger @ A4 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4418_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( uminus_uminus_rat @ A4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4419_length__induct,axiom,
    ! [P2: 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 ) )
             => ( P2 @ Ys ) )
         => ( P2 @ Xs2 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_4420_length__induct,axiom,
    ! [P2: 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 ) )
             => ( P2 @ Ys ) )
         => ( P2 @ Xs2 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_4421_length__induct,axiom,
    ! [P2: list_int > $o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs2 ) )
             => ( P2 @ Ys ) )
         => ( P2 @ Xs2 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_4422_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_4423_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_4424_zero__induct__lemma,axiom,
    ! [P2: nat > $o,K2: nat,I2: nat] :
      ( ( P2 @ K2 )
     => ( ! [N5: nat] :
            ( ( P2 @ ( suc @ N5 ) )
           => ( P2 @ N5 ) )
       => ( P2 @ ( minus_minus_nat @ K2 @ I2 ) ) ) ) ).

% zero_induct_lemma
thf(fact_4425_less__imp__diff__less,axiom,
    ! [J2: nat,K2: nat,N2: nat] :
      ( ( ord_less_nat @ J2 @ K2 )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K2 ) ) ).

% less_imp_diff_less
thf(fact_4426_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_4427_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_4428_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_4429_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_4430_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_4431_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
          = ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_4432_le__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
          = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_4433_eq__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N2 )
       => ( ( ( minus_minus_nat @ M @ K2 )
            = ( minus_minus_nat @ N2 @ K2 ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_4434_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_4435_diff__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_cancel2
thf(fact_4436_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_4437_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_4438_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K2 )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N2 @ K2 ) ) ) ).

% diff_mult_distrib
thf(fact_4439_diff__mult__distrib2,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_4440_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_4441_nat__mult__distrib__neg,axiom,
    ! [Z3: int,Z7: int] :
      ( ( ord_less_eq_int @ Z3 @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z3 @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_4442_zdiff__int__split,axiom,
    ! [P2: int > $o,X: nat,Y: nat] :
      ( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P2 @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_4443_le__iff__diff__le__0,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A5 @ B4 ) @ zero_z3403309356797280102nteger ) ) ) ).

% le_iff_diff_le_0
thf(fact_4444_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_4445_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_4446_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_4447_less__iff__diff__less__0,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A5 @ B4 ) @ zero_z3403309356797280102nteger ) ) ) ).

% less_iff_diff_less_0
thf(fact_4448_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_4449_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_4450_diff__le__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_4451_diff__le__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_4452_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_4453_le__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_4454_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_4455_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).

% le_add_diff
thf(fact_4456_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_4457_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_4458_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
        = ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_4459_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_4460_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_4461_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_4462_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_4463_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C2 )
          = ( B
            = ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_4464_add__le__imp__le__diff,axiom,
    ! [I2: rat,K2: rat,N2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N2 )
     => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N2 @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_4465_add__le__imp__le__diff,axiom,
    ! [I2: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N2 )
     => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_4466_add__le__imp__le__diff,axiom,
    ! [I2: int,K2: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N2 )
     => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N2 @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_4467_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K2: rat,N2: rat,J2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N2 )
     => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J2 @ K2 ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N2 )
         => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J2 @ K2 ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K2 ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4468_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K2: nat,N2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J2 @ K2 ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N2 )
         => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J2 @ K2 ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K2 ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4469_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K2: int,N2: int,J2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J2 @ K2 ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N2 )
         => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J2 @ K2 ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K2 ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4470_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_4471_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_4472_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_4473_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_4474_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_4475_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_4476_less__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_4477_diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_4478_diff__less__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_4479_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_4480_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_4481_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_4482_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_4483_eq__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_4484_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( C2
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_4485_eq__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( C2
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_4486_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_4487_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_4488_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_4489_ex__nat,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X5: nat] : ( P4 @ X5 ) )
    = ( ^ [P5: nat > $o] :
        ? [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
          & ( P5 @ ( nat2 @ X4 ) ) ) ) ) ).

% ex_nat
thf(fact_4490_all__nat,axiom,
    ( ( ^ [P4: nat > $o] :
        ! [X5: nat] : ( P4 @ X5 ) )
    = ( ^ [P5: nat > $o] :
        ! [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( P5 @ ( nat2 @ X4 ) ) ) ) ) ).

% all_nat
thf(fact_4491_eq__nat__nat__iff,axiom,
    ! [Z3: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( ( nat2 @ Z3 )
            = ( nat2 @ Z7 ) )
          = ( Z3 = Z7 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_4492_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K2 ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K2 @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_4493_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_4494_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4495_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4496_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4497_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4498_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4499_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4500_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_4501_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_4502_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_4503_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_4504_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_4505_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_4506_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4507_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4508_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4509_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_4510_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_4511_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_4512_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_4513_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_4514_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_4515_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_4516_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_4517_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_4518_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_4519_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_4520_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_4521_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_4522_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_4523_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_4524_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_4525_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_4526_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_4527_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_4528_less__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
          = ( ord_less_nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_4529_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_4530_less__diff__conv,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J2 ) ) ).

% less_diff_conv
thf(fact_4531_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_4532_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ( minus_minus_nat @ J2 @ I2 )
          = K2 )
        = ( J2
          = ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_4533_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I2 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_4534_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K2 )
        = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_4535_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J2 ) ) ) ).

% Nat.le_diff_conv2
thf(fact_4536_le__diff__conv,axiom,
    ! [J2: nat,K2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I2 )
      = ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K2 ) ) ) ).

% le_diff_conv
thf(fact_4537_int__le__induct,axiom,
    ! [I2: int,K2: int,P2: int > $o] :
      ( ( ord_less_eq_int @ I2 @ K2 )
     => ( ( P2 @ K2 )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K2 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_le_induct
thf(fact_4538_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_4539_int__less__induct,axiom,
    ! [I2: int,K2: int,P2: int > $o] :
      ( ( ord_less_int @ I2 @ K2 )
     => ( ( P2 @ ( minus_minus_int @ K2 @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K2 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_less_induct
thf(fact_4540_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_4541_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_4542_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_4543_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_4544_le__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 ) @ ( 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 ) @ C2 ) @ D2 ) ) ).

% le_add_iff1
thf(fact_4545_le__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( 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 ) @ C2 ) @ D2 ) ) ).

% le_add_iff1
thf(fact_4546_le__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_4547_le__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_4548_less__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4549_less__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4550_less__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4551_less__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4552_nat__mono__iff,axiom,
    ! [Z3: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
        = ( ord_less_int @ W2 @ Z3 ) ) ) ).

% nat_mono_iff
thf(fact_4553_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_4554_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_4555_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_4556_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_4557_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_4558_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_4559_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_4560_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_4561_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_4562_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z3: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z3 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z3 ) ) ).

% zless_nat_eq_int_zless
thf(fact_4563_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_4564_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_4565_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_4566_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_4567_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_4568_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_4569_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_4570_int__eq__iff,axiom,
    ! [M: nat,Z3: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z3 )
      = ( ( M
          = ( nat2 @ Z3 ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).

% int_eq_iff
thf(fact_4571_nat__0__le,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
        = Z3 ) ) ).

% nat_0_le
thf(fact_4572_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_4573_inf__shunt,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( ( inf_in3604695632404883862_int_o @ X @ Y )
        = bot_bo8147686125503663512_int_o )
      = ( ord_le8369615600986905444_int_o @ X @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).

% inf_shunt
thf(fact_4574_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_4575_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_4576_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_4577_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_4578_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_4579_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_4580_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z3 )
      = ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_4581_shunt1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Z3 )
      = ( ord_le8369615600986905444_int_o @ X @ ( sup_su8463660629351352368_int_o @ ( uminus7117520113953359693_int_o @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_4582_shunt1,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ X @ Y ) @ Z3 )
      = ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_4583_shunt1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ Z3 )
      = ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_4584_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z3 )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_4585_shunt1,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Z3 )
      = ( ord_less_eq_assn @ X @ ( sup_sup_assn @ ( uminus_uminus_assn @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_4586_shunt1,axiom,
    ! [X: product_unit,Y: product_unit,Z3: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z3 )
      = ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ ( uminus2952777764628376836t_unit @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_4587_shunt1,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X @ Y ) @ Z3 )
      = ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_4588_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) @ Z3 )
      = ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_4589_shunt2,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z3: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ ( uminus7117520113953359693_int_o @ Y ) ) @ Z3 )
      = ( ord_le8369615600986905444_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_4590_shunt2,axiom,
    ! [X: set_Pr8551490117392284871at_nat,Y: set_Pr8551490117392284871at_nat,Z3: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ X @ ( uminus2330091110623919550at_nat @ Y ) ) @ Z3 )
      = ( ord_le8081472938463900775at_nat @ X @ ( sup_su3035147773818789531at_nat @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_4591_shunt2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X @ ( uminus935396558254630718at_nat @ Y ) ) @ Z3 )
      = ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_4592_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) @ Z3 )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_4593_shunt2,axiom,
    ! [X: assn,Y: assn,Z3: assn] :
      ( ( ord_less_eq_assn @ ( inf_inf_assn @ X @ ( uminus_uminus_assn @ Y ) ) @ Z3 )
      = ( ord_less_eq_assn @ X @ ( sup_sup_assn @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_4594_shunt2,axiom,
    ! [X: product_unit,Y: product_unit,Z3: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) @ Z3 )
      = ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_4595_shunt2,axiom,
    ! [X: set_int,Y: set_int,Z3: set_int] :
      ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) @ Z3 )
      = ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_4596_sup__neg__inf,axiom,
    ! [P3: set_Pr1261947904930325089at_nat,Q6: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ P3 @ ( sup_su6327502436637775413at_nat @ Q6 @ R2 ) )
      = ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P3 @ ( uminus6524753893492686040at_nat @ Q6 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_4597_sup__neg__inf,axiom,
    ! [P3: product_prod_int_int > $o,Q6: product_prod_int_int > $o,R2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ P3 @ ( sup_su8463660629351352368_int_o @ Q6 @ R2 ) )
      = ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ P3 @ ( uminus7117520113953359693_int_o @ Q6 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_4598_sup__neg__inf,axiom,
    ! [P3: set_Pr8551490117392284871at_nat,Q6: set_Pr8551490117392284871at_nat,R2: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ P3 @ ( sup_su3035147773818789531at_nat @ Q6 @ R2 ) )
      = ( ord_le8081472938463900775at_nat @ ( inf_in1697001100524423349at_nat @ P3 @ ( uminus2330091110623919550at_nat @ Q6 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_4599_sup__neg__inf,axiom,
    ! [P3: set_Pr4329608150637261639at_nat,Q6: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ P3 @ ( sup_su5525570899277871387at_nat @ Q6 @ R2 ) )
      = ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ P3 @ ( uminus935396558254630718at_nat @ Q6 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_4600_sup__neg__inf,axiom,
    ! [P3: set_nat,Q6: set_nat,R2: set_nat] :
      ( ( ord_less_eq_set_nat @ P3 @ ( sup_sup_set_nat @ Q6 @ R2 ) )
      = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P3 @ ( uminus5710092332889474511et_nat @ Q6 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_4601_sup__neg__inf,axiom,
    ! [P3: assn,Q6: assn,R2: assn] :
      ( ( ord_less_eq_assn @ P3 @ ( sup_sup_assn @ Q6 @ R2 ) )
      = ( ord_less_eq_assn @ ( inf_inf_assn @ P3 @ ( uminus_uminus_assn @ Q6 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_4602_sup__neg__inf,axiom,
    ! [P3: product_unit,Q6: product_unit,R2: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ P3 @ ( sup_sup_Product_unit @ Q6 @ R2 ) )
      = ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ P3 @ ( uminus2952777764628376836t_unit @ Q6 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_4603_sup__neg__inf,axiom,
    ! [P3: set_int,Q6: set_int,R2: set_int] :
      ( ( ord_less_eq_set_int @ P3 @ ( sup_sup_set_int @ Q6 @ R2 ) )
      = ( ord_less_eq_set_int @ ( inf_inf_set_int @ P3 @ ( uminus1532241313380277803et_int @ Q6 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_4604_nat__power__eq,axiom,
    ! [Z3: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( nat2 @ ( power_power_int @ Z3 @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z3 ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_4605_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_4606_nat__diff__split__asm,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P2 @ zero_zero_nat ) )
            | ? [D5: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D5 ) )
                & ~ ( P2 @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_4607_nat__diff__split,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P2 @ zero_zero_nat ) )
        & ! [D5: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D5 ) )
           => ( P2 @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_4608_less__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I2 )
        = ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K2 ) ) ) ) ).

% less_diff_conv2
thf(fact_4609_minusinfinity,axiom,
    ! [D2: int,P1: int > $o,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
       => ( ? [Z6: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z6 )
             => ( ( P2 @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_4610_plusinfinity,axiom,
    ! [D2: int,P6: int > $o,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
       => ( ? [Z6: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z6 @ X3 )
             => ( ( P2 @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_12: int] : ( P6 @ X_12 )
           => ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_4611_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_4612_nat__eq__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_4613_nat__eq__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_4614_nat__le__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_4615_nat__le__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_4616_nat__diff__add__eq1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_4617_nat__diff__add__eq2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_4618_int__induct,axiom,
    ! [P2: int > $o,K2: int,I2: int] :
      ( ( P2 @ K2 )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K2 @ I3 )
           => ( ( P2 @ I3 )
             => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K2 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_induct
thf(fact_4619_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_4620_nonpos__int__cases,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ K2 @ zero_zero_int )
     => ~ ! [N5: nat] :
            ( K2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).

% nonpos_int_cases
thf(fact_4621_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_4622_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_4623_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_4624_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_4625_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_4626_frac__le__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z3 ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z3 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_4627_frac__less__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z3 ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z3 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_4628_nat__less__eq__zless,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
        = ( ord_less_int @ W2 @ Z3 ) ) ) ).

% nat_less_eq_zless
thf(fact_4629_nat__le__eq__zle,axiom,
    ! [W2: int,Z3: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
        = ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).

% nat_le_eq_zle
thf(fact_4630_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_4631_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_4632_split__nat,axiom,
    ! [P2: nat > $o,I2: int] :
      ( ( P2 @ ( nat2 @ I2 ) )
      = ( ! [N: nat] :
            ( ( I2
              = ( semiri1314217659103216013at_int @ N ) )
           => ( P2 @ N ) )
        & ( ( ord_less_int @ I2 @ zero_zero_int )
         => ( P2 @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_4633_le__nat__iff,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K2 ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K2 ) ) ) ).

% le_nat_iff
thf(fact_4634_nat__add__distrib,axiom,
    ! [Z3: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( nat2 @ ( plus_plus_int @ Z3 @ Z7 ) )
          = ( plus_plus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_4635_nat__mult__distrib,axiom,
    ! [Z3: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( nat2 @ ( times_times_int @ Z3 @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_4636_Suc__as__int,axiom,
    ( suc
    = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_4637_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4638_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4639_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4640_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4641_minus__divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4642_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4643_minus__divide__add__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z3 ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4644_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z3 ) ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z3 ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4645_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_4646_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_4647_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_4648_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_4649_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_4650_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_4651_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_4652_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_4653_nz__le__conv__less,axiom,
    ! [K2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_eq_nat @ K2 @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ K2 @ ( suc @ zero_zero_nat ) ) @ M ) ) ) ).

% nz_le_conv_less
thf(fact_4654_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_4655_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_4656_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_4657_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_4658_nat__less__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_4659_nat__less__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_4660_decr__mult__lemma,axiom,
    ! [D2: int,P2: int > $o,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int] :
            ( ( P2 @ X3 )
           => ( P2 @ ( minus_minus_int @ X3 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K2 )
         => ! [X6: int] :
              ( ( P2 @ X6 )
             => ( P2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D2 ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_4661_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_4662_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_4663_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N5: nat] :
            ( ( K2
              = ( semiri1314217659103216013at_int @ N5 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) )
       => ~ ! [N5: nat] :
              ( ( K2
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ) ).

% int_cases3
thf(fact_4664_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_4665_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_4666_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N5: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N5 ) ) ) ) ) ).

% negD
thf(fact_4667_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_4668_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K2 )
       => ( ( modulo_modulo_int @ K2 @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K2 @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_4669_verit__less__mono__div__int2,axiom,
    ! [A4: int,B5: int,N2: int] :
      ( ( ord_less_eq_int @ A4 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N2 ) @ ( divide_divide_int @ A4 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_4670_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_4671_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S2 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4672_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( suc @ ( nat2 @ Z3 ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_4673_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_4674_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4675_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4676_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4677_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4678_minus__divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4679_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4680_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_4681_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_4682_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_4683_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_4684_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_4685_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_4686_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_4687_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_4688_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_4689_neg__int__cases,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ~ ! [N5: nat] :
            ( ( K2
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N5 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).

% neg_int_cases
thf(fact_4690_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K2 ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4691_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K2 ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4692_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K2 ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4693_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_4694_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_4695_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_4696_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_4697_size__union,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( size_s8510653225128441779at_nat @ ( plus_p7104986032573967614at_nat @ M6 @ N7 ) )
      = ( plus_plus_nat @ ( size_s8510653225128441779at_nat @ M6 ) @ ( size_s8510653225128441779at_nat @ N7 ) ) ) ).

% size_union
thf(fact_4698_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_4699_length__rule,axiom,
    ! [A: array_nat,Xs: list_nat] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_nat @ A @ Xs ) @ ( array_len_nat @ A )
      @ ^ [R5: nat] :
          ( times_times_assn @ ( snga_assn_nat @ A @ Xs )
          @ ( pure_assn
            @ ( R5
              = ( size_size_list_nat @ Xs ) ) ) ) ) ).

% length_rule
thf(fact_4700_length__rule,axiom,
    ! [A: array_o,Xs: list_o] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_o @ A @ Xs ) @ ( array_len_o @ A )
      @ ^ [R5: nat] :
          ( times_times_assn @ ( snga_assn_o @ A @ Xs )
          @ ( pure_assn
            @ ( R5
              = ( size_size_list_o @ Xs ) ) ) ) ) ).

% length_rule
thf(fact_4701_length__rule,axiom,
    ! [A: array_int,Xs: list_int] :
      ( hoare_3067605981109127869le_nat @ ( snga_assn_int @ A @ Xs ) @ ( array_len_int @ A )
      @ ^ [R5: nat] :
          ( times_times_assn @ ( snga_assn_int @ A @ Xs )
          @ ( pure_assn
            @ ( R5
              = ( size_size_list_int @ Xs ) ) ) ) ) ).

% length_rule
thf(fact_4702_Diff__cancel,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ A4 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_4703_Diff__cancel,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ A4 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_4704_Diff__cancel,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ A4 @ A4 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_4705_Diff__cancel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ A4 )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_cancel
thf(fact_4706_empty__Diff,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A4 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_4707_empty__Diff,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_4708_empty__Diff,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A4 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_4709_empty__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A4 )
      = bot_bo2099793752762293965at_nat ) ).

% empty_Diff
thf(fact_4710_Diff__empty,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ bot_bot_set_o )
      = A4 ) ).

% Diff_empty
thf(fact_4711_Diff__empty,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Diff_empty
thf(fact_4712_Diff__empty,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ A4 @ bot_bot_set_int )
      = A4 ) ).

% Diff_empty
thf(fact_4713_Diff__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = A4 ) ).

% Diff_empty
thf(fact_4714_Compl__anti__mono,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).

% Compl_anti_mono
thf(fact_4715_Compl__subset__Compl__iff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( uminus1532241313380277803et_int @ B5 ) )
      = ( ord_less_eq_set_int @ B5 @ A4 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4716_Diff__Compl,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ B5 ) )
      = ( inf_inf_set_nat @ A4 @ B5 ) ) ).

% Diff_Compl
thf(fact_4717_Diff__Compl,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( uminus6524753893492686040at_nat @ B5 ) )
      = ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ).

% Diff_Compl
thf(fact_4718_inter__compl__diff__conv,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ B5 ) )
      = ( minus_minus_set_nat @ A4 @ B5 ) ) ).

% inter_compl_diff_conv
thf(fact_4719_inter__compl__diff__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( uminus6524753893492686040at_nat @ B5 ) )
      = ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ).

% inter_compl_diff_conv
thf(fact_4720_Compl__Diff__eq,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( uminus2330091110623919550at_nat @ ( minus_5060654252129873198at_nat @ A4 @ B5 ) )
      = ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ A4 ) @ B5 ) ) ).

% Compl_Diff_eq
thf(fact_4721_Compl__Diff__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ B5 ) ) ).

% Compl_Diff_eq
thf(fact_4722_Compl__Diff__eq,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A4 @ B5 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ B5 ) ) ).

% Compl_Diff_eq
thf(fact_4723_Compl__Diff__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ B5 ) ) ).

% Compl_Diff_eq
thf(fact_4724_Un__Diff__cancel,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A4 @ ( minus_5060654252129873198at_nat @ B5 @ A4 ) )
      = ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) ).

% Un_Diff_cancel
thf(fact_4725_Un__Diff__cancel,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( minus_3314409938677909166at_nat @ B5 @ A4 ) )
      = ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) ).

% Un_Diff_cancel
thf(fact_4726_Un__Diff__cancel,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ A4 ) )
      = ( sup_sup_set_nat @ A4 @ B5 ) ) ).

% Un_Diff_cancel
thf(fact_4727_Un__Diff__cancel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) )
      = ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ).

% Un_Diff_cancel
thf(fact_4728_Un__Diff__cancel2,axiom,
    ! [B5: set_Pr8551490117392284871at_nat,A4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ B5 @ A4 ) @ A4 )
      = ( sup_su3035147773818789531at_nat @ B5 @ A4 ) ) ).

% Un_Diff_cancel2
thf(fact_4729_Un__Diff__cancel2,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ B5 @ A4 ) @ A4 )
      = ( sup_su5525570899277871387at_nat @ B5 @ A4 ) ) ).

% Un_Diff_cancel2
thf(fact_4730_Un__Diff__cancel2,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B5 @ A4 ) @ A4 )
      = ( sup_sup_set_nat @ B5 @ A4 ) ) ).

% Un_Diff_cancel2
thf(fact_4731_Un__Diff__cancel2,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) @ A4 )
      = ( sup_su6327502436637775413at_nat @ B5 @ A4 ) ) ).

% Un_Diff_cancel2
thf(fact_4732_diff__diff__add__mset,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat,P2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( minus_4286766774447292334at_nat @ M6 @ N7 ) @ P2 )
      = ( minus_4286766774447292334at_nat @ M6 @ ( plus_p7104986032573967614at_nat @ N7 @ P2 ) ) ) ).

% diff_diff_add_mset
thf(fact_4733_union__eq__empty,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ M6 @ N7 )
        = zero_z1048942125864253310at_nat )
      = ( ( M6 = zero_z1048942125864253310at_nat )
        & ( N7 = zero_z1048942125864253310at_nat ) ) ) ).

% union_eq_empty
thf(fact_4734_empty__eq__union,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat
        = ( plus_p7104986032573967614at_nat @ M6 @ N7 ) )
      = ( ( M6 = zero_z1048942125864253310at_nat )
        & ( N7 = zero_z1048942125864253310at_nat ) ) ) ).

% empty_eq_union
thf(fact_4735_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat
        = ( plus_p7104986032573967614at_nat @ X @ Y ) )
      = ( ( X = zero_z1048942125864253310at_nat )
        & ( Y = zero_z1048942125864253310at_nat ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_4736_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ X @ Y )
        = zero_z1048942125864253310at_nat )
      = ( ( X = zero_z1048942125864253310at_nat )
        & ( Y = zero_z1048942125864253310at_nat ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_4737_Diff__eq__empty__iff,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ( minus_minus_set_o @ A4 @ B5 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A4 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_4738_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( minus_minus_set_nat @ A4 @ B5 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_4739_Diff__eq__empty__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_4740_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( minus_minus_set_int @ A4 @ B5 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A4 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_4741_Diff__disjoint,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( inf_inf_set_o @ A4 @ ( minus_minus_set_o @ B5 @ A4 ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_4742_Diff__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ A4 ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_4743_Diff__disjoint,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( inf_inf_set_int @ A4 @ ( minus_minus_set_int @ B5 @ A4 ) )
      = bot_bot_set_int ) ).

% Diff_disjoint
thf(fact_4744_Diff__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_4745_Compl__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( uminus6524753893492686040at_nat @ A4 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint
thf(fact_4746_Compl__disjoint,axiom,
    ! [A4: set_o] :
      ( ( inf_inf_set_o @ A4 @ ( uminus_uminus_set_o @ A4 ) )
      = bot_bot_set_o ) ).

% Compl_disjoint
thf(fact_4747_Compl__disjoint,axiom,
    ! [A4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
      = bot_bot_set_nat ) ).

% Compl_disjoint
thf(fact_4748_Compl__disjoint,axiom,
    ! [A4: set_int] :
      ( ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
      = bot_bot_set_int ) ).

% Compl_disjoint
thf(fact_4749_Compl__disjoint2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ A4 )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint2
thf(fact_4750_Compl__disjoint2,axiom,
    ! [A4: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A4 ) @ A4 )
      = bot_bot_set_o ) ).

% Compl_disjoint2
thf(fact_4751_Compl__disjoint2,axiom,
    ! [A4: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ A4 )
      = bot_bot_set_nat ) ).

% Compl_disjoint2
thf(fact_4752_Compl__disjoint2,axiom,
    ! [A4: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ A4 )
      = bot_bot_set_int ) ).

% Compl_disjoint2
thf(fact_4753_mod__not__dist,axiom,
    ! [P2: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( uminus_uminus_assn @ P2 ) @ H )
      = ( ( in_range @ H )
        & ~ ( rep_assn @ P2 @ H ) ) ) ).

% mod_not_dist
thf(fact_4754_minus__assn__def,axiom,
    ( minus_minus_assn
    = ( ^ [A5: assn,B4: assn] : ( inf_inf_assn @ A5 @ ( uminus_uminus_assn @ B4 ) ) ) ) ).

% minus_assn_def
thf(fact_4755_Multiset_Odiff__add,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat,Q2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ M6 @ ( plus_p7104986032573967614at_nat @ N7 @ Q2 ) )
      = ( minus_4286766774447292334at_nat @ ( minus_4286766774447292334at_nat @ M6 @ N7 ) @ Q2 ) ) ).

% Multiset.diff_add
thf(fact_4756_diff__union__cancelL,axiom,
    ! [N7: multis2468970476368604999at_nat,M6: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ N7 @ M6 ) @ N7 )
      = M6 ) ).

% diff_union_cancelL
thf(fact_4757_diff__union__cancelR,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ M6 @ N7 ) @ N7 )
      = M6 ) ).

% diff_union_cancelR
thf(fact_4758_union__diff__assoc,axiom,
    ! [C4: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat,A4: multis2468970476368604999at_nat] :
      ( ( ( minus_4286766774447292334at_nat @ C4 @ B5 )
        = zero_z1048942125864253310at_nat )
     => ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A4 @ B5 ) @ C4 )
        = ( plus_p7104986032573967614at_nat @ A4 @ ( minus_4286766774447292334at_nat @ B5 @ C4 ) ) ) ) ).

% union_diff_assoc
thf(fact_4759_empty__neutral_I1_J,axiom,
    ! [X: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ X )
      = X ) ).

% empty_neutral(1)
thf(fact_4760_empty__neutral_I2_J,axiom,
    ! [X: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ X @ zero_z1048942125864253310at_nat )
      = X ) ).

% empty_neutral(2)
thf(fact_4761_mset__distrib,axiom,
    ! [A4: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat,M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A4 @ B5 )
        = ( plus_p7104986032573967614at_nat @ M6 @ N7 ) )
     => ~ ! [Am: multis2468970476368604999at_nat,An: multis2468970476368604999at_nat] :
            ( ( A4
              = ( plus_p7104986032573967614at_nat @ Am @ An ) )
           => ! [Bm: multis2468970476368604999at_nat,Bn: multis2468970476368604999at_nat] :
                ( ( B5
                  = ( plus_p7104986032573967614at_nat @ Bm @ Bn ) )
               => ( ( M6
                    = ( plus_p7104986032573967614at_nat @ Am @ Bm ) )
                 => ( N7
                   != ( plus_p7104986032573967614at_nat @ An @ Bn ) ) ) ) ) ) ).

% mset_distrib
thf(fact_4762_multi__union__self__other__eq,axiom,
    ! [A4: multis2468970476368604999at_nat,X7: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A4 @ X7 )
        = ( plus_p7104986032573967614at_nat @ A4 @ Y7 ) )
     => ( X7 = Y7 ) ) ).

% multi_union_self_other_eq
thf(fact_4763_union__right__cancel,axiom,
    ! [M6: multis2468970476368604999at_nat,K5: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ M6 @ K5 )
        = ( plus_p7104986032573967614at_nat @ N7 @ K5 ) )
      = ( M6 = N7 ) ) ).

% union_right_cancel
thf(fact_4764_union__left__cancel,axiom,
    ! [K5: multis2468970476368604999at_nat,M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ K5 @ M6 )
        = ( plus_p7104986032573967614at_nat @ K5 @ N7 ) )
      = ( M6 = N7 ) ) ).

% union_left_cancel
thf(fact_4765_union__commute,axiom,
    ( plus_p7104986032573967614at_nat
    = ( ^ [M9: multis2468970476368604999at_nat,N8: multis2468970476368604999at_nat] : ( plus_p7104986032573967614at_nat @ N8 @ M9 ) ) ) ).

% union_commute
thf(fact_4766_union__lcomm,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat,K5: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ M6 @ ( plus_p7104986032573967614at_nat @ N7 @ K5 ) )
      = ( plus_p7104986032573967614at_nat @ N7 @ ( plus_p7104986032573967614at_nat @ M6 @ K5 ) ) ) ).

% union_lcomm
thf(fact_4767_union__assoc,axiom,
    ! [M6: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat,K5: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ M6 @ N7 ) @ K5 )
      = ( plus_p7104986032573967614at_nat @ M6 @ ( plus_p7104986032573967614at_nat @ N7 @ K5 ) ) ) ).

% union_assoc
thf(fact_4768_Diff__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ C4 )
     => ( ( ord_le3146513528884898305at_nat @ D4 @ B5 )
       => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ ( minus_1356011639430497352at_nat @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_4769_Diff__mono,axiom,
    ! [A4: set_int,C4: set_int,D4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ C4 )
     => ( ( ord_less_eq_set_int @ D4 @ B5 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_4770_Diff__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ A4 ) ).

% Diff_subset
thf(fact_4771_Diff__subset,axiom,
    ! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ A4 ) ).

% Diff_subset
thf(fact_4772_double__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
     => ( ( ord_le3146513528884898305at_nat @ B5 @ C4 )
       => ( ( minus_1356011639430497352at_nat @ B5 @ ( minus_1356011639430497352at_nat @ C4 @ A4 ) )
          = A4 ) ) ) ).

% double_diff
thf(fact_4773_double__diff,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A4 ) )
          = A4 ) ) ) ).

% double_diff
thf(fact_4774_Diff__Int__distrib2,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ C4 )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C4 ) @ ( inf_inf_set_nat @ B5 @ C4 ) ) ) ).

% Diff_Int_distrib2
thf(fact_4775_Diff__Int__distrib2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ C4 )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ C4 ) @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) ) ) ).

% Diff_Int_distrib2
thf(fact_4776_Diff__Int__distrib,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat] :
      ( ( inf_inf_set_nat @ C4 @ ( minus_minus_set_nat @ A4 @ B5 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C4 @ A4 ) @ ( inf_inf_set_nat @ C4 @ B5 ) ) ) ).

% Diff_Int_distrib
thf(fact_4777_Diff__Int__distrib,axiom,
    ! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ C4 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C4 @ A4 ) @ ( inf_in2572325071724192079at_nat @ C4 @ B5 ) ) ) ).

% Diff_Int_distrib
thf(fact_4778_Diff__Diff__Int,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( minus_minus_set_nat @ A4 @ B5 ) )
      = ( inf_inf_set_nat @ A4 @ B5 ) ) ).

% Diff_Diff_Int
thf(fact_4779_Diff__Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
      = ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ).

% Diff_Diff_Int
thf(fact_4780_Diff__Int2,axiom,
    ! [A4: set_nat,C4: set_nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C4 ) @ ( inf_inf_set_nat @ B5 @ C4 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C4 ) @ B5 ) ) ).

% Diff_Int2
thf(fact_4781_Diff__Int2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ C4 ) @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ C4 ) @ B5 ) ) ).

% Diff_Int2
thf(fact_4782_Int__Diff,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ C4 )
      = ( inf_inf_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ C4 ) ) ) ).

% Int_Diff
thf(fact_4783_Int__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ C4 )
      = ( inf_in2572325071724192079at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B5 @ C4 ) ) ) ).

% Int_Diff
thf(fact_4784_Un__Diff,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( minus_5060654252129873198at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) @ C4 )
      = ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A4 @ C4 ) @ ( minus_5060654252129873198at_nat @ B5 @ C4 ) ) ) ).

% Un_Diff
thf(fact_4785_Un__Diff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) @ C4 )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A4 @ C4 ) @ ( minus_3314409938677909166at_nat @ B5 @ C4 ) ) ) ).

% Un_Diff
thf(fact_4786_Un__Diff,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A4 @ B5 ) @ C4 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ C4 ) @ ( minus_minus_set_nat @ B5 @ C4 ) ) ) ).

% Un_Diff
thf(fact_4787_Un__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) @ C4 )
      = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A4 @ C4 ) @ ( minus_1356011639430497352at_nat @ B5 @ C4 ) ) ) ).

% Un_Diff
thf(fact_4788_set__diff__diff__left,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( minus_5060654252129873198at_nat @ ( minus_5060654252129873198at_nat @ A4 @ B5 ) @ C4 )
      = ( minus_5060654252129873198at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) ) ) ).

% set_diff_diff_left
thf(fact_4789_set__diff__diff__left,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) @ C4 )
      = ( minus_3314409938677909166at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) ) ) ).

% set_diff_diff_left
thf(fact_4790_set__diff__diff__left,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ C4 )
      = ( minus_minus_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C4 ) ) ) ).

% set_diff_diff_left
thf(fact_4791_set__diff__diff__left,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ C4 )
      = ( minus_1356011639430497352at_nat @ A4 @ ( sup_su6327502436637775413at_nat @ B5 @ C4 ) ) ) ).

% set_diff_diff_left
thf(fact_4792_psubset__imp__ex__mem,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ord_less_set_o @ A4 @ B5 )
     => ? [B3: $o] : ( member_o @ B3 @ ( minus_minus_set_o @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4793_psubset__imp__ex__mem,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( ord_le1924305788584680229nt_int @ A4 @ B5 )
     => ? [B3: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ B3 @ ( minus_2612819937483484256nt_int @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4794_psubset__imp__ex__mem,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] :
      ( ( ord_less_set_set_nat @ A4 @ B5 )
     => ? [B3: set_nat] : ( member_set_nat @ B3 @ ( minus_2163939370556025621et_nat @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4795_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B5 )
     => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4796_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4797_psubset__imp__ex__mem,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
     => ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4798_uminus__unit__def,axiom,
    ( uminus2952777764628376836t_unit
    = ( ^ [Uu3: product_unit] : product_Unity ) ) ).

% uminus_unit_def
thf(fact_4799_Collect__imp__eq,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) )
      = ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P2 ) ) @ ( collect_list_nat @ Q2 ) ) ) ).

% Collect_imp_eq
thf(fact_4800_Collect__imp__eq,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ( collec5210948495886036740nt_int
        @ ^ [X4: set_Pr958786334691620121nt_int] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) )
      = ( sup_su2047564715030645325nt_int @ ( uminus6423885277529793776nt_int @ ( collec5210948495886036740nt_int @ P2 ) ) @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ).

% Collect_imp_eq
thf(fact_4801_Collect__imp__eq,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) )
      = ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P2 ) ) @ ( collect_set_nat @ Q2 ) ) ) ).

% Collect_imp_eq
thf(fact_4802_Collect__imp__eq,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) )
      = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) @ ( collect_int @ Q2 ) ) ) ).

% Collect_imp_eq
thf(fact_4803_Collect__imp__eq,axiom,
    ! [P2: produc4166570645942440679at_nat > $o,Q2: produc4166570645942440679at_nat > $o] :
      ( ( collec5204685387357076818at_nat
        @ ^ [X4: produc4166570645942440679at_nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) )
      = ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ ( collec5204685387357076818at_nat @ P2 ) ) @ ( collec5204685387357076818at_nat @ Q2 ) ) ) ).

% Collect_imp_eq
thf(fact_4804_Collect__imp__eq,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,Q2: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ ( collec6321179662152712658at_nat @ P2 ) ) @ ( collec6321179662152712658at_nat @ Q2 ) ) ) ).

% Collect_imp_eq
thf(fact_4805_Collect__imp__eq,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P2 @ X4 )
           => ( Q2 @ X4 ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_imp_eq
thf(fact_4806_subset__minus__empty,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ B5 )
     => ( ( minus_minus_set_o @ A4 @ B5 )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_4807_subset__minus__empty,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( minus_minus_set_nat @ A4 @ B5 )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_4808_subset__minus__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
     => ( ( minus_1356011639430497352at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
thf(fact_4809_subset__minus__empty,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( minus_minus_set_int @ A4 @ B5 )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_4810_subset__Compl__self__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ A4 ) )
      = ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
thf(fact_4811_subset__Compl__self__eq,axiom,
    ! [A4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_4812_subset__Compl__self__eq,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_4813_subset__Compl__self__eq,axiom,
    ! [A4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_4814_disjoint__alt__simp2,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ( minus_minus_set_o @ A4 @ B5 )
       != A4 )
      = ( ( inf_inf_set_o @ A4 @ B5 )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
thf(fact_4815_disjoint__alt__simp2,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( minus_minus_set_nat @ A4 @ B5 )
       != A4 )
      = ( ( inf_inf_set_nat @ A4 @ B5 )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
thf(fact_4816_disjoint__alt__simp2,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( minus_minus_set_int @ A4 @ B5 )
       != A4 )
      = ( ( inf_inf_set_int @ A4 @ B5 )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
thf(fact_4817_disjoint__alt__simp2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A4 @ B5 )
       != A4 )
      = ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
thf(fact_4818_disjoint__alt__simp1,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ( minus_minus_set_o @ A4 @ B5 )
        = A4 )
      = ( ( inf_inf_set_o @ A4 @ B5 )
        = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
thf(fact_4819_disjoint__alt__simp1,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( minus_minus_set_nat @ A4 @ B5 )
        = A4 )
      = ( ( inf_inf_set_nat @ A4 @ B5 )
        = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
thf(fact_4820_disjoint__alt__simp1,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( minus_minus_set_int @ A4 @ B5 )
        = A4 )
      = ( ( inf_inf_set_int @ A4 @ B5 )
        = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
thf(fact_4821_disjoint__alt__simp1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A4 @ B5 )
        = A4 )
      = ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
thf(fact_4822_Int__Diff__disjoint,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A4 @ B5 ) @ ( minus_minus_set_o @ A4 @ B5 ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_4823_Int__Diff__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ A4 @ B5 ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_4824_Int__Diff__disjoint,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ A4 @ B5 ) @ ( minus_minus_set_int @ A4 @ B5 ) )
      = bot_bot_set_int ) ).

% Int_Diff_disjoint
thf(fact_4825_Int__Diff__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_4826_Diff__triv,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ A4 @ B5 )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A4 @ B5 )
        = A4 ) ) ).

% Diff_triv
thf(fact_4827_Diff__triv,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( inf_inf_set_nat @ A4 @ B5 )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A4 @ B5 )
        = A4 ) ) ).

% Diff_triv
thf(fact_4828_Diff__triv,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( inf_inf_set_int @ A4 @ B5 )
        = bot_bot_set_int )
     => ( ( minus_minus_set_int @ A4 @ B5 )
        = A4 ) ) ).

% Diff_triv
thf(fact_4829_Diff__triv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A4 @ B5 )
        = A4 ) ) ).

% Diff_triv
thf(fact_4830_Diff__subset__conv,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ ( minus_5060654252129873198at_nat @ A4 @ B5 ) @ C4 )
      = ( ord_le8081472938463900775at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) ) ) ).

% Diff_subset_conv
thf(fact_4831_Diff__subset__conv,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) @ C4 )
      = ( ord_le1268244103169919719at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) ) ) ).

% Diff_subset_conv
thf(fact_4832_Diff__subset__conv,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ C4 )
      = ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C4 ) ) ) ).

% Diff_subset_conv
thf(fact_4833_Diff__subset__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ C4 )
      = ( ord_le3146513528884898305at_nat @ A4 @ ( sup_su6327502436637775413at_nat @ B5 @ C4 ) ) ) ).

% Diff_subset_conv
thf(fact_4834_Diff__subset__conv,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ C4 )
      = ( ord_less_eq_set_int @ A4 @ ( sup_sup_set_int @ B5 @ C4 ) ) ) ).

% Diff_subset_conv
thf(fact_4835_Diff__partition,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( ord_le8081472938463900775at_nat @ A4 @ B5 )
     => ( ( sup_su3035147773818789531at_nat @ A4 @ ( minus_5060654252129873198at_nat @ B5 @ A4 ) )
        = B5 ) ) ).

% Diff_partition
thf(fact_4836_Diff__partition,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B5 )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ ( minus_3314409938677909166at_nat @ B5 @ A4 ) )
        = B5 ) ) ).

% Diff_partition
thf(fact_4837_Diff__partition,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( sup_sup_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ A4 ) )
        = B5 ) ) ).

% Diff_partition
thf(fact_4838_Diff__partition,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
     => ( ( sup_su6327502436637775413at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) )
        = B5 ) ) ).

% Diff_partition
thf(fact_4839_Diff__partition,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( sup_sup_set_int @ A4 @ ( minus_minus_set_int @ B5 @ A4 ) )
        = B5 ) ) ).

% Diff_partition
thf(fact_4840_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_4841_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_4842_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_4843_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_4844_Un__Diff__Int,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A4 @ B5 ) @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) )
      = A4 ) ).

% Un_Diff_Int
thf(fact_4845_Un__Diff__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) )
      = A4 ) ).

% Un_Diff_Int
thf(fact_4846_Un__Diff__Int,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( inf_inf_set_nat @ A4 @ B5 ) )
      = A4 ) ).

% Un_Diff_Int
thf(fact_4847_Un__Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
      = A4 ) ).

% Un_Diff_Int
thf(fact_4848_Int__Diff__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) @ ( minus_5060654252129873198at_nat @ A4 @ B5 ) )
      = A4 ) ).

% Int_Diff_Un
thf(fact_4849_Int__Diff__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) )
      = A4 ) ).

% Int_Diff_Un
thf(fact_4850_Int__Diff__Un,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ A4 @ B5 ) )
      = A4 ) ).

% Int_Diff_Un
thf(fact_4851_Int__Diff__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
      = A4 ) ).

% Int_Diff_Un
thf(fact_4852_Diff__Int,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( minus_5060654252129873198at_nat @ A4 @ ( inf_in1697001100524423349at_nat @ B5 @ C4 ) )
      = ( sup_su3035147773818789531at_nat @ ( minus_5060654252129873198at_nat @ A4 @ B5 ) @ ( minus_5060654252129873198at_nat @ A4 @ C4 ) ) ) ).

% Diff_Int
thf(fact_4853_Diff__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A4 @ ( inf_in7913087082777306421at_nat @ B5 @ C4 ) )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) @ ( minus_3314409938677909166at_nat @ A4 @ C4 ) ) ) ).

% Diff_Int
thf(fact_4854_Diff__Int,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( inf_inf_set_nat @ B5 @ C4 ) )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ A4 @ C4 ) ) ) ).

% Diff_Int
thf(fact_4855_Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) )
      = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ ( minus_1356011639430497352at_nat @ A4 @ C4 ) ) ) ).

% Diff_Int
thf(fact_4856_Diff__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( minus_5060654252129873198at_nat @ A4 @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) )
      = ( inf_in1697001100524423349at_nat @ ( minus_5060654252129873198at_nat @ A4 @ B5 ) @ ( minus_5060654252129873198at_nat @ A4 @ C4 ) ) ) ).

% Diff_Un
thf(fact_4857_Diff__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) )
      = ( inf_in7913087082777306421at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) @ ( minus_3314409938677909166at_nat @ A4 @ C4 ) ) ) ).

% Diff_Un
thf(fact_4858_Diff__Un,axiom,
    ! [A4: set_nat,B5: set_nat,C4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( sup_sup_set_nat @ B5 @ C4 ) )
      = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ A4 @ C4 ) ) ) ).

% Diff_Un
thf(fact_4859_Diff__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( sup_su6327502436637775413at_nat @ B5 @ C4 ) )
      = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ ( minus_1356011639430497352at_nat @ A4 @ C4 ) ) ) ).

% Diff_Un
thf(fact_4860_Compl__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ ( uminus6524753893492686040at_nat @ B5 ) ) ) ).

% Compl_Int
thf(fact_4861_Compl__Int,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( uminus2330091110623919550at_nat @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) )
      = ( sup_su3035147773818789531at_nat @ ( uminus2330091110623919550at_nat @ A4 ) @ ( uminus2330091110623919550at_nat @ B5 ) ) ) ).

% Compl_Int
thf(fact_4862_Compl__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ ( uminus935396558254630718at_nat @ B5 ) ) ) ).

% Compl_Int
thf(fact_4863_Compl__Int,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A4 @ B5 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( uminus5710092332889474511et_nat @ B5 ) ) ) ).

% Compl_Int
thf(fact_4864_Compl__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
      = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ ( uminus6524753893492686040at_nat @ B5 ) ) ) ).

% Compl_Un
thf(fact_4865_Compl__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( uminus2330091110623919550at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
      = ( inf_in1697001100524423349at_nat @ ( uminus2330091110623919550at_nat @ A4 ) @ ( uminus2330091110623919550at_nat @ B5 ) ) ) ).

% Compl_Un
thf(fact_4866_Compl__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
      = ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ ( uminus935396558254630718at_nat @ B5 ) ) ) ).

% Compl_Un
thf(fact_4867_Compl__Un,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A4 @ B5 ) )
      = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( uminus5710092332889474511et_nat @ B5 ) ) ) ).

% Compl_Un
thf(fact_4868_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_4869_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_4870_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ B5 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_4871_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ A4 @ B5 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ B5 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_4872_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( inf_inf_set_nat @ A4 @ B5 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ B5 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_4873_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( inf_inf_set_int @ A4 @ B5 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ B5 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_4874_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_4875_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_4876_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_4877_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_4878_disjoint__alt__simp3,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( ord_less_set_o @ ( minus_minus_set_o @ A4 @ B5 ) @ A4 )
      = ( ( inf_inf_set_o @ A4 @ B5 )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
thf(fact_4879_disjoint__alt__simp3,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ A4 )
      = ( ( inf_inf_set_nat @ A4 @ B5 )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
thf(fact_4880_disjoint__alt__simp3,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ A4 )
      = ( ( inf_inf_set_int @ A4 @ B5 )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
thf(fact_4881_disjoint__alt__simp3,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ A4 )
      = ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
thf(fact_4882_uminus__assn__def,axiom,
    ( uminus_uminus_assn
    = ( ^ [P5: assn] :
          ( abs_assn
          @ ^ [H6: produc3658429121746597890et_nat] :
              ( ( in_range @ H6 )
              & ~ ( rep_assn @ P5 @ H6 ) ) ) ) ) ).

% uminus_assn_def
thf(fact_4883_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_4884_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_4885_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_4886_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_4887_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_4888_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_4889_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_4890_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_4891_pochhammer__rec_H,axiom,
    ! [Z3: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z3 @ ( suc @ N2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z3 @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4892_pochhammer__rec_H,axiom,
    ! [Z3: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z3 @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4893_pochhammer__rec_H,axiom,
    ! [Z3: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z3 @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4894_pochhammer__rec_H,axiom,
    ! [Z3: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ Z3 @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z3 @ ( semiri4939895301339042750nteger @ N2 ) ) @ ( comm_s8582702949713902594nteger @ Z3 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4895_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_4896_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K2 )
        = zero_zero_rat )
      = ( ord_less_nat @ N2 @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_4897_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K2 )
        = zero_zero_int )
      = ( ord_less_nat @ N2 @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_4898_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K2 )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N2 @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_4899_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ord_less_nat @ N2 @ K2 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K2 )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_4900_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ord_less_nat @ N2 @ K2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K2 )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_4901_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ord_less_nat @ N2 @ K2 )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K2 )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_4902_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K2 )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_4903_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K2 )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_4904_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K2 )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_4905_pochhammer__product_H,axiom,
    ! [Z3: rat,N2: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z3 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z3 @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z3 @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4906_pochhammer__product_H,axiom,
    ! [Z3: int,N2: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z3 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z3 @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4907_pochhammer__product_H,axiom,
    ! [Z3: nat,N2: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z3 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z3 @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4908_pochhammer__product_H,axiom,
    ! [Z3: code_integer,N2: nat,M: nat] :
      ( ( comm_s8582702949713902594nteger @ Z3 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z3 @ N2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z3 @ ( semiri4939895301339042750nteger @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_4909_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q6: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P3 @ Q6 ) )
     => ( ord_less_int @ zero_zero_int @ Q6 ) ) ).

% normalize_denom_pos
thf(fact_4910_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z3: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ Z3 @ N2 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z3 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z3 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4911_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z3: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z3 @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z3 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4912_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z3 @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z3 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4913_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z3: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s8582702949713902594nteger @ Z3 @ N2 )
        = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z3 @ M ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z3 @ ( semiri4939895301339042750nteger @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_4914_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K2 ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4915_pochhammer__absorb__comp,axiom,
    ! [R2: int,K2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K2 ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K2 ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4916_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K2: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K2 ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K2 ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4917_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_4918_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_4919_pochhammer__minus,axiom,
    ! [B: int,K2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_4920_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K2 ) ) @ one_one_Code_integer ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_4921_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_4922_pochhammer__minus_H,axiom,
    ! [B: int,K2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_4923_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K2 ) ) @ one_one_Code_integer ) @ K2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K2 ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_4924_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K2: 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 ) @ K2 )
       => ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_4925_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_4926_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_4927_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs ) )
       => ( ( size_size_list_int @ ( slice_int @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_4928_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K2: 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 ) @ K2 )
       => ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K2 ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_4929_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_4930_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_rat @ A @ K2 )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_4931_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_4932_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_4933_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_4934_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_4935_DiffI,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ A4 )
     => ( ~ ( member_o @ C2 @ B5 )
       => ( member_o @ C2 @ ( minus_minus_set_o @ A4 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4936_DiffI,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ A4 )
     => ( ~ ( member2340774599025711042nt_int @ C2 @ B5 )
       => ( member2340774599025711042nt_int @ C2 @ ( minus_2612819937483484256nt_int @ A4 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4937_DiffI,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ A4 )
     => ( ~ ( member_set_nat @ C2 @ B5 )
       => ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4938_DiffI,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ A4 )
     => ( ~ ( member_nat @ C2 @ B5 )
       => ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4939_DiffI,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ A4 )
     => ( ~ ( member_int @ C2 @ B5 )
       => ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4940_DiffI,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ C2 @ B5 )
       => ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ) ).

% DiffI
thf(fact_4941_Diff__iff,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( minus_minus_set_o @ A4 @ B5 ) )
      = ( ( member_o @ C2 @ A4 )
        & ~ ( member_o @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_4942_Diff__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( minus_2612819937483484256nt_int @ A4 @ B5 ) )
      = ( ( member2340774599025711042nt_int @ C2 @ A4 )
        & ~ ( member2340774599025711042nt_int @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_4943_Diff__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) )
      = ( ( member_set_nat @ C2 @ A4 )
        & ~ ( member_set_nat @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_4944_Diff__iff,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B5 ) )
      = ( ( member_nat @ C2 @ A4 )
        & ~ ( member_nat @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_4945_Diff__iff,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B5 ) )
      = ( ( member_int @ C2 @ A4 )
        & ~ ( member_int @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_4946_Diff__iff,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ C2 @ A4 )
        & ~ ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_4947_Diff__idemp,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ B5 )
      = ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ).

% Diff_idemp
thf(fact_4948_ComplI,axiom,
    ! [C2: $o,A4: set_o] :
      ( ~ ( member_o @ C2 @ A4 )
     => ( member_o @ C2 @ ( uminus_uminus_set_o @ A4 ) ) ) ).

% ComplI
thf(fact_4949_ComplI,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ C2 @ A4 )
     => ( member2340774599025711042nt_int @ C2 @ ( uminus6423885277529793776nt_int @ A4 ) ) ) ).

% ComplI
thf(fact_4950_ComplI,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( ~ ( member_set_nat @ C2 @ A4 )
     => ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A4 ) ) ) ).

% ComplI
thf(fact_4951_ComplI,axiom,
    ! [C2: nat,A4: set_nat] :
      ( ~ ( member_nat @ C2 @ A4 )
     => ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A4 ) ) ) ).

% ComplI
thf(fact_4952_ComplI,axiom,
    ! [C2: int,A4: set_int] :
      ( ~ ( member_int @ C2 @ A4 )
     => ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).

% ComplI
thf(fact_4953_Compl__iff,axiom,
    ! [C2: $o,A4: set_o] :
      ( ( member_o @ C2 @ ( uminus_uminus_set_o @ A4 ) )
      = ( ~ ( member_o @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_4954_Compl__iff,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( uminus6423885277529793776nt_int @ A4 ) )
      = ( ~ ( member2340774599025711042nt_int @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_4955_Compl__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A4 ) )
      = ( ~ ( member_set_nat @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_4956_Compl__iff,axiom,
    ! [C2: nat,A4: set_nat] :
      ( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A4 ) )
      = ( ~ ( member_nat @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_4957_Compl__iff,axiom,
    ! [C2: int,A4: set_int] :
      ( ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A4 ) )
      = ( ~ ( member_int @ C2 @ A4 ) ) ) ).

% Compl_iff
thf(fact_4958_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_4959_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_4960_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_4961_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% abs_of_nat
thf(fact_4962_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_4963_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% abs_of_nat
thf(fact_4964_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_4965_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_4966_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_4967_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_4968_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_4969_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_4970_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4971_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4972_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4973_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_4974_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_4975_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_4976_slice__complete,axiom,
    ! [Xs: list_nat] :
      ( ( slice_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_4977_slice__complete,axiom,
    ! [Xs: list_o] :
      ( ( slice_o @ zero_zero_nat @ ( size_size_list_o @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_4978_slice__complete,axiom,
    ! [Xs: list_int] :
      ( ( slice_int @ zero_zero_nat @ ( size_size_list_int @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_4979_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4980_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_4981_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_4982_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_4983_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_4984_zabs__less__one__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
      = ( Z3 = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_4985_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_4986_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_4987_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_4988_DiffE,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( minus_minus_set_o @ A4 @ B5 ) )
     => ~ ( ( member_o @ C2 @ A4 )
         => ( member_o @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_4989_DiffE,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( minus_2612819937483484256nt_int @ A4 @ B5 ) )
     => ~ ( ( member2340774599025711042nt_int @ C2 @ A4 )
         => ( member2340774599025711042nt_int @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_4990_DiffE,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) )
     => ~ ( ( member_set_nat @ C2 @ A4 )
         => ( member_set_nat @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_4991_DiffE,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B5 ) )
     => ~ ( ( member_nat @ C2 @ A4 )
         => ( member_nat @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_4992_DiffE,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B5 ) )
     => ~ ( ( member_int @ C2 @ A4 )
         => ( member_int @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_4993_DiffE,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ C2 @ A4 )
         => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_4994_ComplD,axiom,
    ! [C2: $o,A4: set_o] :
      ( ( member_o @ C2 @ ( uminus_uminus_set_o @ A4 ) )
     => ~ ( member_o @ C2 @ A4 ) ) ).

% ComplD
thf(fact_4995_ComplD,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( uminus6423885277529793776nt_int @ A4 ) )
     => ~ ( member2340774599025711042nt_int @ C2 @ A4 ) ) ).

% ComplD
thf(fact_4996_ComplD,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( uminus613421341184616069et_nat @ A4 ) )
     => ~ ( member_set_nat @ C2 @ A4 ) ) ).

% ComplD
thf(fact_4997_ComplD,axiom,
    ! [C2: nat,A4: set_nat] :
      ( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A4 ) )
     => ~ ( member_nat @ C2 @ A4 ) ) ).

% ComplD
thf(fact_4998_ComplD,axiom,
    ! [C2: int,A4: set_int] :
      ( ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A4 ) )
     => ~ ( member_int @ C2 @ A4 ) ) ).

% ComplD
thf(fact_4999_DiffD1,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( minus_minus_set_o @ A4 @ B5 ) )
     => ( member_o @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_5000_DiffD1,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( minus_2612819937483484256nt_int @ A4 @ B5 ) )
     => ( member2340774599025711042nt_int @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_5001_DiffD1,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) )
     => ( member_set_nat @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_5002_DiffD1,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B5 ) )
     => ( member_nat @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_5003_DiffD1,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B5 ) )
     => ( member_int @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_5004_DiffD1,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
     => ( member8440522571783428010at_nat @ C2 @ A4 ) ) ).

% DiffD1
thf(fact_5005_DiffD2,axiom,
    ! [C2: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ C2 @ ( minus_minus_set_o @ A4 @ B5 ) )
     => ~ ( member_o @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_5006_DiffD2,axiom,
    ! [C2: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ C2 @ ( minus_2612819937483484256nt_int @ A4 @ B5 ) )
     => ~ ( member2340774599025711042nt_int @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_5007_DiffD2,axiom,
    ! [C2: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ C2 @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) )
     => ~ ( member_set_nat @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_5008_DiffD2,axiom,
    ! [C2: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B5 ) )
     => ~ ( member_nat @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_5009_DiffD2,axiom,
    ! [C2: int,A4: set_int,B5: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A4 @ B5 ) )
     => ~ ( member_int @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_5010_DiffD2,axiom,
    ! [C2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
     => ~ ( member8440522571783428010at_nat @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_5011_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_5012_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_5013_uminus__set__def,axiom,
    ( uminus6423885277529793776nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ( uminus8147837162492574189_int_o
            @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_5014_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_5015_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_5016_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_5017_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_5018_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_5019_minus__set__def,axiom,
    ( minus_2612819937483484256nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ( minus_357216186751819389_int_o
            @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ A6 )
            @ ^ [X4: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_5020_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_5021_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_5022_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_5023_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_5024_Collect__neg__eq,axiom,
    ! [P2: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ~ ( P2 @ X4 ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_5025_Collect__neg__eq,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o] :
      ( ( collec5210948495886036740nt_int
        @ ^ [X4: set_Pr958786334691620121nt_int] :
            ~ ( P2 @ X4 ) )
      = ( uminus6423885277529793776nt_int @ ( collec5210948495886036740nt_int @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_5026_Collect__neg__eq,axiom,
    ! [P2: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ~ ( P2 @ X4 ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_5027_Collect__neg__eq,axiom,
    ! [P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ~ ( P2 @ X4 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_5028_Collect__neg__eq,axiom,
    ! [P2: int > $o] :
      ( ( collect_int
        @ ^ [X4: int] :
            ~ ( P2 @ X4 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_5029_Compl__eq,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ~ ( member_o @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5030_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ~ ( member_list_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5031_Compl__eq,axiom,
    ( uminus6423885277529793776nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] :
              ~ ( member2340774599025711042nt_int @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5032_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ~ ( member_set_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5033_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ~ ( member_nat @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5034_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ~ ( member_int @ X4 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_5035_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_5036_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_5037_set__diff__eq,axiom,
    ( minus_2612819937483484256nt_int
    = ( ^ [A6: set_se6260736226359567993nt_int,B6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] :
              ( ( member2340774599025711042nt_int @ X4 @ A6 )
              & ~ ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5038_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_5039_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_5040_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_5041_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_5042_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_5043_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_5044_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_5045_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_5046_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_5047_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_5048_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_5049_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_5050_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_5051_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_5052_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_5053_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_5054_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_5055_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_5056_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5057_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5058_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5059_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_5060_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_5061_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_5062_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_5063_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_5064_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_5065_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_5066_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_5067_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_5068_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_5069_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_5070_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_5071_abs__mult__less,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C2 )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5072_abs__mult__less,axiom,
    ! [A: rat,C2: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C2 )
     => ( ( 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 @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5073_abs__mult__less,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C2 )
     => ( ( 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 @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5074_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_5075_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_5076_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_5077_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_5078_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_5079_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_5080_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_5081_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_5082_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_5083_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_5084_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_5085_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_5086_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_5087_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_5088_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_5089_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_5090_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K2 ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_5091_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri3624122377584611663nteger @ N2 ) ) ).

% fact_ge_zero
thf(fact_5092_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_zero
thf(fact_5093_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_5094_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_5095_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri3624122377584611663nteger @ N2 ) @ zero_z3403309356797280102nteger ) ).

% fact_not_neg
thf(fact_5096_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_5097_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_5098_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_5099_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri3624122377584611663nteger @ N2 ) ) ).

% fact_gt_zero
thf(fact_5100_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_gt_zero
thf(fact_5101_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_5102_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_5103_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) ) ).

% fact_ge_1
thf(fact_5104_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_1
thf(fact_5105_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_5106_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_5107_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_5108_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_5109_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_5110_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_5111_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K2 ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_addition_formula
thf(fact_5112_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_5113_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_5114_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_5115_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_5116_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_5117_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_5118_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_5119_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_5120_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_5121_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_5122_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_5123_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_5124_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_5125_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_5126_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_5127_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_5128_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_5129_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_5130_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_5131_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_5132_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_5133_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_5134_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C2 @ D2 ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5135_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C2 @ D2 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5136_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C2: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C2 @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5137_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5138_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5139_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5140_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_5141_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_5142_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_5143_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_5144_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_5145_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_5146_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_5147_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_5148_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_5149_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5150_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5151_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5152_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_5153_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_5154_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_5155_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_5156_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_5157_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I: int] : ( if_int @ ( ord_less_int @ I @ zero_zero_int ) @ ( uminus_uminus_int @ I ) @ I ) ) ) ).

% zabs_def
thf(fact_5158_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_5159_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_5160_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_5161_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K2 @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_5162_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_5163_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_5164_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo8411746178871703098atural @ ( semiri2447717529341329178atural @ N2 ) @ ( semiri2447717529341329178atural @ M ) )
        = zero_z2226904508553997617atural ) ) ).

% fact_mod
thf(fact_5165_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_5166_fact__le__power,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri4939895301339042750nteger @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_5167_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_5168_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_5169_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_5170_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% Suc_times_gbinomial
thf(fact_5171_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,M: nat,A: rat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K2 ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( minus_minus_nat @ M @ K2 ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_5172_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_5173_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_5174_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_5175_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_5176_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K2 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_5177_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R2 ) ) ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_5178_gbinomial__factors,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% gbinomial_factors
thf(fact_5179_gbinomial__rec,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_rec
thf(fact_5180_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_minus
thf(fact_5181_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_rat @ A @ K2 )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_5182_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_5183_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K2: 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 ) @ K2 )
         => ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M @ I3 )
                & ( ord_less_eq_nat @ I3 @ N2 )
                & ( ( F @ I3 )
                  = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_5184_decr__lemma,axiom,
    ! [D2: int,X: int,Z3: 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 @ Z3 ) ) @ one_one_int ) @ D2 ) ) @ Z3 ) ) ).

% decr_lemma
thf(fact_5185_incr__lemma,axiom,
    ! [D2: int,Z3: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ Z3 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z3 ) ) @ one_one_int ) @ D2 ) ) ) ) ).

% incr_lemma
thf(fact_5186_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,Q8: int] :
              ( ( A12 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q8 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q8 @ L2 ) ) )
          | ? [R5: int,L2: int,K3: int,Q8: int] :
              ( ( A12 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q8 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q8 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_5187_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 ) ) ) )
         => ~ ! [R4: int,Q5: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q5 @ R4 ) )
               => ( ( ( sgn_sgn_int @ R4 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A23 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q5 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_5188_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_5189_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_5190_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( ( rotate1_int @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_5191_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_5192_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_5193_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_5194_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q6: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K2
            = ( plus_plus_int @ ( times_times_int @ Q6 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q6 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_5195_dbl__dec__def,axiom,
    ( neg_nu7757733837767384882nteger
    = ( ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X4 @ X4 ) @ one_one_Code_integer ) ) ) ).

% dbl_dec_def
thf(fact_5196_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X4: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_5197_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_5198_sgn__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_less
thf(fact_5199_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_5200_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_5201_sgn__greater,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_greater
thf(fact_5202_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_5203_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_5204_sgn__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_5205_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_5206_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_5207_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_5208_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_5209_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_5210_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( sgn_sgn_Code_integer @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_5211_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
        = ( sgn_sgn_rat @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_5212_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
        = ( sgn_sgn_int @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_5213_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z4: list_nat] : Y6 = Z4 )
    = ( ^ [Xs3: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I )
                = ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_5214_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z4: list_o] : Y6 = Z4 )
    = ( ^ [Xs3: list_o,Ys2: list_o] :
          ( ( ( size_size_list_o @ Xs3 )
            = ( size_size_list_o @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs3 ) )
             => ( ( nth_o @ Xs3 @ I )
                = ( nth_o @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_5215_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z4: list_int] : Y6 = Z4 )
    = ( ^ [Xs3: list_int,Ys2: list_int] :
          ( ( ( size_size_list_int @ Xs3 )
            = ( size_size_list_int @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs3 ) )
             => ( ( nth_int @ Xs3 @ I )
                = ( nth_int @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_5216_Skolem__list__nth,axiom,
    ! [K2: nat,P2: nat > nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X11: nat] : ( P2 @ I @ X11 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P2 @ I @ ( nth_nat @ Xs3 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_5217_Skolem__list__nth,axiom,
    ! [K2: nat,P2: nat > $o > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X11: $o] : ( P2 @ I @ X11 ) ) )
      = ( ? [Xs3: list_o] :
            ( ( ( size_size_list_o @ Xs3 )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P2 @ I @ ( nth_o @ Xs3 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_5218_Skolem__list__nth,axiom,
    ! [K2: nat,P2: nat > int > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X11: int] : ( P2 @ I @ X11 ) ) )
      = ( ? [Xs3: list_int] :
            ( ( ( size_size_list_int @ Xs3 )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P2 @ I @ ( nth_int @ Xs3 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_5219_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_5220_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_5221_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_5222_obtain__list__from__elements,axiom,
    ! [N2: nat,P2: nat > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: nat] : ( P2 @ Li @ I3 ) )
     => ~ ! [L3: list_nat] :
            ( ( ( size_size_list_nat @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P2 @ ( nth_nat @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_5223_obtain__list__from__elements,axiom,
    ! [N2: nat,P2: $o > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: $o] : ( P2 @ Li @ I3 ) )
     => ~ ! [L3: list_o] :
            ( ( ( size_size_list_o @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P2 @ ( nth_o @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_5224_obtain__list__from__elements,axiom,
    ! [N2: nat,P2: int > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ? [Li: int] : ( P2 @ Li @ I3 ) )
     => ~ ! [L3: list_int] :
            ( ( ( size_size_list_int @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N2 )
                 => ( P2 @ ( nth_int @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_5225_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_5226_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_5227_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_5228_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_5229_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_5230_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_5231_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_5232_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_5233_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_5234_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I: int] : ( if_int @ ( I = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_5235_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_5236_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_5237_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_5238_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_5239_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_5240_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_5241_nth__rule,axiom,
    ! [I2: nat,Xs: list_nat,A: array_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( hoare_3067605981109127869le_nat @ ( snga_assn_nat @ A @ Xs ) @ ( array_nth_nat @ A @ I2 )
        @ ^ [R5: nat] :
            ( times_times_assn @ ( snga_assn_nat @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).

% nth_rule
thf(fact_5242_nth__rule,axiom,
    ! [I2: nat,Xs: list_o,A: array_o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
     => ( hoare_hoare_triple_o @ ( snga_assn_o @ A @ Xs ) @ ( array_nth_o @ A @ I2 )
        @ ^ [R5: $o] :
            ( times_times_assn @ ( snga_assn_o @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_o @ Xs @ I2 ) ) ) ) ) ) ).

% nth_rule
thf(fact_5243_nth__rule,axiom,
    ! [I2: nat,Xs: list_int,A: array_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( hoare_3065115510600077593le_int @ ( snga_assn_int @ A @ Xs ) @ ( array_nth_int @ A @ I2 )
        @ ^ [R5: int] :
            ( times_times_assn @ ( snga_assn_int @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_int @ Xs @ I2 ) ) ) ) ) ) ).

% nth_rule
thf(fact_5244_nth__rule,axiom,
    ! [I2: nat,Xs: list_Product_unit,A: array_Product_unit] :
      ( ( ord_less_nat @ I2 @ ( size_s245203480648594047t_unit @ Xs ) )
     => ( hoare_8945653483474564448t_unit @ ( snga_a4522542871529764173t_unit @ A @ Xs ) @ ( array_7872002506669749220t_unit @ A @ I2 )
        @ ^ [R5: product_unit] :
            ( times_times_assn @ ( snga_a4522542871529764173t_unit @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_Product_unit @ Xs @ I2 ) ) ) ) ) ) ).

% nth_rule
thf(fact_5245_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_5246_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_5247_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_5248_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_5249_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_5250_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_5251_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_5252_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_5253_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_5254_product__nth,axiom,
    ! [N2: nat,Xs: list_b,Ys3: list_P131111800688179804it_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_b @ Xs ) @ ( size_s341701280123345136it_nat @ Ys3 ) ) )
     => ( ( nth_Pr3591576263217728691it_nat @ ( produc1870533552098665926it_nat @ Xs @ Ys3 ) @ N2 )
        = ( produc4082563078715348724it_nat @ ( nth_b @ 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_5255_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_o,X: $o] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( 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 @ J2 ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K )
                     => ( ( ord_less_nat @ K @ J2 )
                       => ( ( ord_less_eq_o @ ( nth_o @ L @ K ) @ X )
                         => ~ ( ord_less_o @ X @ ( nth_o @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5256_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_rat,X: rat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( 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 @ J2 ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K )
                     => ( ( ord_less_nat @ K @ J2 )
                       => ( ( ord_less_eq_rat @ ( nth_rat @ L @ K ) @ X )
                         => ~ ( ord_less_rat @ X @ ( nth_rat @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5257_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_num,X: num] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( 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 @ J2 ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K )
                     => ( ( ord_less_nat @ K @ J2 )
                       => ( ( ord_less_eq_num @ ( nth_num @ L @ K ) @ X )
                         => ~ ( ord_less_num @ X @ ( nth_num @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5258_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( 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 @ J2 ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K )
                     => ( ( ord_less_nat @ K @ J2 )
                       => ( ( ord_less_eq_nat @ ( nth_nat @ L @ K ) @ X )
                         => ~ ( ord_less_nat @ X @ ( nth_nat @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5259_sorted__in__between,axiom,
    ! [I2: nat,J2: nat,L: list_int,X: int] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( 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 @ J2 ) )
               => ~ ! [K: nat] :
                      ( ( ord_less_eq_nat @ I2 @ K )
                     => ( ( ord_less_nat @ K @ J2 )
                       => ( ( ord_less_eq_int @ ( nth_int @ L @ K ) @ X )
                         => ~ ( ord_less_int @ X @ ( nth_int @ L @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
thf(fact_5260_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_5261_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_5262_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_5263_array__of__list__make,axiom,
    ( array_of_list_nat
    = ( ^ [Xs3: list_nat] : ( array_make_nat @ ( size_size_list_nat @ Xs3 ) @ ( nth_nat @ Xs3 ) ) ) ) ).

% array_of_list_make
thf(fact_5264_array__of__list__make,axiom,
    ( array_of_list_o
    = ( ^ [Xs3: list_o] : ( array_make_o @ ( size_size_list_o @ Xs3 ) @ ( nth_o @ Xs3 ) ) ) ) ).

% array_of_list_make
thf(fact_5265_array__of__list__make,axiom,
    ( array_of_list_int
    = ( ^ [Xs3: list_int] : ( array_make_int @ ( size_size_list_int @ Xs3 ) @ ( nth_int @ Xs3 ) ) ) ) ).

% array_of_list_make
thf(fact_5266_nth__step__trancl,axiom,
    ! [Xs: list_o,R3: set_Product_prod_o_o,N2: nat,M: nat] :
      ( ! [N5: nat] :
          ( ( ord_less_nat @ N5 @ ( minus_minus_nat @ ( size_size_list_o @ Xs ) @ one_one_nat ) )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ ( suc @ N5 ) ) @ ( nth_o @ Xs @ N5 ) ) @ R3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
       => ( ( ord_less_nat @ M @ N2 )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ N2 ) @ ( nth_o @ Xs @ M ) ) @ ( transitive_trancl_o @ R3 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5267_nth__step__trancl,axiom,
    ! [Xs: list_int,R3: set_Pr958786334691620121nt_int,N2: nat,M: nat] :
      ( ! [N5: nat] :
          ( ( ord_less_nat @ N5 @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ ( suc @ N5 ) ) @ ( nth_int @ Xs @ N5 ) ) @ R3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ M @ N2 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N2 ) @ ( nth_int @ Xs @ M ) ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5268_nth__step__trancl,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat,N2: nat,M: nat] :
      ( ! [N5: nat] :
          ( ( ord_less_nat @ N5 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( suc @ N5 ) ) @ ( nth_nat @ Xs @ N5 ) ) @ R3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ M @ N2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Xs @ M ) ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).

% nth_step_trancl
thf(fact_5269_nth__rotate,axiom,
    ! [N2: nat,Xs: list_nat,M: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rotate_nat @ M @ Xs ) @ N2 )
        = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% nth_rotate
thf(fact_5270_nth__rotate,axiom,
    ! [N2: nat,Xs: list_o,M: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( rotate_o @ M @ Xs ) @ N2 )
        = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N2 ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).

% nth_rotate
thf(fact_5271_nth__rotate,axiom,
    ! [N2: nat,Xs: list_int,M: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( rotate_int @ M @ Xs ) @ N2 )
        = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N2 ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).

% nth_rotate
thf(fact_5272_rotate__length01,axiom,
    ! [Xs: list_nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate_nat @ N2 @ Xs )
        = Xs ) ) ).

% rotate_length01
thf(fact_5273_rotate__length01,axiom,
    ! [Xs: list_o,N2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( ( rotate_o @ N2 @ Xs )
        = Xs ) ) ).

% rotate_length01
thf(fact_5274_rotate__length01,axiom,
    ! [Xs: list_int,N2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( ( rotate_int @ N2 @ Xs )
        = Xs ) ) ).

% rotate_length01
thf(fact_5275_sorted__wrt__true,axiom,
    ! [Xs: list_nat] :
      ( sorted_wrt_nat
      @ ^ [Uu2: nat,Uv2: nat] : $true
      @ Xs ) ).

% sorted_wrt_true
thf(fact_5276_sorted__wrt__true,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int
      @ ^ [Uu2: int,Uv2: int] : $true
      @ Xs ) ).

% sorted_wrt_true
thf(fact_5277_trancl_Ocases,axiom,
    ! [A1: int,A23: int,R2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A23 ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A23 ) @ R2 )
       => ~ ! [B3: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
             => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B3 @ A23 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_5278_trancl_Ocases,axiom,
    ! [A1: nat,A23: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A23 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A23 ) @ R2 )
       => ~ ! [B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A23 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_5279_trancl_Osimps,axiom,
    ! [A1: int,A23: int,R2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A23 ) @ ( transi6261509568448316235cl_int @ R2 ) )
      = ( ? [A5: int,B4: int] :
            ( ( A1 = A5 )
            & ( A23 = B4 )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A5 @ B4 ) @ R2 ) )
        | ? [A5: int,B4: int,C3: int] :
            ( ( A1 = A5 )
            & ( A23 = C3 )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A5 @ B4 ) @ ( transi6261509568448316235cl_int @ R2 ) )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B4 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_5280_trancl_Osimps,axiom,
    ! [A1: nat,A23: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A23 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
      = ( ? [A5: nat,B4: nat] :
            ( ( A1 = A5 )
            & ( A23 = B4 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R2 ) )
        | ? [A5: nat,B4: nat,C3: nat] :
            ( ( A1 = A5 )
            & ( A23 = C3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_5281_trancl_Or__into__trancl,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_5282_trancl_Or__into__trancl,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_5283_tranclE,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
       => ~ ! [C: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ C ) @ ( transi6261509568448316235cl_int @ R2 ) )
             => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ C @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_5284_tranclE,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ~ ! [C: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_5285_trancl__trans,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z3: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_5286_trancl__trans,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_5287_trancl__induct,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,P2: int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ! [Y3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ Y3 ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: int,Z2: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ Y3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R2 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_5288_trancl__induct,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: nat,Z2: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R2 )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z2 ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% trancl_induct
thf(fact_5289_r__r__into__trancl,axiom,
    ! [A: int,B: int,R3: set_Pr958786334691620121nt_int,C2: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R3 )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C2 ) @ R3 )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ C2 ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ).

% r_r_into_trancl
thf(fact_5290_r__r__into__trancl,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).

% r_r_into_trancl
thf(fact_5291_converse__tranclE,axiom,
    ! [X: int,Z3: int,R2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ R2 )
       => ~ ! [Y3: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ R2 )
             => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z3 ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_5292_converse__tranclE,axiom,
    ! [X: nat,Z3: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ R2 )
       => ~ ! [Y3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_5293_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ! [X3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_5294_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
thf(fact_5295_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,C2: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C2 ) @ R2 )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ C2 ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_5296_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ R2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_5297_trancl__into__trancl2,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,C2: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ C2 ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_5298_trancl__into__trancl2,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_5299_trancl__trans__induct,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,P2: int > int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ! [X3: int,Y3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R2 )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
                 => ( ( P2 @ Y3 @ Z2 )
                   => ( P2 @ X3 @ Z2 ) ) ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_5300_trancl__trans__induct,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
           => ( P2 @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
             => ( ( P2 @ X3 @ Y3 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
                 => ( ( P2 @ Y3 @ Z2 )
                   => ( P2 @ X3 @ Z2 ) ) ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% trancl_trans_induct
thf(fact_5301_converse__trancl__induct,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,P2: int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
     => ( ! [Y3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ B ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: int,Z2: int] :
              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R2 )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_5302_converse__trancl__induct,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ! [Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R2 )
           => ( P2 @ Y3 ) )
       => ( ! [Y3: nat,Z2: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R2 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
               => ( ( P2 @ Z2 )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_5303_trancl__induct2,axiom,
    ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P2: int > int > $o] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi6288783178788033498nt_int @ R2 ) )
     => ( ! [A3: int,B3: int] :
            ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A3 @ B3 ) ) @ R2 )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: int,B3: int,Aa2: int,Ba: int] :
              ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A3 @ B3 ) ) @ ( transi6288783178788033498nt_int @ R2 ) )
             => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ ( product_Pair_int_int @ Aa2 @ Ba ) ) @ R2 )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5304_trancl__induct2,axiom,
    ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
     => ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
            ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A3 @ B3 ) ) @ R2 )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat,Aa2: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
              ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A3 @ B3 ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
             => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A3 @ B3 ) @ ( produc5001842942810119800et_nat @ Aa2 @ Ba ) ) @ R2 )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5305_trancl__induct2,axiom,
    ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P2: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
     => ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
            ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A3 @ B3 ) ) @ R2 )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat,Aa2: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
              ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A3 @ B3 ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
             => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A3 @ B3 ) @ ( produc2245416461498447860et_nat @ Aa2 @ Ba ) ) @ R2 )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5306_trancl__induct2,axiom,
    ! [Ax: b,Ay: produc6653097349344004940it_nat,Bx: b,By: produc6653097349344004940it_nat,R2: set_Pr5508209795250834101it_nat,P2: b > produc6653097349344004940it_nat > $o] :
      ( ( member6820296301096096254it_nat @ ( produc6853161671299316109it_nat @ ( produc4082563078715348724it_nat @ Ax @ Ay ) @ ( produc4082563078715348724it_nat @ Bx @ By ) ) @ ( transi2518514244971309593it_nat @ R2 ) )
     => ( ! [A3: b,B3: produc6653097349344004940it_nat] :
            ( ( member6820296301096096254it_nat @ ( produc6853161671299316109it_nat @ ( produc4082563078715348724it_nat @ Ax @ Ay ) @ ( produc4082563078715348724it_nat @ A3 @ B3 ) ) @ R2 )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: b,B3: produc6653097349344004940it_nat,Aa2: b,Ba: produc6653097349344004940it_nat] :
              ( ( member6820296301096096254it_nat @ ( produc6853161671299316109it_nat @ ( produc4082563078715348724it_nat @ Ax @ Ay ) @ ( produc4082563078715348724it_nat @ A3 @ B3 ) ) @ ( transi2518514244971309593it_nat @ R2 ) )
             => ( ( member6820296301096096254it_nat @ ( produc6853161671299316109it_nat @ ( produc4082563078715348724it_nat @ A3 @ B3 ) @ ( produc4082563078715348724it_nat @ Aa2 @ Ba ) ) @ R2 )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5307_trancl__induct2,axiom,
    ! [Ax: a,Ay: produc6653097349344004940it_nat,Bx: a,By: produc6653097349344004940it_nat,R2: set_Pr2819221443900773171it_nat,P2: a > produc6653097349344004940it_nat > $o] :
      ( ( member5335690527091456380it_nat @ ( produc1743342482959036555it_nat @ ( produc9178034014595674355it_nat @ Ax @ Ay ) @ ( produc9178034014595674355it_nat @ Bx @ By ) ) @ ( transi7613985180851635224it_nat @ R2 ) )
     => ( ! [A3: a,B3: produc6653097349344004940it_nat] :
            ( ( member5335690527091456380it_nat @ ( produc1743342482959036555it_nat @ ( produc9178034014595674355it_nat @ Ax @ Ay ) @ ( produc9178034014595674355it_nat @ A3 @ B3 ) ) @ R2 )
           => ( P2 @ A3 @ B3 ) )
       => ( ! [A3: a,B3: produc6653097349344004940it_nat,Aa2: a,Ba: produc6653097349344004940it_nat] :
              ( ( member5335690527091456380it_nat @ ( produc1743342482959036555it_nat @ ( produc9178034014595674355it_nat @ Ax @ Ay ) @ ( produc9178034014595674355it_nat @ A3 @ B3 ) ) @ ( transi7613985180851635224it_nat @ R2 ) )
             => ( ( member5335690527091456380it_nat @ ( produc1743342482959036555it_nat @ ( produc9178034014595674355it_nat @ A3 @ B3 ) @ ( produc9178034014595674355it_nat @ Aa2 @ Ba ) ) @ R2 )
               => ( ( P2 @ A3 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_5308_trancl__mono,axiom,
    ! [P3: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ P3 @ ( transi6264000038957366511cl_nat @ R2 ) )
     => ( ( ord_le3146513528884898305at_nat @ R2 @ S2 )
       => ( member8440522571783428010at_nat @ P3 @ ( transi6264000038957366511cl_nat @ S2 ) ) ) ) ).

% trancl_mono
thf(fact_5309_trancl__mono__mp,axiom,
    ! [U2: set_Pr1261947904930325089at_nat,V3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ U2 @ V3 )
     => ( ( member8440522571783428010at_nat @ X @ ( transi6264000038957366511cl_nat @ U2 ) )
       => ( member8440522571783428010at_nat @ X @ ( transi6264000038957366511cl_nat @ V3 ) ) ) ) ).

% trancl_mono_mp
thf(fact_5310_trancl__sub,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ R3 @ ( transi6264000038957366511cl_nat @ R3 ) ) ).

% trancl_sub
thf(fact_5311_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_5312_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_5313_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_5314_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_5315_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_5316_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_nat
    = ( ^ [P5: nat > nat > $o,Xs3: list_nat] :
        ! [I: nat,J: nat] :
          ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs3 ) )
           => ( P5 @ ( nth_nat @ Xs3 @ I ) @ ( nth_nat @ Xs3 @ J ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5317_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_o
    = ( ^ [P5: $o > $o > $o,Xs3: list_o] :
        ! [I: nat,J: nat] :
          ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs3 ) )
           => ( P5 @ ( nth_o @ Xs3 @ I ) @ ( nth_o @ Xs3 @ J ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5318_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_int
    = ( ^ [P5: int > int > $o,Xs3: list_int] :
        ! [I: nat,J: nat] :
          ( ( ord_less_nat @ I @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs3 ) )
           => ( P5 @ ( nth_int @ Xs3 @ I ) @ ( nth_int @ Xs3 @ J ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_5319_sorted__wrt__nth__less,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat,I2: nat,J2: nat] :
      ( ( sorted_wrt_nat @ P2 @ Xs )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
         => ( P2 @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5320_sorted__wrt__nth__less,axiom,
    ! [P2: $o > $o > $o,Xs: list_o,I2: nat,J2: nat] :
      ( ( sorted_wrt_o @ P2 @ Xs )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_o @ Xs ) )
         => ( P2 @ ( nth_o @ Xs @ I2 ) @ ( nth_o @ Xs @ J2 ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5321_sorted__wrt__nth__less,axiom,
    ! [P2: int > int > $o,Xs: list_int,I2: nat,J2: nat] :
      ( ( sorted_wrt_int @ P2 @ Xs )
     => ( ( ord_less_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
         => ( P2 @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_5322_sorted__wrt01,axiom,
    ! [Xs: list_nat,P2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_nat @ P2 @ Xs ) ) ).

% sorted_wrt01
thf(fact_5323_sorted__wrt01,axiom,
    ! [Xs: list_o,P2: $o > $o > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( sorted_wrt_o @ P2 @ Xs ) ) ).

% sorted_wrt01
thf(fact_5324_sorted__wrt01,axiom,
    ! [Xs: list_int,P2: int > int > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( sorted_wrt_int @ P2 @ Xs ) ) ).

% sorted_wrt01
thf(fact_5325_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
             => ( ord_less_eq_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5326_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_rat @ Xs ) )
             => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I ) @ ( nth_rat @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5327_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_num @ Xs ) )
             => ( ord_less_eq_num @ ( nth_num @ Xs @ I ) @ ( nth_num @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5328_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5329_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_5330_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_5331_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_5332_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_5333_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_5334_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_5335_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_o @ Xs ) )
           => ( ord_less_eq_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Xs @ ( suc @ I ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5336_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_rat @ Xs ) )
           => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I ) @ ( nth_rat @ Xs @ ( suc @ I ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5337_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_num @ Xs ) )
           => ( ord_less_eq_num @ ( nth_num @ Xs @ I ) @ ( nth_num @ Xs @ ( suc @ I ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5338_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ ( suc @ I ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5339_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ ( suc @ I ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_5340_sorted__iff__nth__mono,axiom,
    ! [Xs: list_o] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_eq_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
             => ( ord_less_eq_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5341_sorted__iff__nth__mono,axiom,
    ! [Xs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_eq_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_rat @ Xs ) )
             => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I ) @ ( nth_rat @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5342_sorted__iff__nth__mono,axiom,
    ! [Xs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_eq_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_num @ Xs ) )
             => ( ord_less_eq_num @ ( nth_num @ Xs @ I ) @ ( nth_num @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5343_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_eq_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5344_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I: nat,J: nat] :
            ( ( ord_less_eq_nat @ I @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_5345_sorted__nth__mono,axiom,
    ! [Xs: list_o,I2: nat,J2: nat] :
      ( ( sorted_wrt_o @ ord_less_eq_o @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_o @ Xs ) )
         => ( ord_less_eq_o @ ( nth_o @ Xs @ I2 ) @ ( nth_o @ Xs @ J2 ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5346_sorted__nth__mono,axiom,
    ! [Xs: list_rat,I2: nat,J2: nat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_rat @ Xs ) )
         => ( ord_less_eq_rat @ ( nth_rat @ Xs @ I2 ) @ ( nth_rat @ Xs @ J2 ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5347_sorted__nth__mono,axiom,
    ! [Xs: list_num,I2: nat,J2: nat] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_num @ Xs ) )
         => ( ord_less_eq_num @ ( nth_num @ Xs @ I2 ) @ ( nth_num @ Xs @ J2 ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5348_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I2: nat,J2: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5349_sorted__nth__mono,axiom,
    ! [Xs: list_int,I2: nat,J2: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_5350_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys3: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys3 ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5351_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys3: list_o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys3 ) )
       => ( ( nth_Pr112076138515278198_nat_o @ ( zip_nat_o @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ I2 ) @ ( nth_o @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5352_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys3: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys3 ) )
       => ( ( nth_Pr3440142176431000676at_int @ ( zip_nat_int @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_nat_int @ ( nth_nat @ Xs @ I2 ) @ ( nth_int @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5353_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys3: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys3 ) )
       => ( ( nth_Pr5826913651314560976_o_nat @ ( zip_o_nat @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_o_nat @ ( nth_o @ Xs @ I2 ) @ ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5354_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys3: list_o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys3 ) )
       => ( ( nth_Product_prod_o_o @ ( zip_o_o @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_o_o @ ( nth_o @ Xs @ I2 ) @ ( nth_o @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5355_nth__zip,axiom,
    ! [I2: nat,Xs: list_o,Ys3: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys3 ) )
       => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_o_int @ ( nth_o @ Xs @ I2 ) @ ( nth_int @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5356_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys3: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys3 ) )
       => ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_int_nat @ ( nth_int @ Xs @ I2 ) @ ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5357_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys3: list_o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys3 ) )
       => ( ( nth_Pr7514405829937366042_int_o @ ( zip_int_o @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_int_o @ ( nth_int @ Xs @ I2 ) @ ( nth_o @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5358_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys3: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys3 ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys3 ) @ I2 )
          = ( product_Pair_int_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5359_nth__zip,axiom,
    ! [I2: nat,Xs: list_b,Ys3: list_P131111800688179804it_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s341701280123345136it_nat @ Ys3 ) )
       => ( ( nth_Pr3591576263217728691it_nat @ ( zip_b_5987488365222434772it_nat @ Xs @ Ys3 ) @ I2 )
          = ( produc4082563078715348724it_nat @ ( nth_b @ Xs @ I2 ) @ ( nth_Pr4561283279332054789it_nat @ Ys3 @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5360_nth__Cons__pos,axiom,
    ! [N2: nat,X: int,Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N2 )
        = ( nth_int @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_5361_nth__Cons__pos,axiom,
    ! [N2: nat,X: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_5362_find__Some__iff,axiom,
    ! [P2: produc7388388658123137530it_nat > $o,Xs: list_P7438302566501821706it_nat,X: produc7388388658123137530it_nat] :
      ( ( ( find_P8943372834735446238it_nat @ P2 @ Xs )
        = ( some_P2818173045054083285it_nat @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s4572066133376685982it_nat @ Xs ) )
            & ( P2 @ ( nth_Pr3591576263217728691it_nat @ Xs @ I ) )
            & ( X
              = ( nth_Pr3591576263217728691it_nat @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_Pr3591576263217728691it_nat @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5363_find__Some__iff,axiom,
    ! [P2: produc3260487557148687353it_nat > $o,Xs: list_P6935614879863011209it_nat,X: produc3260487557148687353it_nat] :
      ( ( ( find_P4815471733760996061it_nat @ P2 @ Xs )
        = ( some_P7913643980934408916it_nat @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s4069378446737875485it_nat @ Xs ) )
            & ( P2 @ ( nth_Pr8687047199098054322it_nat @ Xs @ I ) )
            & ( X
              = ( nth_Pr8687047199098054322it_nat @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_Pr8687047199098054322it_nat @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5364_find__Some__iff,axiom,
    ! [P2: produc8664842809031399944it_nat > $o,Xs: list_P626663023886443800it_nat,X: produc8664842809031399944it_nat] :
      ( ( ( find_P5317947125398478060it_nat @ P2 @ Xs )
        = ( some_P1914260805536162275it_nat @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s4881146867534228396it_nat @ Xs ) )
            & ( P2 @ ( nth_Pr2956344619228612545it_nat @ Xs @ I ) )
            & ( X
              = ( nth_Pr2956344619228612545it_nat @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_Pr2956344619228612545it_nat @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5365_find__Some__iff,axiom,
    ! [P2: num > $o,Xs: list_num,X: num] :
      ( ( ( find_num @ P2 @ Xs )
        = ( some_num @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_num @ Xs ) )
            & ( P2 @ ( nth_num @ Xs @ I ) )
            & ( X
              = ( nth_num @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_num @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5366_find__Some__iff,axiom,
    ! [P2: nat > $o,Xs: list_nat,X: nat] :
      ( ( ( find_nat @ P2 @ Xs )
        = ( some_nat @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
            & ( P2 @ ( nth_nat @ Xs @ I ) )
            & ( X
              = ( nth_nat @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5367_find__Some__iff,axiom,
    ! [P2: $o > $o,Xs: list_o,X: $o] :
      ( ( ( find_o @ P2 @ Xs )
        = ( some_o @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
            & ( P2 @ ( nth_o @ Xs @ I ) )
            & ( X
              = ( nth_o @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_o @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5368_find__Some__iff,axiom,
    ! [P2: int > $o,Xs: list_int,X: int] :
      ( ( ( find_int @ P2 @ Xs )
        = ( some_int @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
            & ( P2 @ ( nth_int @ Xs @ I ) )
            & ( X
              = ( nth_int @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_5369_find__Some__iff2,axiom,
    ! [X: produc7388388658123137530it_nat,P2: produc7388388658123137530it_nat > $o,Xs: list_P7438302566501821706it_nat] :
      ( ( ( some_P2818173045054083285it_nat @ X )
        = ( find_P8943372834735446238it_nat @ P2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s4572066133376685982it_nat @ Xs ) )
            & ( P2 @ ( nth_Pr3591576263217728691it_nat @ Xs @ I ) )
            & ( X
              = ( nth_Pr3591576263217728691it_nat @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_Pr3591576263217728691it_nat @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5370_find__Some__iff2,axiom,
    ! [X: produc3260487557148687353it_nat,P2: produc3260487557148687353it_nat > $o,Xs: list_P6935614879863011209it_nat] :
      ( ( ( some_P7913643980934408916it_nat @ X )
        = ( find_P4815471733760996061it_nat @ P2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s4069378446737875485it_nat @ Xs ) )
            & ( P2 @ ( nth_Pr8687047199098054322it_nat @ Xs @ I ) )
            & ( X
              = ( nth_Pr8687047199098054322it_nat @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_Pr8687047199098054322it_nat @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5371_find__Some__iff2,axiom,
    ! [X: produc8664842809031399944it_nat,P2: produc8664842809031399944it_nat > $o,Xs: list_P626663023886443800it_nat] :
      ( ( ( some_P1914260805536162275it_nat @ X )
        = ( find_P5317947125398478060it_nat @ P2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s4881146867534228396it_nat @ Xs ) )
            & ( P2 @ ( nth_Pr2956344619228612545it_nat @ Xs @ I ) )
            & ( X
              = ( nth_Pr2956344619228612545it_nat @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_Pr2956344619228612545it_nat @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5372_find__Some__iff2,axiom,
    ! [X: num,P2: num > $o,Xs: list_num] :
      ( ( ( some_num @ X )
        = ( find_num @ P2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_num @ Xs ) )
            & ( P2 @ ( nth_num @ Xs @ I ) )
            & ( X
              = ( nth_num @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_num @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5373_find__Some__iff2,axiom,
    ! [X: nat,P2: nat > $o,Xs: list_nat] :
      ( ( ( some_nat @ X )
        = ( find_nat @ P2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
            & ( P2 @ ( nth_nat @ Xs @ I ) )
            & ( X
              = ( nth_nat @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5374_find__Some__iff2,axiom,
    ! [X: $o,P2: $o > $o,Xs: list_o] :
      ( ( ( some_o @ X )
        = ( find_o @ P2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
            & ( P2 @ ( nth_o @ Xs @ I ) )
            & ( X
              = ( nth_o @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_o @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5375_find__Some__iff2,axiom,
    ! [X: int,P2: int > $o,Xs: list_int] :
      ( ( ( some_int @ X )
        = ( find_int @ P2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
            & ( P2 @ ( nth_int @ Xs @ I ) )
            & ( X
              = ( nth_int @ Xs @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_5376_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5377_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5378_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_5379_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_5380_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_5381_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_5382_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_5383_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_5384_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_5385_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_5386_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_5387_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_5388_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_5389_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_5390_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_5391_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ M ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_5392_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_5393_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_5394_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_5395_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_5396_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ W2 ) @ Z3 ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_5397_add__numeral__left,axiom,
    ! [V: num,W2: num,Z3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W2 ) @ Z3 ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_5398_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_5399_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_5400_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_5401_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_5402_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_5403_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_5404_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_5405_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_5406_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_nat,A2: list_nat,B2: list_nat] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_nat_nat @ A @ B )
            = ( zip_nat_nat @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5407_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_o,A2: list_nat,B2: list_o] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_nat_o @ A @ B )
            = ( zip_nat_o @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5408_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_int,A2: list_nat,B2: list_int] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_size_list_int @ B2 ) )
       => ( ( ( zip_nat_int @ A @ B )
            = ( zip_nat_int @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5409_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_nat,A2: list_o,B2: list_nat] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_o_nat @ A @ B )
            = ( zip_o_nat @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5410_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_o,A2: list_o,B2: list_o] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_o_o @ A @ B )
            = ( zip_o_o @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5411_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_int,A2: list_o,B2: list_int] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_size_list_int @ B2 ) )
       => ( ( ( zip_o_int @ A @ B )
            = ( zip_o_int @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5412_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_nat,A2: list_int,B2: list_nat] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_int @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_int_nat @ A @ B )
            = ( zip_int_nat @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5413_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_o,A2: list_int,B2: list_o] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_int @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_int_o @ A @ B )
            = ( zip_int_o @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5414_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_int,A2: list_int,B2: list_int] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_int @ A2 )
          = ( size_size_list_int @ B2 ) )
       => ( ( ( zip_int_int @ A @ B )
            = ( zip_int_int @ A2 @ B2 ) )
          = ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_5415_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_5416_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_5417_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_5418_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_5419_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_5420_distrib__left__numeral,axiom,
    ! [V: num,B: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_5421_distrib__left__numeral,axiom,
    ! [V: num,B: code_natural,C2: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ B @ C2 ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ B ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_5422_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ B @ C2 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_5423_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_5424_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_5425_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_5426_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_5427_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_5428_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_5429_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_5430_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_5431_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5432_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5433_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5434_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5435_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5436_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5437_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5438_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5439_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5440_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5441_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5442_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5443_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: nat,Ys3: list_nat] :
      ( ( zip_int_nat @ ( cons_int @ X @ Xs ) @ ( cons_nat @ Y @ Ys3 ) )
      = ( cons_P7512249878480867347nt_nat @ ( product_Pair_int_nat @ X @ Y ) @ ( zip_int_nat @ Xs @ Ys3 ) ) ) ).

% zip_Cons_Cons
thf(fact_5444_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: int,Ys3: list_int] :
      ( ( zip_nat_int @ ( cons_nat @ X @ Xs ) @ ( cons_int @ Y @ Ys3 ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ X @ Y ) @ ( zip_nat_int @ Xs @ Ys3 ) ) ) ).

% zip_Cons_Cons
thf(fact_5445_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys3: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys3 ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys3 ) ) ) ).

% zip_Cons_Cons
thf(fact_5446_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys3: list_int] :
      ( ( zip_int_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys3 ) )
      = ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs @ Ys3 ) ) ) ).

% zip_Cons_Cons
thf(fact_5447_zip__Cons__Cons,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o,Y: produc3658429121746597890et_nat,Ys3: list_P9062070895058802706et_nat] :
      ( ( zip_Pr7134870689397686104et_nat @ ( cons_P6219271836124797827_nat_o @ X @ Xs ) @ ( cons_P2869003950965171916et_nat @ Y @ Ys3 ) )
      = ( cons_P3419706843779801160et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys3 ) ) ) ).

% zip_Cons_Cons
thf(fact_5448_zip__Cons__Cons,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o,Y: produc3925858234332021118et_nat,Ys3: list_P2321686559999237006et_nat] :
      ( ( zip_Pr8136144321567152340et_nat @ ( cons_P6219271836124797827_nat_o @ X @ Xs ) @ ( cons_P3419706843779801160et_nat @ Y @ Ys3 ) )
      = ( cons_P6329468272379687876et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys3 ) ) ) ).

% zip_Cons_Cons
thf(fact_5449_zip__Cons__Cons,axiom,
    ! [X: b,Xs: list_b,Y: produc6653097349344004940it_nat,Ys3: list_P131111800688179804it_nat] :
      ( ( zip_b_5987488365222434772it_nat @ ( cons_b @ X @ Xs ) @ ( cons_P7873165156130251286it_nat @ Y @ Ys3 ) )
      = ( cons_P6467324686191779524it_nat @ ( produc4082563078715348724it_nat @ X @ Y ) @ ( zip_b_5987488365222434772it_nat @ Xs @ Ys3 ) ) ) ).

% zip_Cons_Cons
thf(fact_5450_zip__Cons__Cons,axiom,
    ! [X: a,Xs: list_a,Y: produc6653097349344004940it_nat,Ys3: list_P131111800688179804it_nat] :
      ( ( zip_a_1859587264247984595it_nat @ ( cons_a @ X @ Xs ) @ ( cons_P7873165156130251286it_nat @ Y @ Ys3 ) )
      = ( cons_P2339423585217329347it_nat @ ( produc9178034014595674355it_nat @ X @ Y ) @ ( zip_a_1859587264247984595it_nat @ Xs @ Ys3 ) ) ) ).

% zip_Cons_Cons
thf(fact_5451_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_5452_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_5453_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_5454_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_5455_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_5456_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_5457_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_5458_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_5459_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_5460_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_5461_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_5462_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_5463_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_5464_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_5465_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_5466_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_5467_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_5468_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_5469_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_5470_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_5471_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_5472_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_5473_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_5474_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_5475_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_5476_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_5477_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_5478_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_5479_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_5480_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_5481_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_5482_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_5483_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I: num] : ( nat2 @ ( numeral_numeral_int @ I ) ) ) ) ).

% nat_numeral_as_int
thf(fact_5484_int__ops_I3_J,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% int_ops(3)
thf(fact_5485_list__tail__coinc,axiom,
    ! [N1: int,R12: list_int,N22: int,R23: list_int] :
      ( ( ( cons_int @ N1 @ R12 )
        = ( cons_int @ N22 @ R23 ) )
     => ( ( N1 = N22 )
        & ( R12 = R23 ) ) ) ).

% list_tail_coinc
thf(fact_5486_list__tail__coinc,axiom,
    ! [N1: nat,R12: list_nat,N22: nat,R23: list_nat] :
      ( ( ( cons_nat @ N1 @ R12 )
        = ( cons_nat @ N22 @ R23 ) )
     => ( ( N1 = N22 )
        & ( R12 = R23 ) ) ) ).

% list_tail_coinc
thf(fact_5487_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys3: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( ( zip_int_nat @ Xs @ Ys3 )
        = ( cons_P7512249878480867347nt_nat @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs4: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs4 ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys3
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_int_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_5488_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys3: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( ( zip_nat_int @ Xs @ Ys3 )
        = ( cons_P2335045147070616083at_int @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs4 ) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys3
                  = ( cons_int @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_5489_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys3 )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs4 ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys3
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_5490_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys3: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( ( zip_int_int @ Xs @ Ys3 )
        = ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs4: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs4 ) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys3
                  = ( cons_int @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_int_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_5491_zip__eq__ConsE,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,Ys3: list_P9062070895058802706et_nat,Xy: produc3925858234332021118et_nat,Xys: list_P2321686559999237006et_nat] :
      ( ( ( zip_Pr7134870689397686104et_nat @ Xs @ Ys3 )
        = ( cons_P3419706843779801160et_nat @ Xy @ Xys ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Xs4: list_P7985473006766602707_nat_o] :
            ( ( Xs
              = ( cons_P6219271836124797827_nat_o @ X3 @ Xs4 ) )
           => ! [Y3: produc3658429121746597890et_nat,Ys4: list_P9062070895058802706et_nat] :
                ( ( Ys3
                  = ( cons_P2869003950965171916et_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr7134870689397686104et_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_5492_zip__eq__ConsE,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,Ys3: list_P2321686559999237006et_nat,Xy: produc2732055786443039994et_nat,Xys: list_P362550909693114634et_nat] :
      ( ( ( zip_Pr8136144321567152340et_nat @ Xs @ Ys3 )
        = ( cons_P6329468272379687876et_nat @ Xy @ Xys ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Xs4: list_P7985473006766602707_nat_o] :
            ( ( Xs
              = ( cons_P6219271836124797827_nat_o @ X3 @ Xs4 ) )
           => ! [Y3: produc3925858234332021118et_nat,Ys4: list_P2321686559999237006et_nat] :
                ( ( Ys3
                  = ( cons_P3419706843779801160et_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc2245416461498447860et_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr8136144321567152340et_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_5493_zip__eq__ConsE,axiom,
    ! [Xs: list_b,Ys3: list_P131111800688179804it_nat,Xy: produc7388388658123137530it_nat,Xys: list_P7438302566501821706it_nat] :
      ( ( ( zip_b_5987488365222434772it_nat @ Xs @ Ys3 )
        = ( cons_P6467324686191779524it_nat @ Xy @ Xys ) )
     => ~ ! [X3: b,Xs4: list_b] :
            ( ( Xs
              = ( cons_b @ X3 @ Xs4 ) )
           => ! [Y3: produc6653097349344004940it_nat,Ys4: list_P131111800688179804it_nat] :
                ( ( Ys3
                  = ( cons_P7873165156130251286it_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc4082563078715348724it_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_b_5987488365222434772it_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_5494_zip__eq__ConsE,axiom,
    ! [Xs: list_a,Ys3: list_P131111800688179804it_nat,Xy: produc3260487557148687353it_nat,Xys: list_P6935614879863011209it_nat] :
      ( ( ( zip_a_1859587264247984595it_nat @ Xs @ Ys3 )
        = ( cons_P2339423585217329347it_nat @ Xy @ Xys ) )
     => ~ ! [X3: a,Xs4: list_a] :
            ( ( Xs
              = ( cons_a @ X3 @ Xs4 ) )
           => ! [Y3: produc6653097349344004940it_nat,Ys4: list_P131111800688179804it_nat] :
                ( ( Ys3
                  = ( cons_P7873165156130251286it_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_a_1859587264247984595it_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_5495_find_Osimps_I2_J,axiom,
    ! [P2: int > $o,X: int,Xs: list_int] :
      ( ( ( P2 @ X )
       => ( ( find_int @ P2 @ ( cons_int @ X @ Xs ) )
          = ( some_int @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_int @ P2 @ ( cons_int @ X @ Xs ) )
          = ( find_int @ P2 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5496_find_Osimps_I2_J,axiom,
    ! [P2: nat > $o,X: nat,Xs: list_nat] :
      ( ( ( P2 @ X )
       => ( ( find_nat @ P2 @ ( cons_nat @ X @ Xs ) )
          = ( some_nat @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_nat @ P2 @ ( cons_nat @ X @ Xs ) )
          = ( find_nat @ P2 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5497_find_Osimps_I2_J,axiom,
    ! [P2: produc7388388658123137530it_nat > $o,X: produc7388388658123137530it_nat,Xs: list_P7438302566501821706it_nat] :
      ( ( ( P2 @ X )
       => ( ( find_P8943372834735446238it_nat @ P2 @ ( cons_P6467324686191779524it_nat @ X @ Xs ) )
          = ( some_P2818173045054083285it_nat @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_P8943372834735446238it_nat @ P2 @ ( cons_P6467324686191779524it_nat @ X @ Xs ) )
          = ( find_P8943372834735446238it_nat @ P2 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5498_find_Osimps_I2_J,axiom,
    ! [P2: produc3260487557148687353it_nat > $o,X: produc3260487557148687353it_nat,Xs: list_P6935614879863011209it_nat] :
      ( ( ( P2 @ X )
       => ( ( find_P4815471733760996061it_nat @ P2 @ ( cons_P2339423585217329347it_nat @ X @ Xs ) )
          = ( some_P7913643980934408916it_nat @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_P4815471733760996061it_nat @ P2 @ ( cons_P2339423585217329347it_nat @ X @ Xs ) )
          = ( find_P4815471733760996061it_nat @ P2 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5499_find_Osimps_I2_J,axiom,
    ! [P2: produc8664842809031399944it_nat > $o,X: produc8664842809031399944it_nat,Xs: list_P626663023886443800it_nat] :
      ( ( ( P2 @ X )
       => ( ( find_P5317947125398478060it_nat @ P2 @ ( cons_P4136552807766418642it_nat @ X @ Xs ) )
          = ( some_P1914260805536162275it_nat @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_P5317947125398478060it_nat @ P2 @ ( cons_P4136552807766418642it_nat @ X @ Xs ) )
          = ( find_P5317947125398478060it_nat @ P2 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5500_find_Osimps_I2_J,axiom,
    ! [P2: num > $o,X: num,Xs: list_num] :
      ( ( ( P2 @ X )
       => ( ( find_num @ P2 @ ( cons_num @ X @ Xs ) )
          = ( some_num @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_num @ P2 @ ( cons_num @ X @ Xs ) )
          = ( find_num @ P2 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5501_pair__list__split,axiom,
    ! [L: list_P6011104703257516679at_nat] :
      ~ ! [L1: list_nat,L22: list_nat] :
          ( ( L
            = ( zip_nat_nat @ L1 @ L22 ) )
         => ( ( ( size_size_list_nat @ L1 )
              = ( size_size_list_nat @ L22 ) )
           => ( ( size_s5460976970255530739at_nat @ L )
             != ( size_size_list_nat @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5502_pair__list__split,axiom,
    ! [L: list_P7333126701944960589_nat_o] :
      ~ ! [L1: list_nat,L22: list_o] :
          ( ( L
            = ( zip_nat_o @ L1 @ L22 ) )
         => ( ( ( size_size_list_nat @ L1 )
              = ( size_size_list_o @ L22 ) )
           => ( ( size_s6491369823275344609_nat_o @ L )
             != ( size_size_list_o @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5503_pair__list__split,axiom,
    ! [L: list_P3521021558325789923at_int] :
      ~ ! [L1: list_nat,L22: list_int] :
          ( ( L
            = ( zip_nat_int @ L1 @ L22 ) )
         => ( ( ( size_size_list_nat @ L1 )
              = ( size_size_list_int @ L22 ) )
           => ( ( size_s2970893825323803983at_int @ L )
             != ( size_size_list_int @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5504_pair__list__split,axiom,
    ! [L: list_P6285523579766656935_o_nat] :
      ~ ! [L1: list_o,L22: list_nat] :
          ( ( L
            = ( zip_o_nat @ L1 @ L22 ) )
         => ( ( ( size_size_list_o @ L1 )
              = ( size_size_list_nat @ L22 ) )
           => ( ( size_s5443766701097040955_o_nat @ L )
             != ( size_size_list_nat @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5505_pair__list__split,axiom,
    ! [L: list_P4002435161011370285od_o_o] :
      ~ ! [L1: list_o,L22: list_o] :
          ( ( L
            = ( zip_o_o @ L1 @ L22 ) )
         => ( ( ( size_size_list_o @ L1 )
              = ( size_size_list_o @ L22 ) )
           => ( ( size_s1515746228057227161od_o_o @ L )
             != ( size_size_list_o @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5506_pair__list__split,axiom,
    ! [L: list_P3795440434834930179_o_int] :
      ~ ! [L1: list_o,L22: list_int] :
          ( ( L
            = ( zip_o_int @ L1 @ L22 ) )
         => ( ( ( size_size_list_o @ L1 )
              = ( size_size_list_int @ L22 ) )
           => ( ( size_s2953683556165314199_o_int @ L )
             != ( size_size_list_int @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5507_pair__list__split,axiom,
    ! [L: list_P8198026277950538467nt_nat] :
      ~ ! [L1: list_int,L22: list_nat] :
          ( ( L
            = ( zip_int_nat @ L1 @ L22 ) )
         => ( ( ( size_size_list_int @ L1 )
              = ( size_size_list_nat @ L22 ) )
           => ( ( size_s7647898544948552527nt_nat @ L )
             != ( size_size_list_nat @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5508_pair__list__split,axiom,
    ! [L: list_P5087981734274514673_int_o] :
      ~ ! [L1: list_int,L22: list_o] :
          ( ( L
            = ( zip_int_o @ L1 @ L22 ) )
         => ( ( ( size_size_list_int @ L1 )
              = ( size_size_list_o @ L22 ) )
           => ( ( size_s4246224855604898693_int_o @ L )
             != ( size_size_list_o @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5509_pair__list__split,axiom,
    ! [L: list_P5707943133018811711nt_int] :
      ~ ! [L1: list_int,L22: list_int] :
          ( ( L
            = ( zip_int_int @ L1 @ L22 ) )
         => ( ( ( size_size_list_int @ L1 )
              = ( size_size_list_int @ L22 ) )
           => ( ( size_s5157815400016825771nt_int @ L )
             != ( size_size_list_int @ L22 ) ) ) ) ).

% pair_list_split
thf(fact_5510_zip__inj,axiom,
    ! [A: list_nat,B: list_nat,A2: list_nat,B2: list_nat] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_nat_nat @ A @ B )
            = ( zip_nat_nat @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5511_zip__inj,axiom,
    ! [A: list_nat,B: list_o,A2: list_nat,B2: list_o] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_nat_o @ A @ B )
            = ( zip_nat_o @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5512_zip__inj,axiom,
    ! [A: list_nat,B: list_int,A2: list_nat,B2: list_int] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_nat @ A2 )
          = ( size_size_list_int @ B2 ) )
       => ( ( ( zip_nat_int @ A @ B )
            = ( zip_nat_int @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5513_zip__inj,axiom,
    ! [A: list_o,B: list_nat,A2: list_o,B2: list_nat] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_o_nat @ A @ B )
            = ( zip_o_nat @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5514_zip__inj,axiom,
    ! [A: list_o,B: list_o,A2: list_o,B2: list_o] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_o_o @ A @ B )
            = ( zip_o_o @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5515_zip__inj,axiom,
    ! [A: list_o,B: list_int,A2: list_o,B2: list_int] :
      ( ( ( size_size_list_o @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_o @ A2 )
          = ( size_size_list_int @ B2 ) )
       => ( ( ( zip_o_int @ A @ B )
            = ( zip_o_int @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5516_zip__inj,axiom,
    ! [A: list_int,B: list_nat,A2: list_int,B2: list_nat] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_int @ A2 )
          = ( size_size_list_nat @ B2 ) )
       => ( ( ( zip_int_nat @ A @ B )
            = ( zip_int_nat @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5517_zip__inj,axiom,
    ! [A: list_int,B: list_o,A2: list_int,B2: list_o] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_o @ B ) )
     => ( ( ( size_size_list_int @ A2 )
          = ( size_size_list_o @ B2 ) )
       => ( ( ( zip_int_o @ A @ B )
            = ( zip_int_o @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5518_zip__inj,axiom,
    ! [A: list_int,B: list_int,A2: list_int,B2: list_int] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_int @ A2 )
          = ( size_size_list_int @ B2 ) )
       => ( ( ( zip_int_int @ A @ B )
            = ( zip_int_int @ A2 @ B2 ) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
thf(fact_5519_find__SomeD_I1_J,axiom,
    ! [P2: produc7388388658123137530it_nat > $o,Xs: list_P7438302566501821706it_nat,X: produc7388388658123137530it_nat] :
      ( ( ( find_P8943372834735446238it_nat @ P2 @ Xs )
        = ( some_P2818173045054083285it_nat @ X ) )
     => ( P2 @ X ) ) ).

% find_SomeD(1)
thf(fact_5520_find__SomeD_I1_J,axiom,
    ! [P2: produc3260487557148687353it_nat > $o,Xs: list_P6935614879863011209it_nat,X: produc3260487557148687353it_nat] :
      ( ( ( find_P4815471733760996061it_nat @ P2 @ Xs )
        = ( some_P7913643980934408916it_nat @ X ) )
     => ( P2 @ X ) ) ).

% find_SomeD(1)
thf(fact_5521_find__SomeD_I1_J,axiom,
    ! [P2: produc8664842809031399944it_nat > $o,Xs: list_P626663023886443800it_nat,X: produc8664842809031399944it_nat] :
      ( ( ( find_P5317947125398478060it_nat @ P2 @ Xs )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( P2 @ X ) ) ).

% find_SomeD(1)
thf(fact_5522_find__SomeD_I1_J,axiom,
    ! [P2: num > $o,Xs: list_num,X: num] :
      ( ( ( find_num @ P2 @ Xs )
        = ( some_num @ X ) )
     => ( P2 @ X ) ) ).

% find_SomeD(1)
thf(fact_5523_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N2 ) @ zero_z2226904508553997617atural ) ).

% not_numeral_le_zero
thf(fact_5524_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N2 ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_le_zero
thf(fact_5525_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_5526_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_5527_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_5528_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( numera5444537566228673987atural @ N2 ) ) ).

% zero_le_numeral
thf(fact_5529_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N2 ) ) ).

% zero_le_numeral
thf(fact_5530_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_5531_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_5532_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_5533_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_5534_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_5535_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_5536_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N2 ) @ zero_z2226904508553997617atural ) ).

% not_numeral_less_zero
thf(fact_5537_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N2 ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_less_zero
thf(fact_5538_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_5539_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_5540_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_5541_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( numera5444537566228673987atural @ N2 ) ) ).

% zero_less_numeral
thf(fact_5542_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N2 ) ) ).

% zero_less_numeral
thf(fact_5543_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N2 ) ) ).

% one_le_numeral
thf(fact_5544_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) ) ).

% one_le_numeral
thf(fact_5545_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_5546_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_5547_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_5548_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_5549_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_5550_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_5551_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N2 ) @ one_one_Code_natural ) ).

% not_numeral_less_one
thf(fact_5552_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N2 ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_5553_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5554_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_5555_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_5556_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5557_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_5558_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_5559_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_5560_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_5561_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_5562_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).

% one_plus_numeral_commute
thf(fact_5563_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% one_plus_numeral_commute
thf(fact_5564_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_5565_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5566_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_5567_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_5568_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5569_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_5570_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_5571_length__Cons,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X @ Xs ) )
      = ( suc @ ( size_size_list_o @ Xs ) ) ) ).

% length_Cons
thf(fact_5572_length__Cons,axiom,
    ! [X: int,Xs: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
      = ( suc @ ( size_size_list_int @ Xs ) ) ) ).

% length_Cons
thf(fact_5573_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys3: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys3 ) )
     => ( Xs
       != ( cons_nat @ X @ Ys3 ) ) ) ).

% impossible_Cons
thf(fact_5574_impossible__Cons,axiom,
    ! [Xs: list_o,Ys3: list_o,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys3 ) )
     => ( Xs
       != ( cons_o @ X @ Ys3 ) ) ) ).

% impossible_Cons
thf(fact_5575_impossible__Cons,axiom,
    ! [Xs: list_int,Ys3: list_int,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys3 ) )
     => ( Xs
       != ( cons_int @ X @ Ys3 ) ) ) ).

% impossible_Cons
thf(fact_5576_sorted2,axiom,
    ! [X: rat,Y: rat,Zs: list_rat] :
      ( ( sorted_wrt_rat @ ord_less_eq_rat @ ( cons_rat @ X @ ( cons_rat @ Y @ Zs ) ) )
      = ( ( ord_less_eq_rat @ X @ Y )
        & ( sorted_wrt_rat @ ord_less_eq_rat @ ( cons_rat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5577_sorted2,axiom,
    ! [X: num,Y: num,Zs: list_num] :
      ( ( sorted_wrt_num @ ord_less_eq_num @ ( cons_num @ X @ ( cons_num @ Y @ Zs ) ) )
      = ( ( ord_less_eq_num @ X @ Y )
        & ( sorted_wrt_num @ ord_less_eq_num @ ( cons_num @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5578_sorted2,axiom,
    ! [X: nat,Y: nat,Zs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs ) ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5579_sorted2,axiom,
    ! [X: int,Y: int,Zs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X @ ( cons_int @ Y @ Zs ) ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_5580_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5581_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_5582_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_5583_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5584_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_5585_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_5586_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_5587_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5588_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_5589_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_5590_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5591_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_5592_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5593_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_5594_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_5595_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5596_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_5597_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_5598_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_5599_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_5600_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_5601_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5602_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_5603_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_5604_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5605_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_5606_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_5607_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_5608_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5609_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_5610_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_5611_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5612_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_5613_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_5614_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5615_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_5616_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_5617_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5618_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_5619_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_5620_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_5621_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_5622_Suc__le__length__iff,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X4: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Ys2 ) )
            & ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5623_Suc__le__length__iff,axiom,
    ! [N2: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_o @ Xs ) )
      = ( ? [X4: $o,Ys2: list_o] :
            ( ( Xs
              = ( cons_o @ X4 @ Ys2 ) )
            & ( ord_less_eq_nat @ N2 @ ( size_size_list_o @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5624_Suc__le__length__iff,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( size_size_list_int @ Xs ) )
      = ( ? [X4: int,Ys2: list_int] :
            ( ( Xs
              = ( cons_int @ X4 @ Ys2 ) )
            & ( ord_less_eq_nat @ N2 @ ( size_size_list_int @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5625_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_5626_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_5627_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5628_list_Osize_I4_J,axiom,
    ! [X21: $o,X222: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_o @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5629_list_Osize_I4_J,axiom,
    ! [X21: int,X222: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_int @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5630_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_5631_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_5632_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5633_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5634_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N2: nat] :
      ( ( X != Y )
     => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N2 )
          = Y )
        = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5635_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N2: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
          = Y )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5636_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( cons_int @ X @ ( slice_int @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( slice_int @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_5637_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( slice_nat @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( slice_nat @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_5638_same__fstI,axiom,
    ! [P2: ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,Y8: produc3658429121746597890et_nat,Y: produc3658429121746597890et_nat,R3: ( produc3658429121746597890et_nat > $o ) > set_Pr719794911490849221et_nat] :
      ( ( P2 @ X )
     => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ Y8 @ Y ) @ ( R3 @ X ) )
       => ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ X @ Y8 ) @ ( produc5001842942810119800et_nat @ X @ Y ) ) @ ( same_f7251492184213700963et_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_5639_same__fstI,axiom,
    ! [P2: ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,Y8: produc3925858234332021118et_nat,Y: produc3925858234332021118et_nat,R3: ( produc3658429121746597890et_nat > $o ) > set_Pr7928877670098842301et_nat] :
      ( ( P2 @ X )
     => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ Y8 @ Y ) @ ( R3 @ X ) )
       => ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ X @ Y8 ) @ ( produc2245416461498447860et_nat @ X @ Y ) ) @ ( same_f1912051055550046943et_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_5640_same__fstI,axiom,
    ! [P2: b > $o,X: b,Y8: produc6653097349344004940it_nat,Y: produc6653097349344004940it_nat,R3: b > set_Pr4389693562480114009it_nat] :
      ( ( P2 @ X )
     => ( ( member5617269971687963298it_nat @ ( produc3130510018828335921it_nat @ Y8 @ Y ) @ ( R3 @ X ) )
       => ( member6820296301096096254it_nat @ ( produc6853161671299316109it_nat @ ( produc4082563078715348724it_nat @ X @ Y8 ) @ ( produc4082563078715348724it_nat @ X @ Y ) ) @ ( same_f3722904161141537503it_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_5641_same__fstI,axiom,
    ! [P2: a > $o,X: a,Y8: produc6653097349344004940it_nat,Y: produc6653097349344004940it_nat,R3: a > set_Pr4389693562480114009it_nat] :
      ( ( P2 @ X )
     => ( ( member5617269971687963298it_nat @ ( produc3130510018828335921it_nat @ Y8 @ Y ) @ ( R3 @ X ) )
       => ( member5335690527091456380it_nat @ ( produc1743342482959036555it_nat @ ( produc9178034014595674355it_nat @ X @ Y8 ) @ ( produc9178034014595674355it_nat @ X @ Y ) ) @ ( same_f8818375097021863134it_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_5642_same__fstI,axiom,
    ! [P2: int > $o,X: int,Y8: int,Y: int,R3: int > set_Pr958786334691620121nt_int] :
      ( ( P2 @ X )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y8 @ Y ) @ ( R3 @ X ) )
       => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ X @ Y8 ) @ ( product_Pair_int_int @ X @ Y ) ) @ ( same_fst_int_int @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_5643_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I: int,J: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J @ I ) @ Js @ ( upto_aux @ I @ ( minus_minus_int @ J @ one_one_int ) @ ( cons_int @ J @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_5644_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_5645_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_5646_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_5647_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_5648_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_5649_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_5650_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_5651_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_5652_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_5653_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_5654_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_5655_of__int__le__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z3 ) )
      = ( ord_less_eq_int @ W2 @ Z3 ) ) ).

% of_int_le_iff
thf(fact_5656_of__int__le__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_eq_int @ W2 @ Z3 ) ) ).

% of_int_le_iff
thf(fact_5657_of__int__less__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z3 ) )
      = ( ord_less_int @ W2 @ Z3 ) ) ).

% of_int_less_iff
thf(fact_5658_of__int__less__iff,axiom,
    ! [W2: int,Z3: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_int @ W2 @ Z3 ) ) ).

% of_int_less_iff
thf(fact_5659_of__int__add,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z3 ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_add
thf(fact_5660_of__int__add,axiom,
    ! [W2: int,Z3: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z3 ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z3 ) ) ) ).

% of_int_add
thf(fact_5661_ceiling__add__of__int,axiom,
    ! [X: rat,Z3: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z3 ) ) ).

% ceiling_add_of_int
thf(fact_5662_of__int__le__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z3 ) @ zero_z3403309356797280102nteger )
      = ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5663_of__int__le__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5664_of__int__le__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5665_of__int__0__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_le_iff
thf(fact_5666_of__int__0__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_le_iff
thf(fact_5667_of__int__0__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_le_iff
thf(fact_5668_of__int__less__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z3 ) @ zero_z3403309356797280102nteger )
      = ( ord_less_int @ Z3 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5669_of__int__less__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ zero_zero_rat )
      = ( ord_less_int @ Z3 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5670_of__int__less__0__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
      = ( ord_less_int @ Z3 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5671_of__int__0__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z3 ) )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_less_iff
thf(fact_5672_of__int__0__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z3 ) )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_less_iff
thf(fact_5673_of__int__0__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% of_int_0_less_iff
thf(fact_5674_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( ring_18347121197199848620nteger @ Z3 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_le_iff
thf(fact_5675_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z3 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_le_iff
thf(fact_5676_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_le_iff
thf(fact_5677_of__int__le__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z3 ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_eq_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5678_of__int__le__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5679_of__int__le__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5680_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z3 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_less_iff
thf(fact_5681_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_less_iff
thf(fact_5682_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z3: int] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( ring_18347121197199848620nteger @ Z3 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z3 ) ) ).

% of_int_numeral_less_iff
thf(fact_5683_of__int__less__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5684_of__int__less__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5685_of__int__less__numeral__iff,axiom,
    ! [Z3: int,N2: num] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z3 ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_int @ Z3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5686_of__int__le__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z3 ) @ one_one_Code_integer )
      = ( ord_less_eq_int @ Z3 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5687_of__int__le__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ one_one_rat )
      = ( ord_less_eq_int @ Z3 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5688_of__int__le__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ one_one_int )
      = ( ord_less_eq_int @ Z3 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5689_of__int__1__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z3 ) )
      = ( ord_less_eq_int @ one_one_int @ Z3 ) ) ).

% of_int_1_le_iff
thf(fact_5690_of__int__1__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z3 ) )
      = ( ord_less_eq_int @ one_one_int @ Z3 ) ) ).

% of_int_1_le_iff
thf(fact_5691_of__int__1__le__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_eq_int @ one_one_int @ Z3 ) ) ).

% of_int_1_le_iff
thf(fact_5692_of__int__less__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z3 ) @ one_one_Code_integer )
      = ( ord_less_int @ Z3 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5693_of__int__less__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ one_one_rat )
      = ( ord_less_int @ Z3 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5694_of__int__less__1__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ one_one_int )
      = ( ord_less_int @ Z3 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5695_of__int__1__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z3 ) )
      = ( ord_less_int @ one_one_int @ Z3 ) ) ).

% of_int_1_less_iff
thf(fact_5696_of__int__1__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z3 ) )
      = ( ord_less_int @ one_one_int @ Z3 ) ) ).

% of_int_1_less_iff
thf(fact_5697_of__int__1__less__iff,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z3 ) )
      = ( ord_less_int @ one_one_int @ Z3 ) ) ).

% of_int_1_less_iff
thf(fact_5698_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_5699_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_5700_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_5701_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_5702_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_5703_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_5704_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_5705_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_5706_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_5707_ceiling__add__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_5708_of__nat__nat,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z3 ) )
        = ( ring_1_of_int_rat @ Z3 ) ) ) ).

% of_nat_nat
thf(fact_5709_of__nat__nat,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
        = ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_nat_nat
thf(fact_5710_of__nat__nat,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( semiri4939895301339042750nteger @ ( nat2 @ Z3 ) )
        = ( ring_18347121197199848620nteger @ Z3 ) ) ) ).

% of_nat_nat
thf(fact_5711_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_5712_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_5713_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_5714_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_5715_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_5716_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_5717_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_5718_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_5719_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_5720_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_5721_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_5722_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_5723_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_5724_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_5725_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_5726_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_5727_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_5728_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_5729_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_5730_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_5731_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_5732_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_5733_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_5734_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_5735_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_5736_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_5737_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_5738_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_5739_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_5740_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_5741_ceiling__le__iff,axiom,
    ! [X: rat,Z3: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z3 )
      = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ) ).

% ceiling_le_iff
thf(fact_5742_less__ceiling__iff,axiom,
    ! [Z3: int,X: rat] :
      ( ( ord_less_int @ Z3 @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ) ).

% less_ceiling_iff
thf(fact_5743_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_5744_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_5745_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_5746_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_5747_ceiling__unique,axiom,
    ! [Z3: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z3 ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z3 ) ) ) ).

% ceiling_unique
thf(fact_5748_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_5749_ceiling__split,axiom,
    ! [P2: int > $o,T6: rat] :
      ( ( P2 @ ( archim2889992004027027881ng_rat @ T6 ) )
      = ( ! [I: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I ) @ one_one_rat ) @ T6 )
              & ( ord_less_eq_rat @ T6 @ ( ring_1_of_int_rat @ I ) ) )
           => ( P2 @ I ) ) ) ) ).

% ceiling_split
thf(fact_5750_ceiling__less__iff,axiom,
    ! [X: rat,Z3: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z3 )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z3 ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_5751_le__ceiling__iff,axiom,
    ! [Z3: int,X: rat] :
      ( ( ord_less_eq_int @ Z3 @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z3 ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_5752_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_5753_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_5754_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_5755_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_5756_of__nat__ceiling,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).

% of_nat_ceiling
thf(fact_5757_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_5758_of__int__nonneg,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z3 ) ) ) ).

% of_int_nonneg
thf(fact_5759_of__int__nonneg,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z3 ) ) ) ).

% of_int_nonneg
thf(fact_5760_of__int__nonneg,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_nonneg
thf(fact_5761_of__int__pos,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z3 ) ) ) ).

% of_int_pos
thf(fact_5762_of__int__pos,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z3 ) ) ) ).

% of_int_pos
thf(fact_5763_of__int__pos,axiom,
    ! [Z3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).

% of_int_pos
thf(fact_5764_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_5765_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_5766_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_5767_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_5768_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_5769_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_5770_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_5771_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 ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X3 ) ) ) ).

% floor_exists1
thf(fact_5772_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_5773_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_5774_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_5775_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_5776_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_5777_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_5778_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_5779_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_5780_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_5781_in__lex__prod,axiom,
    ! [A: int,B: int,A2: int,B2: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A @ B ) @ ( product_Pair_int_int @ A2 @ B2 ) ) @ ( lex_prod_int_int @ R2 @ S2 ) )
      = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B2 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_5782_in__lex__prod,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr719794911490849221et_nat] :
      ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ ( produc5001842942810119800et_nat @ A2 @ B2 ) ) @ ( lex_pr5340572901959109728et_nat @ R2 @ S2 ) )
      = ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B @ B2 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_5783_in__lex__prod,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr7928877670098842301et_nat] :
      ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ ( produc2245416461498447860et_nat @ A2 @ B2 ) ) @ ( lex_pr4722427456421979612et_nat @ R2 @ S2 ) )
      = ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B @ B2 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_5784_in__lex__prod,axiom,
    ! [A: b,B: produc6653097349344004940it_nat,A2: b,B2: produc6653097349344004940it_nat,R2: set_Product_prod_b_b,S2: set_Pr4389693562480114009it_nat] :
      ( ( member6820296301096096254it_nat @ ( produc6853161671299316109it_nat @ ( produc4082563078715348724it_nat @ A @ B ) @ ( produc4082563078715348724it_nat @ A2 @ B2 ) ) @ ( lex_pr2603589247317408988it_nat @ R2 @ S2 ) )
      = ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member5617269971687963298it_nat @ ( produc3130510018828335921it_nat @ B @ B2 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_5785_in__lex__prod,axiom,
    ! [A: a,B: produc6653097349344004940it_nat,A2: a,B2: produc6653097349344004940it_nat,R2: set_Product_prod_a_a,S2: set_Pr4389693562480114009it_nat] :
      ( ( member5335690527091456380it_nat @ ( produc1743342482959036555it_nat @ ( produc9178034014595674355it_nat @ A @ B ) @ ( produc9178034014595674355it_nat @ A2 @ B2 ) ) @ ( lex_pr7699060183197734619it_nat @ R2 @ S2 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member5617269971687963298it_nat @ ( produc3130510018828335921it_nat @ B @ B2 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_5786_in__lex__prod,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) @ ( lex_prod_nat_nat @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_5787_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_5788_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_5789_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_5790_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_5791_floor__add2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( member_rat @ X @ ring_1_Ints_rat )
        | ( member_rat @ Y @ ring_1_Ints_rat ) )
     => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
        = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ) ).

% floor_add2
thf(fact_5792_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_5793_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_5794_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_5795_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_5796_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_5797_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_5798_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_5799_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_5800_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_5801_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_5802_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_add
thf(fact_5803_Ints__add,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_add
thf(fact_5804_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_5805_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_5806_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_5807_floor__le__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).

% floor_le_ceiling
thf(fact_5808_Ints__double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( ( plus_p5714425477246183910nteger @ A @ A )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_5809_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ( plus_plus_rat @ A @ A )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_5810_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ( plus_plus_int @ A @ A )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_5811_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_5812_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_5813_le__floor__iff,axiom,
    ! [Z3: int,X: rat] :
      ( ( ord_less_eq_int @ Z3 @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ) ).

% le_floor_iff
thf(fact_5814_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_5815_floor__less__iff,axiom,
    ! [X: rat,Z3: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z3 )
      = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ) ).

% floor_less_iff
thf(fact_5816_floor__add__int,axiom,
    ! [X: rat,Z3: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z3 )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ) ) ).

% floor_add_int
thf(fact_5817_int__add__floor,axiom,
    ! [Z3: int,X: rat] :
      ( ( plus_plus_int @ Z3 @ ( archim3151403230148437115or_rat @ X ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ) ) ).

% int_add_floor
thf(fact_5818_Ints__odd__nonzero,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ A )
       != zero_z3403309356797280102nteger ) ) ).

% Ints_odd_nonzero
thf(fact_5819_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
       != zero_zero_rat ) ) ).

% Ints_odd_nonzero
thf(fact_5820_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
       != zero_zero_int ) ) ).

% Ints_odd_nonzero
thf(fact_5821_of__nat__floor,axiom,
    ! [R2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).

% of_nat_floor
thf(fact_5822_one__add__floor,axiom,
    ! [X: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_5823_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_5824_ceiling__altdef,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X4: rat] :
          ( if_int
          @ ( X4
            = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
          @ ( archim3151403230148437115or_rat @ X4 )
          @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_5825_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_5826_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_5827_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_5828_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_5829_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_5830_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_5831_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_5832_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_5833_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_5834_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_5835_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_5836_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_5837_floor__unique,axiom,
    ! [Z3: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z3 ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z3 ) ) ) ).

% floor_unique
thf(fact_5838_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_5839_floor__split,axiom,
    ! [P2: int > $o,T6: rat] :
      ( ( P2 @ ( archim3151403230148437115or_rat @ T6 ) )
      = ( ! [I: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I ) @ T6 )
              & ( ord_less_rat @ T6 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I ) @ one_one_rat ) ) )
           => ( P2 @ I ) ) ) ) ).

% floor_split
thf(fact_5840_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_5841_less__floor__iff,axiom,
    ! [Z3: int,X: rat] :
      ( ( ord_less_int @ Z3 @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z3 ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_5842_floor__le__iff,axiom,
    ! [X: rat,Z3: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z3 )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z3 ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_5843_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_5844_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_5845_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_5846_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_5847_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_5848_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5849_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5850_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5851_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_5852_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_5853_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_5854_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5855_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5856_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5857_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5858_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5859_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5860_semiring__norm_I68_J,axiom,
    ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).

% semiring_norm(68)
thf(fact_5861_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_5862_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_5863_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_5864_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_5865_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_5866_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_5867_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_5868_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_5869_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_5870_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_5871_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_5872_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_5873_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N2 ) )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_5874_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_5875_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_5876_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_5877_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_5878_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_5879_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_5880_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_5881_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_5882_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_5883_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_5884_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_5885_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_5886_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_5887_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_5888_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_5889_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_5890_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_5891_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ N2 )
      = ( plus_plus_num @ N2 @ one ) ) ).

% add_One_commute
thf(fact_5892_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_5893_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_5894_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_5895_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_5896_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_5897_frac__1__eq,axiom,
    ! [X: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( archimedean_frac_rat @ X ) ) ).

% frac_1_eq
thf(fact_5898_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat @ ( inc @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_5899_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_5900_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_5901_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera5444537566228673987atural @ ( inc @ X ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).

% numeral_inc
thf(fact_5902_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6620942414471956472nteger @ ( inc @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% numeral_inc
thf(fact_5903_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_5904_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_5905_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_5906_pochhammer__double,axiom,
    ! [Z3: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z3 ) @ ( 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 @ Z3 @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_5907_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q6: int,R2: 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 @ R2 ) )
       => ( 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 ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5908_execute__change,axiom,
    ! [F: product_unit > product_unit,R2: ref_Product_unit,H: heap_e7401611519738050253t_unit] :
      ( ( heap_T875086893843062177t_unit @ ( ref_ch7259622376331601608t_unit @ F @ R2 ) @ H )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ ( F @ ( ref_get_Product_unit @ H @ R2 ) ) @ ( produc584006145561248582it_nat @ ( ref_set_Product_unit @ R2 @ ( F @ ( ref_get_Product_unit @ H @ R2 ) ) @ H ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ) ).

% execute_change
thf(fact_5909_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_5910_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_5911_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_5912_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q6: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R2 ) )
       => ( 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 ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_5913_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_5914_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( bit0 @ X2 )
        = ( bit0 @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_5915_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( ( bit1 @ X32 )
        = ( bit1 @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_5916_semiring__norm_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(6)
thf(fact_5917_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_5918_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_5919_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_5920_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_5921_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_5922_semiring__norm_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(9)
thf(fact_5923_semiring__norm_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(7)
thf(fact_5924_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_5925_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_5926_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_5927_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_5928_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_5929_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_5930_semiring__norm_I10_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_5931_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_5932_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_5933_semiring__norm_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_5934_semiring__norm_I3_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% semiring_norm(3)
thf(fact_5935_semiring__norm_I16_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_5936_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_5937_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_5938_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5939_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5940_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5941_one__add__one,axiom,
    ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural )
    = ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5942_one__add__one,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_5943_add__2__eq__Suc_H,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc'
thf(fact_5944_add__2__eq__Suc,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc
thf(fact_5945_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_5946_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_5947_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_5948_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_5949_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_5950_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_5951_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_5952_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_5953_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_5954_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_5955_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_5956_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_5957_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_5958_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_5959_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_5960_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_5961_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_5962_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_5963_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_5964_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_5965_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_5966_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_5967_zmod__numeral__Bit1,axiom,
    ! [V: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_5968_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] :
      ( ( bit0 @ X2 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_5969_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_5970_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] :
      ( one
     != ( bit0 @ X2 ) ) ).

% verit_eq_simplify(10)
thf(fact_5971_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_5972_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_5973_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_5974_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit0 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_5975_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_5976_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_5977_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_5978_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_5979_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit0 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_5980_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_5981_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_5982_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_5983_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_5984_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit1 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) @ one_one_Code_natural ) ) ).

% numeral_Bit1
thf(fact_5985_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ one_one_Code_integer ) ) ).

% numeral_Bit1
thf(fact_5986_num_Osize_I6_J,axiom,
    ! [X32: num] :
      ( ( size_size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_5987_num_Osize_I5_J,axiom,
    ! [X2: num] :
      ( ( size_size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_5988_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_5989_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_5990_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_5991_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_5992_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_5993_power__numeral__odd,axiom,
    ! [Z3: assn,W2: num] :
      ( ( power_power_assn @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_assn @ ( times_times_assn @ Z3 @ ( power_power_assn @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_assn @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_5994_power__numeral__odd,axiom,
    ! [Z3: rat,W2: num] :
      ( ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z3 @ ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_5995_power__numeral__odd,axiom,
    ! [Z3: nat,W2: num] :
      ( ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z3 @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_5996_power__numeral__odd,axiom,
    ! [Z3: int,W2: num] :
      ( ( power_power_int @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_int @ ( times_times_int @ Z3 @ ( power_power_int @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_int @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_5997_power__numeral__even,axiom,
    ! [Z3: assn,W2: num] :
      ( ( power_power_assn @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_assn @ ( power_power_assn @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_assn @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_5998_power__numeral__even,axiom,
    ! [Z3: rat,W2: num] :
      ( ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_5999_power__numeral__even,axiom,
    ! [Z3: nat,W2: num] :
      ( ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_6000_power__numeral__even,axiom,
    ! [Z3: int,W2: num] :
      ( ( power_power_int @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_int @ ( power_power_int @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_int @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_6001_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_6002_mult__2,axiom,
    ! [Z3: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_rat @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_6003_mult__2,axiom,
    ! [Z3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_nat @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_6004_mult__2,axiom,
    ! [Z3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_int @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_6005_mult__2,axiom,
    ! [Z3: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_p4538020629002901425atural @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_6006_mult__2,axiom,
    ! [Z3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_p5714425477246183910nteger @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_6007_mult__2__right,axiom,
    ! [Z3: rat] :
      ( ( times_times_rat @ Z3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_6008_mult__2__right,axiom,
    ! [Z3: nat] :
      ( ( times_times_nat @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_6009_mult__2__right,axiom,
    ! [Z3: int] :
      ( ( times_times_int @ Z3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_6010_mult__2__right,axiom,
    ! [Z3: code_natural] :
      ( ( times_2397367101498566445atural @ Z3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_6011_mult__2__right,axiom,
    ! [Z3: code_integer] :
      ( ( times_3573771949741848930nteger @ Z3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_6012_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_6013_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_6014_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_6015_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_6016_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_6017_nat__1__add__1,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% nat_1_add_1
thf(fact_6018_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_6019_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_6020_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_6021_self__le__ge2__pow,axiom,
    ! [K2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K2 @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_6022_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_6023_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_6024_zero__le__power2,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_6025_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_6026_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_6027_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_6028_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_6029_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_6030_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_6031_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_6032_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_6033_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_6034_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_6035_power2__less__0,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger ) ).

% power2_less_0
thf(fact_6036_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_6037_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_6038_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_6039_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_6040_sum__power2__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( 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_eq_zero_iff
thf(fact_6041_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: 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_eq_zero_iff
thf(fact_6042_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: 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_eq_zero_iff
thf(fact_6043_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_6044_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_6045_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_6046_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_6047_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_6048_nat__induct2,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ( P2 @ one_one_nat )
       => ( ! [N5: nat] :
              ( ( P2 @ N5 )
             => ( P2 @ ( plus_plus_nat @ N5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P2 @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_6049_diff__le__diff__pow,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K2 @ M ) @ ( power_power_nat @ K2 @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_6050_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_6051_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_6052_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_6053_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_6054_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_6055_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_6056_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_6057_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_6058_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_6059_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_6060_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_6061_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_6062_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_6063_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_6064_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_6065_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_6066_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_6067_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_6068_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_6069_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_6070_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_6071_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_6072_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_6073_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_6074_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_6075_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_6076_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_6077_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_6078_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_6079_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_6080_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_6081_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_6082_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_6083_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_6084_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_6085_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_6086_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_6087_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_6088_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_6089_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_6090_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_6091_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_6092_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_6093_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_6094_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_6095_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_6096_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_6097_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_6098_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_6099_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_6100_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_6101_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_6102_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_6103_ex__power__ivl1,axiom,
    ! [B: nat,K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K2 )
       => ? [N5: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N5 ) @ K2 )
            & ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_6104_ex__power__ivl2,axiom,
    ! [B: nat,K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
       => ? [N5: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N5 ) @ K2 )
            & ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_6105_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_6106_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_6107_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_6108_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_6109_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_6110_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_6111_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_6112_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_6113_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_6114_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_6115_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% half_nonnegative_int_iff
thf(fact_6116_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q6: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q6 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q6 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_6117_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q6: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q6 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q6 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_6118_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q6: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q6 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q6 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_6119_int__bit__induct,axiom,
    ! [P2: int > $o,K2: int] :
      ( ( P2 @ zero_zero_int )
     => ( ( P2 @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K: int] :
              ( ( P2 @ K )
             => ( ( K != zero_zero_int )
               => ( P2 @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K: int] :
                ( ( P2 @ K )
               => ( ( K
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P2 @ K2 ) ) ) ) ) ).

% int_bit_induct
thf(fact_6120_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_6121_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_6122_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_6123_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_6124_nat__bit__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ( P2 @ N5 )
           => ( ( ord_less_nat @ zero_zero_nat @ N5 )
             => ( P2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) )
       => ( ! [N5: nat] :
              ( ( P2 @ N5 )
             => ( P2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) )
         => ( P2 @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_6125_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_6126_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_6127_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_z2226904508553997617atural )
     => ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 )
       != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_right
thf(fact_6128_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_6129_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_6130_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_6131_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_z2226904508553997617atural )
     => ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M )
       != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_left
thf(fact_6132_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_6133_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_6134_div__exp__eq,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_6135_div__exp__eq,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_6136_div__exp__eq,axiom,
    ! [A: code_natural,M: nat,N2: nat] :
      ( ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_6137_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_6138_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_6139_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_6140_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_6141_bits__stable__imp__add__self,axiom,
    ! [A: code_natural] :
      ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
        = zero_z2226904508553997617atural ) ) ).

% bits_stable_imp_add_self
thf(fact_6142_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_6143_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_6144_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_6145_div__exp__mod__exp__eq,axiom,
    ! [A: code_natural,N2: nat,M: nat] :
      ( ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M ) )
      = ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_6146_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_6147_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_6148_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_6149_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_6150_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_6151_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_6152_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_6153_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_6154_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_6155_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_6156_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_6157_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_6158_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_6159_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_6160_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_6161_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_6162_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_6163_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_6164_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_6165_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_6166_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_6167_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_6168_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_6169_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_6170_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_6171_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_6172_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_6173_signed__take__bit__add,axiom,
    ! [N2: nat,K2: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K2 @ L ) ) ) ).

% signed_take_bit_add
thf(fact_6174_unset__bit__less__eq,axiom,
    ! [N2: nat,K2: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K2 ) @ K2 ) ).

% unset_bit_less_eq
thf(fact_6175_set__bit__greater__eq,axiom,
    ! [K2: int,N2: nat] : ( ord_less_eq_int @ K2 @ ( bit_se7879613467334960850it_int @ N2 @ K2 ) ) ).

% set_bit_greater_eq
thf(fact_6176_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_6177_floor__le__round,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).

% floor_le_round
thf(fact_6178_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K2: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_6179_round__diff__minimal,axiom,
    ! [Z3: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z3 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z3 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z3 @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_6180_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_eq_int @ K2 @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) )
      = ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_6181_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K2 ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_6182_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: nat,K2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_6183_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K2 ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_6184_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_int @ K2 @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) )
      = ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_6185_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K2 )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_6186_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ K2 )
        = K2 )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K2 )
        & ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_6187_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K2 )
     => ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_ri631733984087533419it_int @ N2 @ K2 )
          = K2 ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_6188_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_6189_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X4: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6190_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_6191_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_6192_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_6193_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_6194_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_6195_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_6196_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_6197_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_6198_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_6199_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_6200_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_6201_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_6202_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_6203_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_6204_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_6205_concat__bit__Suc,axiom,
    ! [N2: nat,K2: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K2 @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_6206_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_6207_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_6208_binomial__eq__0__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ( binomial @ N2 @ K2 )
        = zero_zero_nat )
      = ( ord_less_nat @ N2 @ K2 ) ) ).

% binomial_eq_0_iff
thf(fact_6209_binomial__Suc__Suc,axiom,
    ! [N2: nat,K2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K2 ) )
      = ( plus_plus_nat @ ( binomial @ N2 @ K2 ) @ ( binomial @ N2 @ ( suc @ K2 ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_6210_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K2 @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_6211_concat__bit__negative__iff,axiom,
    ! [N2: nat,K2: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K2 @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_6212_zero__less__binomial__iff,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K2 ) )
      = ( ord_less_eq_nat @ K2 @ N2 ) ) ).

% zero_less_binomial_iff
thf(fact_6213_less__numeral__Suc,axiom,
    ! [K2: num,N2: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ ( pred_numeral @ K2 ) @ N2 ) ) ).

% less_numeral_Suc
thf(fact_6214_less__Suc__numeral,axiom,
    ! [N2: nat,K2: num] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K2 ) )
      = ( ord_less_nat @ N2 @ ( pred_numeral @ K2 ) ) ) ).

% less_Suc_numeral
thf(fact_6215_le__numeral__Suc,axiom,
    ! [K2: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K2 ) @ N2 ) ) ).

% le_numeral_Suc
thf(fact_6216_le__Suc__numeral,axiom,
    ! [N2: nat,K2: num] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K2 ) )
      = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K2 ) ) ) ).

% le_Suc_numeral
thf(fact_6217_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_6218_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_6219_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_6220_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_6221_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_6222_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6223_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_6224_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_6225_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6226_concat__bit__assoc,axiom,
    ! [N2: nat,K2: int,M: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N2 @ K2 @ ( bit_concat_bit @ M @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K2 @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_6227_binomial__eq__0,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ord_less_nat @ N2 @ K2 )
     => ( ( binomial @ N2 @ K2 )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_6228_binomial__symmetric,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( binomial @ N2 @ K2 )
        = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ).

% binomial_symmetric
thf(fact_6229_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K2: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K2 ) @ ( plus_plus_nat @ M @ K2 ) ) @ ( binomial @ ( plus_plus_nat @ M @ K2 ) @ K2 ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_6230_binomial__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R2 ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_6231_zero__less__binomial,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K2 ) ) ) ).

% zero_less_binomial
thf(fact_6232_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_6233_choose__mult,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K2 ) )
          = ( times_times_nat @ ( binomial @ N2 @ K2 ) @ ( binomial @ ( minus_minus_nat @ N2 @ K2 ) @ ( minus_minus_nat @ M @ K2 ) ) ) ) ) ) ).

% choose_mult
thf(fact_6234_binomial__fact__lemma,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K2 ) ) ) @ ( binomial @ N2 @ K2 ) )
        = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% binomial_fact_lemma
thf(fact_6235_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K2 ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_6236_choose__reduce__nat,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ( binomial @ N2 @ K2 )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K2 ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_6237_times__binomial__minus1__eq,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( times_times_nat @ K2 @ ( binomial @ N2 @ K2 ) )
        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_6238_binomial__le__pow2,axiom,
    ! [N2: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% binomial_le_pow2
thf(fact_6239_binomial__altdef__nat,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( binomial @ N2 @ K2 )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_6240_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_def
thf(fact_6241_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_6242_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_6243_binomial__addition__formula,axiom,
    ! [N2: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( binomial @ N2 @ ( suc @ K2 ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K2 ) ) ) ) ).

% binomial_addition_formula
thf(fact_6244_fact__binomial,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K2 ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ) ).

% fact_binomial
thf(fact_6245_binomial__fact,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K2 ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ) ) ).

% binomial_fact
thf(fact_6246_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_6247_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_6248_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_6249_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_6250_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_6251_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_6252_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_6253_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_6254_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_6255_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_6256_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_6257_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_6258_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q8: nat,R5: nat] : ( product_Pair_nat_nat @ Q8 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6259_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q8: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q8 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6260_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q8: int,R5: int] : ( product_Pair_int_int @ Q8 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6261_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_6262_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_6263_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_6264_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_6265_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_6266_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_6267_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_6268_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_6269_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_6270_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6271_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6272_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_6273_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6274_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6275_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_6276_dvd__1__left,axiom,
    ! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).

% dvd_1_left
thf(fact_6277_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_6278_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_6279_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_6280_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_6281_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_6282_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_6283_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_6284_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_6285_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_6286_curry__case__prod,axiom,
    ! [F: int > int > product_prod_int_int] :
      ( ( produc8249235968001453780nt_int @ ( produc4245557441103728435nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6287_curry__case__prod,axiom,
    ! [F: int > int > $o] :
      ( ( produc175634133007206835_int_o @ ( produc4947309494688390418_int_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6288_curry__case__prod,axiom,
    ! [F: int > int > int] :
      ( ( produc1016772743285680337nt_int @ ( produc8211389475949308722nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6289_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_6290_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_6291_case__prod__curry,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ ( produc8249235968001453780nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6292_case__prod__curry,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o @ ( produc175634133007206835_int_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6293_case__prod__curry,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int @ ( produc1016772743285680337nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6294_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C2 @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6295_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C2 @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6296_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C2 @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_6297_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C2 @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6298_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C2 @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6299_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C2 @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_6300_div__add,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C2 @ A )
     => ( ( dvd_dvd_nat @ C2 @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ) ).

% div_add
thf(fact_6301_div__add,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C2 @ A )
     => ( ( dvd_dvd_int @ C2 @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ) ).

% div_add
thf(fact_6302_div__add,axiom,
    ! [C2: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ A )
     => ( ( dvd_dvd_Code_natural @ C2 @ B )
       => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C2 )
          = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C2 ) @ ( divide5121882707175180666atural @ B @ C2 ) ) ) ) ) ).

% div_add
thf(fact_6303_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_6304_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_6305_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6306_odd__add,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6307_odd__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6308_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_6309_even__add,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6310_even__add,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6311_even__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6312_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_6313_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6314_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6315_even__plus__one__iff,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) )
      = ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6316_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_6317_even__diff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).

% even_diff
thf(fact_6318_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).

% even_diff
thf(fact_6319_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_6320_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_6321_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_6322_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_6323_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_6324_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_6325_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_6326_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_6327_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_6328_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_6329_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_6330_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_6331_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_6332_odd__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ) ).

% odd_succ_div_two
thf(fact_6333_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6334_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6335_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6336_even__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6337_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6338_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6339_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6340_even__succ__div__2,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6341_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_6342_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_6343_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_6344_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_6345_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_6346_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_6347_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_6348_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q8: nat,R5: nat] : ( product_Pair_nat_nat @ Q8 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6349_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q8: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q8 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6350_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q8: int,R5: int] : ( product_Pair_int_int @ Q8 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6351_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_6352_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_6353_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_6354_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_6355_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_6356_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_6357_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_6358_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,R5: nat] : ( F3 @ L2 @ R5 @ Y4 )
          @ X4 ) ) ) ).

% case_prod_app
thf(fact_6359_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,R5: nat] : ( F3 @ L2 @ R5 @ Y4 )
          @ X4 ) ) ) ).

% case_prod_app
thf(fact_6360_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,B4: nat] : ( F3 @ A5 @ B4 @ Y4 )
          @ X4 ) ) ) ).

% nested_case_prod_simp
thf(fact_6361_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,B4: nat] : ( F3 @ A5 @ B4 @ Y4 )
          @ X4 ) ) ) ).

% nested_case_prod_simp
thf(fact_6362_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
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6363_prod_Ocase__distrib,axiom,
    ! [H: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6364_prod_Ocase__distrib,axiom,
    ! [H: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6365_prod_Ocase__distrib,axiom,
    ! [H: int > int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6366_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
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6367_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
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6368_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
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6369_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
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6370_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
        @ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6371_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
        @ ^ [X12: nat,X23: nat] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6372_take__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B ) ) ) ).

% take_bit_add
thf(fact_6373_take__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) )
      = ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B ) ) ) ).

% take_bit_add
thf(fact_6374_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_6375_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_6376_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_6377_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_6378_dvd__add,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C2 )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ) ).

% dvd_add
thf(fact_6379_dvd__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C2 )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ) ).

% dvd_add
thf(fact_6380_dvd__add,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C2 )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ) ).

% dvd_add
thf(fact_6381_dvd__add__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C2 )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6382_dvd__add__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C2 )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6383_dvd__add__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( dvd_dvd_int @ A @ C2 )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_6384_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
        = ( dvd_dvd_rat @ A @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_6385_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
        = ( dvd_dvd_nat @ A @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_6386_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
        = ( dvd_dvd_int @ A @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_6387_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_6388_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_6389_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_6390_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_6391_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_6392_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_6393_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_6394_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_6395_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_6396_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_6397_dvd__diff__nat,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K2 @ M )
     => ( ( dvd_dvd_nat @ K2 @ N2 )
       => ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_6398_uminus__dvd__conv_I1_J,axiom,
    ( dvd_dvd_int
    = ( ^ [D5: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D5 ) ) ) ) ).

% uminus_dvd_conv(1)
thf(fact_6399_uminus__dvd__conv_I2_J,axiom,
    ( dvd_dvd_int
    = ( ^ [D5: int,T4: int] : ( dvd_dvd_int @ D5 @ ( uminus_uminus_int @ T4 ) ) ) ) ).

% uminus_dvd_conv(2)
thf(fact_6400_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_6401_case__prod__Pair__iden,axiom,
    ! [P3: produc3925858234332021118et_nat] :
      ( ( produc4058941399401191971et_nat @ produc5001842942810119800et_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_6402_case__prod__Pair__iden,axiom,
    ! [P3: produc2732055786443039994et_nat] :
      ( ( produc2377985495875741467et_nat @ produc2245416461498447860et_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_6403_case__prod__Pair__iden,axiom,
    ! [P3: produc7388388658123137530it_nat] :
      ( ( produc7293086598567236123it_nat @ produc4082563078715348724it_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_6404_case__prod__Pair__iden,axiom,
    ! [P3: produc3260487557148687353it_nat] :
      ( ( produc2183267410226956569it_nat @ produc9178034014595674355it_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_6405_case__prod__Pair__iden,axiom,
    ! [P3: product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_6406_nat__take__bit__eq,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) )
        = ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K2 ) ) ) ) ).

% nat_take_bit_eq
thf(fact_6407_take__bit__nat__eq,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K2 ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) ) ) ) ).

% take_bit_nat_eq
thf(fact_6408_case__prodE2,axiom,
    ! [Q2: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ( Q2 @ ( produc27273713700761075at_nat @ P2 @ Z3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_6409_case__prodE2,axiom,
    ! [Q2: ( product_prod_nat_nat > $o ) > $o,P2: nat > nat > product_prod_nat_nat > $o,Z3: product_prod_nat_nat] :
      ( ( Q2 @ ( produc8739625826339149834_nat_o @ P2 @ Z3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_6410_case__prodE2,axiom,
    ! [Q2: product_prod_int_int > $o,P2: int > int > product_prod_int_int,Z3: product_prod_int_int] :
      ( ( Q2 @ ( produc4245557441103728435nt_int @ P2 @ Z3 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_6411_case__prodE2,axiom,
    ! [Q2: $o > $o,P2: int > int > $o,Z3: product_prod_int_int] :
      ( ( Q2 @ ( produc4947309494688390418_int_o @ P2 @ Z3 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_6412_case__prodE2,axiom,
    ! [Q2: int > $o,P2: int > int > int,Z3: product_prod_int_int] :
      ( ( Q2 @ ( produc8211389475949308722nt_int @ P2 @ Z3 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_6413_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_6414_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_6415_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_6416_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_6417_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_6418_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_6419_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_6420_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_6421_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_6422_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_6423_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6424_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6425_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K2 )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K2 @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_6426_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6427_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6428_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N2: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K2 )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( plus_plus_int @ K2 @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_6429_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K2: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K2 ) @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_6430_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ K2 )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_6431_take__bit__nonnegative,axiom,
    ! [N2: nat,K2: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) ) ).

% take_bit_nonnegative
thf(fact_6432_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_int @ K2 @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_6433_not__take__bit__negative,axiom,
    ! [N2: nat,K2: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_6434_pinf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z2 @ X6 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ S2 ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_6435_pinf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X6 @ S2 ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_6436_pinf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z2 @ X6 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ S2 ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_6437_pinf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z2 @ X6 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_6438_pinf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z2 @ X6 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_6439_pinf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z2 @ X6 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_6440_minf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z2 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ S2 ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_6441_minf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X6 @ S2 ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_6442_minf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z2 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ S2 ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_6443_minf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z2 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_6444_minf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z2 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_6445_minf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z2 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_6446_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_6447_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C2 @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6448_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C2 @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6449_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C2 )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C2 ) @ ( divide5121882707175180666atural @ B @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6450_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C2 @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6451_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C2 @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6452_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ B )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C2 )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C2 ) @ ( divide5121882707175180666atural @ B @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6453_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_6454_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_6455_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_6456_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_6457_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_6458_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_6459_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_6460_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_6461_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_6462_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_6463_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_6464_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_6465_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_6466_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_6467_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_6468_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_6469_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_6470_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_6471_dvd__diffD1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K2 @ M )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K2 @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_6472_dvd__diffD,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K2 @ N2 )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K2 @ M ) ) ) ) ).

% dvd_diffD
thf(fact_6473_zdvd__mono,axiom,
    ! [K2: int,M: int,T6: int] :
      ( ( K2 != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T6 )
        = ( dvd_dvd_int @ ( times_times_int @ K2 @ M ) @ ( times_times_int @ K2 @ T6 ) ) ) ) ).

% zdvd_mono
thf(fact_6474_bezout__lemma__nat,axiom,
    ! [D2: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D2 @ A )
     => ( ( dvd_dvd_nat @ D2 @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D2 ) ) )
         => ? [X3: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D2 @ A )
              & ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D2 ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D2 ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_6475_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D @ A )
      & ( dvd_dvd_nat @ D @ B )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ).

% bezout_add_nat
thf(fact_6476_zdvd__reduce,axiom,
    ! [K2: int,N2: int,M: int] :
      ( ( dvd_dvd_int @ K2 @ ( plus_plus_int @ N2 @ ( times_times_int @ K2 @ M ) ) )
      = ( dvd_dvd_int @ K2 @ N2 ) ) ).

% zdvd_reduce
thf(fact_6477_zdvd__period,axiom,
    ! [A: int,D2: int,X: int,T6: int,C2: int] :
      ( ( dvd_dvd_int @ A @ D2 )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T6 ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C2 @ D2 ) ) @ T6 ) ) ) ) ).

% zdvd_period
thf(fact_6478_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_6479_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_6480_uncurry__def,axiom,
    uncurr8011562610307062878at_nat = produc27273713700761075at_nat ).

% uncurry_def
thf(fact_6481_uncurry__def,axiom,
    uncurr7511940902602773877_nat_o = produc8739625826339149834_nat_o ).

% uncurry_def
thf(fact_6482_uncurry__def,axiom,
    uncurr7650761721940715016nt_int = produc4245557441103728435nt_int ).

% uncurry_def
thf(fact_6483_uncurry__def,axiom,
    uncurry_int_int_o = produc4947309494688390418_int_o ).

% uncurry_def
thf(fact_6484_uncurry__def,axiom,
    uncurry_int_int_int = produc8211389475949308722nt_int ).

% uncurry_def
thf(fact_6485_internal__case__prod__def,axiom,
    produc1854806715440696265at_nat = produc27273713700761075at_nat ).

% internal_case_prod_def
thf(fact_6486_internal__case__prod__def,axiom,
    produc4780622933104268256_nat_o = produc8739625826339149834_nat_o ).

% internal_case_prod_def
thf(fact_6487_internal__case__prod__def,axiom,
    produc297006045350968285nt_int = produc4245557441103728435nt_int ).

% internal_case_prod_def
thf(fact_6488_internal__case__prod__def,axiom,
    produc8005341501107743676_int_o = produc4947309494688390418_int_o ).

% internal_case_prod_def
thf(fact_6489_internal__case__prod__def,axiom,
    produc7926200574084438792nt_int = produc8211389475949308722nt_int ).

% internal_case_prod_def
thf(fact_6490_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_6491_not__mask__negative__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_6492_nat__dvd__iff,axiom,
    ! [Z3: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z3 ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
         => ( dvd_dvd_int @ Z3 @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_6493_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_6494_unity__coeff__ex,axiom,
    ! [P2: code_integer > $o,L: code_integer] :
      ( ( ? [X4: code_integer] : ( P2 @ ( times_3573771949741848930nteger @ L @ X4 ) ) )
      = ( ? [X4: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X4 @ zero_z3403309356797280102nteger ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6495_unity__coeff__ex,axiom,
    ! [P2: rat > $o,L: rat] :
      ( ( ? [X4: rat] : ( P2 @ ( times_times_rat @ L @ X4 ) ) )
      = ( ? [X4: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X4 @ zero_zero_rat ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6496_unity__coeff__ex,axiom,
    ! [P2: nat > $o,L: nat] :
      ( ( ? [X4: nat] : ( P2 @ ( times_times_nat @ L @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6497_unity__coeff__ex,axiom,
    ! [P2: int > $o,L: int] :
      ( ( ? [X4: int] : ( P2 @ ( times_times_int @ L @ X4 ) ) )
      = ( ? [X4: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6498_inf__period_I3_J,axiom,
    ! [D2: rat,D4: rat,T6: rat] :
      ( ( dvd_dvd_rat @ D2 @ D4 )
     => ! [X6: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ T6 ) )
          = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) @ T6 ) ) ) ) ).

% inf_period(3)
thf(fact_6499_inf__period_I3_J,axiom,
    ! [D2: int,D4: int,T6: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X6: int,K4: int] :
          ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T6 ) )
          = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) @ T6 ) ) ) ) ).

% inf_period(3)
thf(fact_6500_inf__period_I4_J,axiom,
    ! [D2: rat,D4: rat,T6: rat] :
      ( ( dvd_dvd_rat @ D2 @ D4 )
     => ! [X6: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ T6 ) ) )
          = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) @ T6 ) ) ) ) ) ).

% inf_period(4)
thf(fact_6501_inf__period_I4_J,axiom,
    ! [D2: int,D4: int,T6: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X6: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T6 ) ) )
          = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) @ T6 ) ) ) ) ) ).

% inf_period(4)
thf(fact_6502_dvd__imp__le,axiom,
    ! [K2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K2 @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).

% dvd_imp_le
thf(fact_6503_dvd__mult__cancel,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_6504_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
        = ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_6505_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D: nat,X3: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D @ A )
          & ( dvd_dvd_nat @ D @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_6506_zdvd__imp__le,axiom,
    ! [Z3: int,N2: int] :
      ( ( dvd_dvd_int @ Z3 @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ Z3 @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_6507_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K2 @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6508_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K2 @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6509_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K2 @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6510_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_6511_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_6512_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_6513_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_6514_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_6515_even__nat__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K2 ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ) ).

% even_nat_iff
thf(fact_6516_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_6517_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6518_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6519_odd__even__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6520_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_6521_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_6522_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_6523_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_6524_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_6525_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_6526_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_6527_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_6528_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_6529_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_6530_choose__dvd,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K2 ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% choose_dvd
thf(fact_6531_choose__dvd,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K2 ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% choose_dvd
thf(fact_6532_choose__dvd,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K2 ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% choose_dvd
thf(fact_6533_even__even__mod__4__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_6534_dvd__minus__add,axiom,
    ! [Q6: nat,N2: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q6 @ N2 )
     => ( ( ord_less_eq_nat @ Q6 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q6 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q6 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_6535_mod__nat__eqI,axiom,
    ! [R2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N2 )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_6536_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) )
      = ( ( dvd_dvd_int @ L @ K2 )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K2 ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_6537_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_6538_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_6539_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_6540_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_6541_take__bit__int__less__exp,axiom,
    ! [N2: nat,K2: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_int_less_exp
thf(fact_6542_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_6543_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_6544_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_6545_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K2 @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_diff_iff
thf(fact_6546_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_add_abs_iff
thf(fact_6547_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K2 ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_abs_add_iff
thf(fact_6548_dvd__power__iff__le,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K2 @ M ) @ ( power_power_nat @ K2 @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_6549_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_6550_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_6551_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_eq_int @ K2 @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) )
      = ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_6552_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K2 ) ) ).

% take_bit_int_less_self_iff
thf(fact_6553_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_6554_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_6555_oddE,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: code_natural] :
            ( A
           != ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_natural ) ) ) ).

% oddE
thf(fact_6556_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_6557_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_6558_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_6559_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_6560_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_6561_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_6562_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_6563_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_6564_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_6565_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_6566_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_6567_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_6568_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_6569_take__bit__int__eq__self,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ K2 )
          = K2 ) ) ) ).

% take_bit_int_eq_self
thf(fact_6570_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K2 )
        = K2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        & ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_6571_take__bit__incr__eq,axiom,
    ! [N2: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K2 )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K2 @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) ) ) ) ).

% take_bit_incr_eq
thf(fact_6572_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_6573_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_6574_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_6575_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N2 ) @ ( numera5444537566228673987atural @ ( bit1 @ K2 ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N2 @ ( numera5444537566228673987atural @ K2 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_Suc_bit1
thf(fact_6576_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( numera6620942414471956472nteger @ ( bit1 @ K2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( numera6620942414471956472nteger @ K2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_bit1
thf(fact_6577_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_6578_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_6579_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_6580_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_6581_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_6582_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_6583_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q8: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q8 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_6584_take__bit__int__less__eq,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_6585_take__bit__int__greater__eq,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) ) ) ).

% take_bit_int_greater_eq
thf(fact_6586_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_6587_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_6588_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_6589_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_6590_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_6591_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_6592_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_6593_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_6594_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_6595_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit1 @ K2 ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K2 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_numeral_bit1
thf(fact_6596_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit1 @ K2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_bit1
thf(fact_6597_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_6598_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_6599_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q8: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q8 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_6600_take__bit__minus__small__eq,axiom,
    ! [K2: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K2 ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K2 ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_6601_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_6602_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_6603_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_6604_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_6605_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_6606_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_6607_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_6608_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_6609_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_6610_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_6611_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q8: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q8 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6612_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q8: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q8 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6613_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q8: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q8 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6614_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
        @ ^ [Q8: 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 @ Q8 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_6615_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
            @ ^ [Q8: nat] : ( product_Pair_nat_nat @ ( suc @ Q8 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_6616_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_6617_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_6618_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_6619_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_6620_num_Osize__gen_I3_J,axiom,
    ! [X32: num] :
      ( ( size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_6621_num_Osize__gen_I2_J,axiom,
    ! [X2: num] :
      ( ( size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_6622_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_6623_case__prodI2,axiom,
    ! [P3: produc3925858234332021118et_nat,C2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
          ( ( P3
            = ( produc5001842942810119800et_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc1437786849005270451_nat_o @ C2 @ P3 ) ) ).

% case_prodI2
thf(fact_6624_case__prodI2,axiom,
    ! [P3: produc2732055786443039994et_nat,C2: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
          ( ( P3
            = ( produc2245416461498447860et_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc838355143741117751_nat_o @ C2 @ P3 ) ) ).

% case_prodI2
thf(fact_6625_case__prodI2,axiom,
    ! [P3: produc7388388658123137530it_nat,C2: b > produc6653097349344004940it_nat > $o] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc4082563078715348724it_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc3971132677695353399_nat_o @ C2 @ P3 ) ) ).

% case_prodI2
thf(fact_6626_case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,C2: a > produc6653097349344004940it_nat > $o] :
      ( ! [A3: a,B3: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc8860486935419594360_nat_o @ C2 @ P3 ) ) ).

% case_prodI2
thf(fact_6627_case__prodI2,axiom,
    ! [P3: product_prod_int_int,C2: int > int > $o] :
      ( ! [A3: int,B3: int] :
          ( ( P3
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc4947309494688390418_int_o @ C2 @ P3 ) ) ).

% case_prodI2
thf(fact_6628_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_6629_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_6630_case__prodI,axiom,
    ! [F: b > produc6653097349344004940it_nat > $o,A: b,B: produc6653097349344004940it_nat] :
      ( ( F @ A @ B )
     => ( produc3971132677695353399_nat_o @ F @ ( produc4082563078715348724it_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_6631_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_6632_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_6633_mem__case__prodI2,axiom,
    ! [P3: product_prod_int_int,Z3: $o,C2: int > int > set_o] :
      ( ! [A3: int,B3: int] :
          ( ( P3
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member_o @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_o @ Z3 @ ( produc4257766111578684402_set_o @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6634_mem__case__prodI2,axiom,
    ! [P3: product_prod_int_int,Z3: nat,C2: int > int > set_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P3
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc4251311855443802252et_nat @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6635_mem__case__prodI2,axiom,
    ! [P3: product_prod_int_int,Z3: int,C2: int > int > set_int] :
      ( ! [A3: int,B3: int] :
          ( ( P3
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member_int @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_int @ Z3 @ ( produc73460835934605544et_int @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6636_mem__case__prodI2,axiom,
    ! [P3: product_prod_int_int,Z3: set_nat,C2: int > int > set_set_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P3
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member_set_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_set_nat @ Z3 @ ( produc5233655623923918146et_nat @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6637_mem__case__prodI2,axiom,
    ! [P3: product_prod_int_int,Z3: set_Pr958786334691620121nt_int,C2: int > int > set_se6260736226359567993nt_int] :
      ( ! [A3: int,B3: int] :
          ( ( P3
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member2340774599025711042nt_int @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member2340774599025711042nt_int @ Z3 @ ( produc7771776839612048883nt_int @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6638_mem__case__prodI2,axiom,
    ! [P3: produc7388388658123137530it_nat,Z3: $o,C2: b > produc6653097349344004940it_nat > set_o] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc4082563078715348724it_nat @ A3 @ B3 ) )
         => ( member_o @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_o @ Z3 @ ( produc8188766211626073367_set_o @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6639_mem__case__prodI2,axiom,
    ! [P3: produc7388388658123137530it_nat,Z3: nat,C2: b > produc6653097349344004940it_nat > set_nat] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc4082563078715348724it_nat @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc8058763169506871975et_nat @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6640_mem__case__prodI2,axiom,
    ! [P3: produc7388388658123137530it_nat,Z3: int,C2: b > produc6653097349344004940it_nat > set_int] :
      ( ! [A3: b,B3: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc4082563078715348724it_nat @ A3 @ B3 ) )
         => ( member_int @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_int @ Z3 @ ( produc3880912149997675267et_int @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6641_mem__case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,Z3: $o,C2: a > produc6653097349344004940it_nat > set_o] :
      ( ! [A3: a,B3: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A3 @ B3 ) )
         => ( member_o @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_o @ Z3 @ ( produc5350075035600711000_set_o @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6642_mem__case__prodI2,axiom,
    ! [P3: produc3260487557148687353it_nat,Z3: nat,C2: a > produc6653097349344004940it_nat > set_nat] :
      ( ! [A3: a,B3: produc6653097349344004940it_nat] :
          ( ( P3
            = ( produc9178034014595674355it_nat @ A3 @ B3 ) )
         => ( member_nat @ Z3 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z3 @ ( produc7315013382793309350et_nat @ C2 @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_6643_mem__case__prodI,axiom,
    ! [Z3: $o,C2: int > int > set_o,A: int,B: int] :
      ( ( member_o @ Z3 @ ( C2 @ A @ B ) )
     => ( member_o @ Z3 @ ( produc4257766111578684402_set_o @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6644_mem__case__prodI,axiom,
    ! [Z3: nat,C2: int > int > set_nat,A: int,B: int] :
      ( ( member_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc4251311855443802252et_nat @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6645_mem__case__prodI,axiom,
    ! [Z3: int,C2: int > int > set_int,A: int,B: int] :
      ( ( member_int @ Z3 @ ( C2 @ A @ B ) )
     => ( member_int @ Z3 @ ( produc73460835934605544et_int @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6646_mem__case__prodI,axiom,
    ! [Z3: set_nat,C2: int > int > set_set_nat,A: int,B: int] :
      ( ( member_set_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member_set_nat @ Z3 @ ( produc5233655623923918146et_nat @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6647_mem__case__prodI,axiom,
    ! [Z3: set_Pr958786334691620121nt_int,C2: int > int > set_se6260736226359567993nt_int,A: int,B: int] :
      ( ( member2340774599025711042nt_int @ Z3 @ ( C2 @ A @ B ) )
     => ( member2340774599025711042nt_int @ Z3 @ ( produc7771776839612048883nt_int @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6648_mem__case__prodI,axiom,
    ! [Z3: $o,C2: b > produc6653097349344004940it_nat > set_o,A: b,B: produc6653097349344004940it_nat] :
      ( ( member_o @ Z3 @ ( C2 @ A @ B ) )
     => ( member_o @ Z3 @ ( produc8188766211626073367_set_o @ C2 @ ( produc4082563078715348724it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6649_mem__case__prodI,axiom,
    ! [Z3: nat,C2: b > produc6653097349344004940it_nat > set_nat,A: b,B: produc6653097349344004940it_nat] :
      ( ( member_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc8058763169506871975et_nat @ C2 @ ( produc4082563078715348724it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6650_mem__case__prodI,axiom,
    ! [Z3: int,C2: b > produc6653097349344004940it_nat > set_int,A: b,B: produc6653097349344004940it_nat] :
      ( ( member_int @ Z3 @ ( C2 @ A @ B ) )
     => ( member_int @ Z3 @ ( produc3880912149997675267et_int @ C2 @ ( produc4082563078715348724it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6651_mem__case__prodI,axiom,
    ! [Z3: $o,C2: a > produc6653097349344004940it_nat > set_o,A: a,B: produc6653097349344004940it_nat] :
      ( ( member_o @ Z3 @ ( C2 @ A @ B ) )
     => ( member_o @ Z3 @ ( produc5350075035600711000_set_o @ C2 @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6652_mem__case__prodI,axiom,
    ! [Z3: nat,C2: a > produc6653097349344004940it_nat > set_nat,A: a,B: produc6653097349344004940it_nat] :
      ( ( member_nat @ Z3 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc7315013382793309350et_nat @ C2 @ ( produc9178034014595674355it_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6653_case__prodI2_H,axiom,
    ! [P3: product_prod_nat_nat,C2: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ( product_Pair_nat_nat @ A3 @ B3 )
            = P3 )
         => ( C2 @ A3 @ B3 @ X ) )
     => ( produc8739625826339149834_nat_o @ C2 @ P3 @ X ) ) ).

% case_prodI2'
thf(fact_6654_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q2 ) )
      = ( P2
       => Q2 ) ) ).

% of_bool_less_eq_iff
thf(fact_6655_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q2 ) )
      = ( P2
       => Q2 ) ) ).

% of_bool_less_eq_iff
thf(fact_6656_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q2 ) )
      = ( P2
       => Q2 ) ) ).

% of_bool_less_eq_iff
thf(fact_6657_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P2 ) @ ( zero_n356916108424825756nteger @ Q2 ) )
      = ( P2
       => Q2 ) ) ).

% of_bool_less_eq_iff
thf(fact_6658_of__bool__less__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q2 ) )
      = ( ~ P2
        & Q2 ) ) ).

% of_bool_less_iff
thf(fact_6659_of__bool__less__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q2 ) )
      = ( ~ P2
        & Q2 ) ) ).

% of_bool_less_iff
thf(fact_6660_of__bool__less__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q2 ) )
      = ( ~ P2
        & Q2 ) ) ).

% of_bool_less_iff
thf(fact_6661_of__bool__less__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P2 ) @ ( zero_n356916108424825756nteger @ Q2 ) )
      = ( ~ P2
        & Q2 ) ) ).

% of_bool_less_iff
thf(fact_6662_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n2687167440665602831ol_nat @ P2 ) ) ).

% of_nat_of_bool
thf(fact_6663_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% of_nat_of_bool
thf(fact_6664_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n356916108424825756nteger @ P2 ) ) ).

% of_nat_of_bool
thf(fact_6665_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6666_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6667_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6668_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6669_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6670_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6671_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6672_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P2 ) @ one_one_Code_integer )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6673_and__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_6674_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K2 @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_6675_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q8: num] : ( some_num @ ( bit0 @ Q8 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_6676_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_6677_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_6678_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q8: num] : ( some_num @ ( bit0 @ Q8 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_6679_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_6680_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_6681_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_6682_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_6683_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_6684_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_6685_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_6686_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_6687_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_6688_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_6689_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_6690_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6691_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6692_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6693_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6694_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_6695_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_6696_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_6697_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_6698_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_6699_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_6700_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_6701_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_6702_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_6703_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_6704_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_6705_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_6706_same__fst__def,axiom,
    ( same_fst_int_int
    = ( ^ [P5: int > $o,R: int > set_Pr958786334691620121nt_int] :
          ( collec7447955092554649554nt_int
          @ ( produc1676490119946744748_int_o
            @ ( produc1459180154077124618_int_o
              @ ^ [X8: int,Y9: int] :
                  ( produc4947309494688390418_int_o
                  @ ^ [X4: int,Y4: int] :
                      ( ( X8 = X4 )
                      & ( P5 @ X4 )
                      & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y9 @ Y4 ) @ ( R @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_6707_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P5: nat > $o,R: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X8: nat,Y9: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X4: nat,Y4: nat] :
                      ( ( X8 = X4 )
                      & ( P5 @ X4 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y9 @ Y4 ) @ ( R @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_6708_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_6709_lex__prod__def,axiom,
    ( lex_prod_int_int
    = ( ^ [Ra: set_Pr958786334691620121nt_int,Rb: set_Pr958786334691620121nt_int] :
          ( collec7447955092554649554nt_int
          @ ( produc1676490119946744748_int_o
            @ ( produc1459180154077124618_int_o
              @ ^ [A5: int,B4: int] :
                  ( produc4947309494688390418_int_o
                  @ ^ [A8: int,B8: int] :
                      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A5 @ A8 ) @ Ra )
                      | ( ( A5 = A8 )
                        & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B4 @ B8 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_6710_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A5: nat,B4: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A8: nat,B8: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A8 ) @ Ra )
                      | ( ( A5 = A8 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B8 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_6711_mem__case__prodE,axiom,
    ! [Z3: $o,C2: int > int > set_o,P3: product_prod_int_int] :
      ( ( member_o @ Z3 @ ( produc4257766111578684402_set_o @ C2 @ P3 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( member_o @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6712_mem__case__prodE,axiom,
    ! [Z3: nat,C2: int > int > set_nat,P3: product_prod_int_int] :
      ( ( member_nat @ Z3 @ ( produc4251311855443802252et_nat @ C2 @ P3 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6713_mem__case__prodE,axiom,
    ! [Z3: int,C2: int > int > set_int,P3: product_prod_int_int] :
      ( ( member_int @ Z3 @ ( produc73460835934605544et_int @ C2 @ P3 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( member_int @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6714_mem__case__prodE,axiom,
    ! [Z3: set_nat,C2: int > int > set_set_nat,P3: product_prod_int_int] :
      ( ( member_set_nat @ Z3 @ ( produc5233655623923918146et_nat @ C2 @ P3 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( member_set_nat @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6715_mem__case__prodE,axiom,
    ! [Z3: set_Pr958786334691620121nt_int,C2: int > int > set_se6260736226359567993nt_int,P3: product_prod_int_int] :
      ( ( member2340774599025711042nt_int @ Z3 @ ( produc7771776839612048883nt_int @ C2 @ P3 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( member2340774599025711042nt_int @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6716_mem__case__prodE,axiom,
    ! [Z3: $o,C2: b > produc6653097349344004940it_nat > set_o,P3: produc7388388658123137530it_nat] :
      ( ( member_o @ Z3 @ ( produc8188766211626073367_set_o @ C2 @ P3 ) )
     => ~ ! [X3: b,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc4082563078715348724it_nat @ X3 @ Y3 ) )
           => ~ ( member_o @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6717_mem__case__prodE,axiom,
    ! [Z3: nat,C2: b > produc6653097349344004940it_nat > set_nat,P3: produc7388388658123137530it_nat] :
      ( ( member_nat @ Z3 @ ( produc8058763169506871975et_nat @ C2 @ P3 ) )
     => ~ ! [X3: b,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc4082563078715348724it_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6718_mem__case__prodE,axiom,
    ! [Z3: int,C2: b > produc6653097349344004940it_nat > set_int,P3: produc7388388658123137530it_nat] :
      ( ( member_int @ Z3 @ ( produc3880912149997675267et_int @ C2 @ P3 ) )
     => ~ ! [X3: b,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc4082563078715348724it_nat @ X3 @ Y3 ) )
           => ~ ( member_int @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6719_mem__case__prodE,axiom,
    ! [Z3: $o,C2: a > produc6653097349344004940it_nat > set_o,P3: produc3260487557148687353it_nat] :
      ( ( member_o @ Z3 @ ( produc5350075035600711000_set_o @ C2 @ P3 ) )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( member_o @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6720_mem__case__prodE,axiom,
    ! [Z3: nat,C2: a > produc6653097349344004940it_nat > set_nat,P3: produc3260487557148687353it_nat] :
      ( ( member_nat @ Z3 @ ( produc7315013382793309350et_nat @ C2 @ P3 ) )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z3 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_6721_case__prodE,axiom,
    ! [C2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P3: produc3925858234332021118et_nat] :
      ( ( produc1437786849005270451_nat_o @ C2 @ P3 )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( ( P3
              = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_6722_case__prodE,axiom,
    ! [C2: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,P3: produc2732055786443039994et_nat] :
      ( ( produc838355143741117751_nat_o @ C2 @ P3 )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
            ( ( P3
              = ( produc2245416461498447860et_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_6723_case__prodE,axiom,
    ! [C2: b > produc6653097349344004940it_nat > $o,P3: produc7388388658123137530it_nat] :
      ( ( produc3971132677695353399_nat_o @ C2 @ P3 )
     => ~ ! [X3: b,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc4082563078715348724it_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_6724_case__prodE,axiom,
    ! [C2: a > produc6653097349344004940it_nat > $o,P3: produc3260487557148687353it_nat] :
      ( ( produc8860486935419594360_nat_o @ C2 @ P3 )
     => ~ ! [X3: a,Y3: produc6653097349344004940it_nat] :
            ( ( P3
              = ( produc9178034014595674355it_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_6725_case__prodE,axiom,
    ! [C2: int > int > $o,P3: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C2 @ P3 )
     => ~ ! [X3: int,Y3: int] :
            ( ( P3
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_6726_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_6727_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_6728_case__prodD,axiom,
    ! [F: b > produc6653097349344004940it_nat > $o,A: b,B: produc6653097349344004940it_nat] :
      ( ( produc3971132677695353399_nat_o @ F @ ( produc4082563078715348724it_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6729_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_6730_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_6731_case__prodE_H,axiom,
    ! [C2: nat > nat > product_prod_nat_nat > $o,P3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C2 @ P3 @ Z3 )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 @ Z3 ) ) ) ).

% case_prodE'
thf(fact_6732_case__prodD_H,axiom,
    ! [R3: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C2: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R3 @ ( product_Pair_nat_nat @ A @ B ) @ C2 )
     => ( R3 @ A @ B @ C2 ) ) ).

% case_prodD'
thf(fact_6733_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6734_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6735_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6736_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6737_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_6738_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_6739_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_6740_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P2 ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_6741_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_6742_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_6743_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_6744_AND__upper1_H,axiom,
    ! [Y: int,Z3: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z3 )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z3 ) ) ) ).

% AND_upper1'
thf(fact_6745_AND__upper2_H,axiom,
    ! [Y: int,Z3: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z3 )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z3 ) ) ) ).

% AND_upper2'
thf(fact_6746_Collect__case__prod__mono,axiom,
    ! [A4: int > int > $o,B5: int > int > $o] :
      ( ( ord_le6741204236512500942_int_o @ A4 @ B5 )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_6747_execute__bind__case,axiom,
    ! [F: heap_Time_Heap_b,G: b > heap_Time_Heap_b,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_execute_b @ ( heap_Time_bind_b_b @ F @ G ) @ H )
      = ( case_o5901184372446974929it_nat @ none_P6779099274072355161it_nat
        @ ( produc3931674313442123243it_nat
          @ ^ [X4: b] :
              ( produc6148253707941125437it_nat
              @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711336rame_b @ N @ ( heap_Time_execute_b @ ( G @ X4 ) @ H7 ) ) ) )
        @ ( heap_Time_execute_b @ F @ H ) ) ) ).

% execute_bind_case
thf(fact_6748_execute__bind__case,axiom,
    ! [F: heap_Time_Heap_a,G: a > heap_Time_Heap_b,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_execute_b @ ( heap_Time_bind_a_b @ F @ G ) @ H )
      = ( case_o1773283271472524752it_nat @ none_P6779099274072355161it_nat
        @ ( produc8352694711413002346it_nat
          @ ^ [X4: a] :
              ( produc6148253707941125437it_nat
              @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711336rame_b @ N @ ( heap_Time_execute_b @ ( G @ X4 ) @ H7 ) ) ) )
        @ ( heap_Time_execute_a @ F @ H ) ) ) ).

% execute_bind_case
thf(fact_6749_execute__bind__case,axiom,
    ! [F: heap_Time_Heap_b,G: b > heap_Time_Heap_a,H: heap_e7401611519738050253t_unit] :
      ( ( heap_Time_execute_a @ ( heap_Time_bind_b_a @ F @ G ) @ H )
      = ( case_o965627003169710416it_nat @ none_P2651198173097904984it_nat
        @ ( produc3428986626803312746it_nat
          @ ^ [X4: b] :
              ( produc5645566021302314940it_nat
              @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711335rame_a @ N @ ( heap_Time_execute_a @ ( G @ X4 ) @ H7 ) ) ) )
        @ ( heap_Time_execute_b @ F @ H ) ) ) ).

% execute_bind_case
thf(fact_6750_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_6751_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_6752_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( 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_rec
thf(fact_6753_AND__upper2_H_H,axiom,
    ! [Y: int,Z3: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z3 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z3 ) ) ) ).

% AND_upper2''
thf(fact_6754_AND__upper1_H_H,axiom,
    ! [Y: int,Z3: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z3 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z3 ) ) ) ).

% AND_upper1''
thf(fact_6755_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ K2 ) ) ).

% and_less_eq
thf(fact_6756_Heap__Time__Monad_Obind__def,axiom,
    ( heap_Time_bind_b_b
    = ( ^ [F3: heap_Time_Heap_b,G3: b > heap_Time_Heap_b] :
          ( heap_Time_Heap_b2
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( case_o5901184372446974929it_nat @ none_P6779099274072355161it_nat
              @ ( produc3931674313442123243it_nat
                @ ^ [R5: b] :
                    ( produc6148253707941125437it_nat
                    @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711336rame_b @ N @ ( heap_Time_execute_b @ ( G3 @ R5 ) @ H7 ) ) ) )
              @ ( heap_Time_execute_b @ F3 @ H6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_6757_Heap__Time__Monad_Obind__def,axiom,
    ( heap_Time_bind_a_b
    = ( ^ [F3: heap_Time_Heap_a,G3: a > heap_Time_Heap_b] :
          ( heap_Time_Heap_b2
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( case_o1773283271472524752it_nat @ none_P6779099274072355161it_nat
              @ ( produc8352694711413002346it_nat
                @ ^ [R5: a] :
                    ( produc6148253707941125437it_nat
                    @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711336rame_b @ N @ ( heap_Time_execute_b @ ( G3 @ R5 ) @ H7 ) ) ) )
              @ ( heap_Time_execute_a @ F3 @ H6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_6758_Heap__Time__Monad_Obind__def,axiom,
    ( heap_Time_bind_b_a
    = ( ^ [F3: heap_Time_Heap_b,G3: b > heap_Time_Heap_a] :
          ( heap_Time_Heap_a2
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( case_o965627003169710416it_nat @ none_P2651198173097904984it_nat
              @ ( produc3428986626803312746it_nat
                @ ^ [R5: b] :
                    ( produc5645566021302314940it_nat
                    @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711335rame_a @ N @ ( heap_Time_execute_a @ ( G3 @ R5 ) @ H7 ) ) ) )
              @ ( heap_Time_execute_b @ F3 @ H6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_6759_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
                @ ^ [R5: a] :
                    ( produc5645566021302314940it_nat
                    @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T7616092557645711335rame_a @ N @ ( heap_Time_execute_a @ ( G3 @ R5 ) @ H7 ) ) ) )
              @ ( heap_Time_execute_a @ F3 @ H6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_6760_Heap__Time__Monad_Obind__def,axiom,
    ( heap_T3413130826733729493t_unit
    = ( ^ [F3: heap_Time_Heap_b,G3: b > heap_T5738788834812785303t_unit] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( case_o2632329678403290591it_nat @ none_P9117596204409417319it_nat
              @ ( produc450363836004716793it_nat
                @ ^ [R5: b] :
                    ( produc7488178964372371019it_nat
                    @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T3616969660504097270t_unit @ N @ ( heap_T875086893843062177t_unit @ ( G3 @ R5 ) @ H7 ) ) ) )
              @ ( heap_Time_execute_b @ F3 @ H6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_6761_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
                @ ^ [R5: a] :
                    ( produc7488178964372371019it_nat
                    @ ^ [H7: heap_e7401611519738050253t_unit,N: nat] : ( heap_T3616969660504097270t_unit @ N @ ( heap_T875086893843062177t_unit @ ( G3 @ R5 ) @ H7 ) ) ) )
              @ ( heap_Time_execute_a @ F3 @ H6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_6762_rel__of__def,axiom,
    ( rel_of4838799251197538391et_nat
    = ( ^ [M3: ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat,P5: produc3925858234332021118et_nat > $o] :
          ( collec1402215087704437587et_nat
          @ ( produc1437786849005270451_nat_o
            @ ^ [K3: produc3658429121746597890et_nat > $o,V2: produc3658429121746597890et_nat] :
                ( ( ( M3 @ K3 )
                  = ( some_P624177172695371229et_nat @ V2 ) )
                & ( P5 @ ( produc5001842942810119800et_nat @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_6763_rel__of__def,axiom,
    ( rel_of7774016450764239315et_nat
    = ( ^ [M3: ( produc3658429121746597890et_nat > $o ) > option5190343406534369742et_nat,P5: produc2732055786443039994et_nat > $o] :
          ( collec5543584681430388431et_nat
          @ ( produc838355143741117751_nat_o
            @ ^ [K3: produc3658429121746597890et_nat > $o,V2: produc3925858234332021118et_nat] :
                ( ( ( M3 @ K3 )
                  = ( some_P750831030444334937et_nat @ V2 ) )
                & ( P5 @ ( produc2245416461498447860et_nat @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_6764_rel__of__def,axiom,
    ( rel_of6889123638293104083it_nat
    = ( ^ [M3: b > option233860712434008220it_nat,P5: produc7388388658123137530it_nat > $o] :
          ( collec8276579174399660751it_nat
          @ ( produc3971132677695353399_nat_o
            @ ^ [K3: b,V2: produc6653097349344004940it_nat] :
                ( ( ( M3 @ K3 )
                  = ( some_P1054239925786823975it_nat @ V2 ) )
                & ( P5 @ ( produc4082563078715348724it_nat @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_6765_rel__of__def,axiom,
    ( rel_of2761222537318653906it_nat
    = ( ^ [M3: a > option233860712434008220it_nat,P5: produc3260487557148687353it_nat > $o] :
          ( collec4148678073425210574it_nat
          @ ( produc8860486935419594360_nat_o
            @ ^ [K3: a,V2: produc6653097349344004940it_nat] :
                ( ( ( M3 @ K3 )
                  = ( some_P1054239925786823975it_nat @ V2 ) )
                & ( P5 @ ( produc9178034014595674355it_nat @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_6766_rel__of__def,axiom,
    ( rel_of_int_int
    = ( ^ [M3: int > option_int,P5: product_prod_int_int > $o] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [K3: int,V2: int] :
                ( ( ( M3 @ K3 )
                  = ( some_int @ V2 ) )
                & ( P5 @ ( product_Pair_int_int @ K3 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_6767_bits__induct,axiom,
    ! [P2: code_natural > $o,A: code_natural] :
      ( ! [A3: code_natural] :
          ( ( ( divide5121882707175180666atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
            = A3 )
         => ( P2 @ A3 ) )
     => ( ! [A3: code_natural,B3: $o] :
            ( ( P2 @ A3 )
           => ( ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B3 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
                = A3 )
             => ( P2 @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B3 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_6768_bits__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [A3: nat] :
          ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A3 )
         => ( P2 @ A3 ) )
     => ( ! [A3: nat,B3: $o] :
            ( ( P2 @ A3 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A3 )
             => ( P2 @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_6769_bits__induct,axiom,
    ! [P2: int > $o,A: int] :
      ( ! [A3: int] :
          ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A3 )
         => ( P2 @ A3 ) )
     => ( ! [A3: int,B3: $o] :
            ( ( P2 @ A3 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A3 )
             => ( P2 @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_6770_bits__induct,axiom,
    ! [P2: code_integer > $o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A3 )
         => ( P2 @ A3 ) )
     => ( ! [A3: code_integer,B3: $o] :
            ( ( P2 @ A3 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A3 )
             => ( P2 @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_6771_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_6772_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_6773_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_6774_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_6775_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_6776_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_6777_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_6778_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_6779_divide__int__unfold,axiom,
    ! [L: int,K2: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K2 )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K2 )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
          & ( ( ( sgn_sgn_int @ K2 )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_6780_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_6781_mlex__eq,axiom,
    ( mlex_prod_int
    = ( ^ [F3: int > nat,R: 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 ) @ R ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_6782_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_6783_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_6784_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_6785_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_6786_Divides_Oadjust__div__eq,axiom,
    ! [Q6: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q6 @ R2 ) )
      = ( plus_plus_int @ Q6 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_6787_insert__absorb2,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ X @ A4 ) )
      = ( insert9069300056098147895at_nat @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_6788_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ X @ A4 ) )
      = ( insert8211810215607154385at_nat @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_6789_insert__absorb2,axiom,
    ! [X: $o,A4: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A4 ) )
      = ( insert_o @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_6790_insert__absorb2,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A4 ) )
      = ( insert_nat @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_6791_insert__absorb2,axiom,
    ! [X: int,A4: set_int] :
      ( ( insert_int @ X @ ( insert_int @ X @ A4 ) )
      = ( insert_int @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_6792_insert__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8757157785044589968at_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_6793_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_6794_insert__iff,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A4 ) )
      = ( ( A = B )
        | ( member_o @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_6795_insert__iff,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ A @ ( insert8897473484851387113nt_int @ B @ A4 ) )
      = ( ( A = B )
        | ( member2340774599025711042nt_int @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_6796_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member_set_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_6797_insert__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_6798_insert__iff,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A4 ) )
      = ( ( A = B )
        | ( member_int @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_6799_insertCI,axiom,
    ! [A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ A @ B5 )
       => ( A = B ) )
     => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_6800_insertCI,axiom,
    ! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ A @ B5 )
       => ( A = B ) )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_6801_insertCI,axiom,
    ! [A: $o,B5: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B5 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B5 ) ) ) ).

% insertCI
thf(fact_6802_insertCI,axiom,
    ! [A: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int] :
      ( ( ~ ( member2340774599025711042nt_int @ A @ B5 )
       => ( A = B ) )
     => ( member2340774599025711042nt_int @ A @ ( insert8897473484851387113nt_int @ B @ B5 ) ) ) ).

% insertCI
thf(fact_6803_insertCI,axiom,
    ! [A: set_nat,B5: set_set_nat,B: set_nat] :
      ( ( ~ ( member_set_nat @ A @ B5 )
       => ( A = B ) )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_6804_insertCI,axiom,
    ! [A: nat,B5: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B5 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_6805_insertCI,axiom,
    ! [A: int,B5: set_int,B: int] :
      ( ( ~ ( member_int @ A @ B5 )
       => ( A = B ) )
     => ( member_int @ A @ ( insert_int @ B @ B5 ) ) ) ).

% insertCI
thf(fact_6806_split__part,axiom,
    ! [P2: $o,Q2: int > int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [A5: int,B4: int] :
            ( P2
            & ( Q2 @ A5 @ B4 ) ) )
      = ( ^ [Ab: product_prod_int_int] :
            ( P2
            & ( produc4947309494688390418_int_o @ Q2 @ Ab ) ) ) ) ).

% split_part
thf(fact_6807_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singletonI
thf(fact_6808_singletonI,axiom,
    ! [A: set_Pr958786334691620121nt_int] : ( member2340774599025711042nt_int @ A @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) ).

% singletonI
thf(fact_6809_singletonI,axiom,
    ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_6810_singletonI,axiom,
    ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_6811_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_6812_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_6813_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_6814_insert__subset,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) @ B5 )
      = ( ( member8757157785044589968at_nat @ X @ B5 )
        & ( ord_le1268244103169919719at_nat @ A4 @ B5 ) ) ) ).

% insert_subset
thf(fact_6815_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ B5 )
      = ( ( member8440522571783428010at_nat @ X @ B5 )
        & ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ) ).

% insert_subset
thf(fact_6816_insert__subset,axiom,
    ! [X: $o,A4: set_o,B5: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X @ A4 ) @ B5 )
      = ( ( member_o @ X @ B5 )
        & ( ord_less_eq_set_o @ A4 @ B5 ) ) ) ).

% insert_subset
thf(fact_6817_insert__subset,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( ord_le483042692224249369nt_int @ ( insert8897473484851387113nt_int @ X @ A4 ) @ B5 )
      = ( ( member2340774599025711042nt_int @ X @ B5 )
        & ( ord_le483042692224249369nt_int @ A4 @ B5 ) ) ) ).

% insert_subset
thf(fact_6818_insert__subset,axiom,
    ! [X: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A4 ) @ B5 )
      = ( ( member_set_nat @ X @ B5 )
        & ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ) ).

% insert_subset
thf(fact_6819_insert__subset,axiom,
    ! [X: nat,A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A4 ) @ B5 )
      = ( ( member_nat @ X @ B5 )
        & ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).

% insert_subset
thf(fact_6820_insert__subset,axiom,
    ! [X: int,A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A4 ) @ B5 )
      = ( ( member_int @ X @ B5 )
        & ( ord_less_eq_set_int @ A4 @ B5 ) ) ) ).

% insert_subset
thf(fact_6821_Int__insert__right__if1,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
        = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6822_Int__insert__right__if1,axiom,
    ! [A: $o,A4: set_o,B5: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B5 ) )
        = ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6823_Int__insert__right__if1,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ A @ A4 )
     => ( ( inf_in8396524679539076455nt_int @ A4 @ ( insert8897473484851387113nt_int @ A @ B5 ) )
        = ( insert8897473484851387113nt_int @ A @ ( inf_in8396524679539076455nt_int @ A4 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6824_Int__insert__right__if1,axiom,
    ! [A: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ A @ A4 )
     => ( ( inf_inf_set_set_nat @ A4 @ ( insert_set_nat @ A @ B5 ) )
        = ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A4 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6825_Int__insert__right__if1,axiom,
    ! [A: int,A4: set_int,B5: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ( inf_inf_set_int @ A4 @ ( insert_int @ A @ B5 ) )
        = ( insert_int @ A @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6826_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
        = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6827_Int__insert__right__if1,axiom,
    ! [A: nat,A4: set_nat,B5: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
        = ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6828_Int__insert__right__if0,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
        = ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_6829_Int__insert__right__if0,axiom,
    ! [A: $o,A4: set_o,B5: set_o] :
      ( ~ ( member_o @ A @ A4 )
     => ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B5 ) )
        = ( inf_inf_set_o @ A4 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_6830_Int__insert__right__if0,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ A @ A4 )
     => ( ( inf_in8396524679539076455nt_int @ A4 @ ( insert8897473484851387113nt_int @ A @ B5 ) )
        = ( inf_in8396524679539076455nt_int @ A4 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_6831_Int__insert__right__if0,axiom,
    ! [A: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ~ ( member_set_nat @ A @ A4 )
     => ( ( inf_inf_set_set_nat @ A4 @ ( insert_set_nat @ A @ B5 ) )
        = ( inf_inf_set_set_nat @ A4 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_6832_Int__insert__right__if0,axiom,
    ! [A: int,A4: set_int,B5: set_int] :
      ( ~ ( member_int @ A @ A4 )
     => ( ( inf_inf_set_int @ A4 @ ( insert_int @ A @ B5 ) )
        = ( inf_inf_set_int @ A4 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_6833_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
        = ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_6834_Int__insert__right__if0,axiom,
    ! [A: nat,A4: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ A @ A4 )
     => ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
        = ( inf_inf_set_nat @ A4 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_6835_insert__inter__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) @ ( insert9069300056098147895at_nat @ A @ B5 ) )
      = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ).

% insert_inter_insert
thf(fact_6836_insert__inter__insert,axiom,
    ! [A: $o,A4: set_o,B5: set_o] :
      ( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ ( insert_o @ A @ B5 ) )
      = ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B5 ) ) ) ).

% insert_inter_insert
thf(fact_6837_insert__inter__insert,axiom,
    ! [A: int,A4: set_int,B5: set_int] :
      ( ( inf_inf_set_int @ ( insert_int @ A @ A4 ) @ ( insert_int @ A @ B5 ) )
      = ( insert_int @ A @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ).

% insert_inter_insert
thf(fact_6838_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) @ ( insert8211810215607154385at_nat @ A @ B5 ) )
      = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ).

% insert_inter_insert
thf(fact_6839_insert__inter__insert,axiom,
    ! [A: nat,A4: set_nat,B5: set_nat] :
      ( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ ( insert_nat @ A @ B5 ) )
      = ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ).

% insert_inter_insert
thf(fact_6840_Int__insert__left__if1,axiom,
    ! [A: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ C4 )
     => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B5 ) @ C4 )
        = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6841_Int__insert__left__if1,axiom,
    ! [A: $o,C4: set_o,B5: set_o] :
      ( ( member_o @ A @ C4 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C4 )
        = ( insert_o @ A @ ( inf_inf_set_o @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6842_Int__insert__left__if1,axiom,
    ! [A: set_Pr958786334691620121nt_int,C4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ A @ C4 )
     => ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ B5 ) @ C4 )
        = ( insert8897473484851387113nt_int @ A @ ( inf_in8396524679539076455nt_int @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6843_Int__insert__left__if1,axiom,
    ! [A: set_nat,C4: set_set_nat,B5: set_set_nat] :
      ( ( member_set_nat @ A @ C4 )
     => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B5 ) @ C4 )
        = ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6844_Int__insert__left__if1,axiom,
    ! [A: int,C4: set_int,B5: set_int] :
      ( ( member_int @ A @ C4 )
     => ( ( inf_inf_set_int @ ( insert_int @ A @ B5 ) @ C4 )
        = ( insert_int @ A @ ( inf_inf_set_int @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6845_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ C4 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C4 )
        = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6846_Int__insert__left__if1,axiom,
    ! [A: nat,C4: set_nat,B5: set_nat] :
      ( ( member_nat @ A @ C4 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C4 )
        = ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6847_Int__insert__left__if0,axiom,
    ! [A: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ C4 )
     => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B5 ) @ C4 )
        = ( inf_in7913087082777306421at_nat @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_6848_Int__insert__left__if0,axiom,
    ! [A: $o,C4: set_o,B5: set_o] :
      ( ~ ( member_o @ A @ C4 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C4 )
        = ( inf_inf_set_o @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_6849_Int__insert__left__if0,axiom,
    ! [A: set_Pr958786334691620121nt_int,C4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ A @ C4 )
     => ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ B5 ) @ C4 )
        = ( inf_in8396524679539076455nt_int @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_6850_Int__insert__left__if0,axiom,
    ! [A: set_nat,C4: set_set_nat,B5: set_set_nat] :
      ( ~ ( member_set_nat @ A @ C4 )
     => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B5 ) @ C4 )
        = ( inf_inf_set_set_nat @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_6851_Int__insert__left__if0,axiom,
    ! [A: int,C4: set_int,B5: set_int] :
      ( ~ ( member_int @ A @ C4 )
     => ( ( inf_inf_set_int @ ( insert_int @ A @ B5 ) @ C4 )
        = ( inf_inf_set_int @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_6852_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ C4 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C4 )
        = ( inf_in2572325071724192079at_nat @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_6853_Int__insert__left__if0,axiom,
    ! [A: nat,C4: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ A @ C4 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C4 )
        = ( inf_inf_set_nat @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_6854_Un__insert__left,axiom,
    ! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C4 )
      = ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_6855_Un__insert__left,axiom,
    ! [A: $o,B5: set_o,C4: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A @ B5 ) @ C4 )
      = ( insert_o @ A @ ( sup_sup_set_o @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_6856_Un__insert__left,axiom,
    ! [A: int,B5: set_int,C4: set_int] :
      ( ( sup_sup_set_int @ ( insert_int @ A @ B5 ) @ C4 )
      = ( insert_int @ A @ ( sup_sup_set_int @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_6857_Un__insert__left,axiom,
    ! [A: produc4166570645942440679at_nat,B5: set_Pr8551490117392284871at_nat,C4: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ ( insert6337962749363155127at_nat @ A @ B5 ) @ C4 )
      = ( insert6337962749363155127at_nat @ A @ ( sup_su3035147773818789531at_nat @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_6858_Un__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ B5 ) @ C4 )
      = ( insert9069300056098147895at_nat @ A @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_6859_Un__insert__left,axiom,
    ! [A: nat,B5: set_nat,C4: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat @ A @ B5 ) @ C4 )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_6860_Un__insert__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
      = ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_6861_Un__insert__right,axiom,
    ! [A4: set_o,A: $o,B5: set_o] :
      ( ( sup_sup_set_o @ A4 @ ( insert_o @ A @ B5 ) )
      = ( insert_o @ A @ ( sup_sup_set_o @ A4 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_6862_Un__insert__right,axiom,
    ! [A4: set_int,A: int,B5: set_int] :
      ( ( sup_sup_set_int @ A4 @ ( insert_int @ A @ B5 ) )
      = ( insert_int @ A @ ( sup_sup_set_int @ A4 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_6863_Un__insert__right,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,A: produc4166570645942440679at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( sup_su3035147773818789531at_nat @ A4 @ ( insert6337962749363155127at_nat @ A @ B5 ) )
      = ( insert6337962749363155127at_nat @ A @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_6864_Un__insert__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
      = ( insert9069300056098147895at_nat @ A @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_6865_Un__insert__right,axiom,
    ! [A4: set_nat,A: nat,B5: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_6866_insert__Diff1,axiom,
    ! [X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ B5 )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) @ B5 )
        = ( minus_3314409938677909166at_nat @ A4 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_6867_insert__Diff1,axiom,
    ! [X: $o,B5: set_o,A4: set_o] :
      ( ( member_o @ X @ B5 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B5 )
        = ( minus_minus_set_o @ A4 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_6868_insert__Diff1,axiom,
    ! [X: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ X @ B5 )
     => ( ( minus_2612819937483484256nt_int @ ( insert8897473484851387113nt_int @ X @ A4 ) @ B5 )
        = ( minus_2612819937483484256nt_int @ A4 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_6869_insert__Diff1,axiom,
    ! [X: set_nat,B5: set_set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ X @ B5 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ B5 )
        = ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_6870_insert__Diff1,axiom,
    ! [X: nat,B5: set_nat,A4: set_nat] :
      ( ( member_nat @ X @ B5 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B5 )
        = ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_6871_insert__Diff1,axiom,
    ! [X: int,B5: set_int,A4: set_int] :
      ( ( member_int @ X @ B5 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ B5 )
        = ( minus_minus_set_int @ A4 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_6872_insert__Diff1,axiom,
    ! [X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ B5 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ B5 )
        = ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_6873_Diff__insert0,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A4 )
     => ( ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B5 ) )
        = ( minus_3314409938677909166at_nat @ A4 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_6874_Diff__insert0,axiom,
    ! [X: $o,A4: set_o,B5: set_o] :
      ( ~ ( member_o @ X @ A4 )
     => ( ( minus_minus_set_o @ A4 @ ( insert_o @ X @ B5 ) )
        = ( minus_minus_set_o @ A4 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_6875_Diff__insert0,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ X @ A4 )
     => ( ( minus_2612819937483484256nt_int @ A4 @ ( insert8897473484851387113nt_int @ X @ B5 ) )
        = ( minus_2612819937483484256nt_int @ A4 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_6876_Diff__insert0,axiom,
    ! [X: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A4 )
     => ( ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ B5 ) )
        = ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_6877_Diff__insert0,axiom,
    ! [X: nat,A4: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ B5 ) )
        = ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_6878_Diff__insert0,axiom,
    ! [X: int,A4: set_int,B5: set_int] :
      ( ~ ( member_int @ X @ A4 )
     => ( ( minus_minus_set_int @ A4 @ ( insert_int @ X @ B5 ) )
        = ( minus_minus_set_int @ A4 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_6879_Diff__insert0,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A4 )
     => ( ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) )
        = ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_6880_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] : X4 = A )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv
thf(fact_6881_singleton__conv,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : X4 = A )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_6882_singleton__conv,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ( collec5210948495886036740nt_int
        @ ^ [X4: set_Pr958786334691620121nt_int] : X4 = A )
      = ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) ).

% singleton_conv
thf(fact_6883_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : X4 = A )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_6884_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : X4 = A )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_6885_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X4: $o] : X4 = A )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_6886_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : X4 = A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_6887_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X4: int] : X4 = A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_6888_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ( ^ [Y6: produc3843707927480180839at_nat,Z4: produc3843707927480180839at_nat] : Y6 = Z4
          @ A ) )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv2
thf(fact_6889_singleton__conv2,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ( ^ [Y6: list_nat,Z4: list_nat] : Y6 = Z4
          @ A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv2
thf(fact_6890_singleton__conv2,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ( collec5210948495886036740nt_int
        @ ( ^ [Y6: set_Pr958786334691620121nt_int,Z4: set_Pr958786334691620121nt_int] : Y6 = Z4
          @ A ) )
      = ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) ).

% singleton_conv2
thf(fact_6891_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y6: set_nat,Z4: set_nat] : Y6 = Z4
          @ A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_6892_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y6: product_prod_nat_nat,Z4: product_prod_nat_nat] : Y6 = Z4
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_6893_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y6: $o,Z4: $o] : Y6 = Z4
          @ A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_6894_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y6: nat,Z4: nat] : Y6 = Z4
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_6895_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y6: int,Z4: int] : Y6 = Z4
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_6896_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ A4 )
        = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
      = ( ( A = B )
        & ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_6897_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ A4 )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_6898_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A4: set_o,B: $o] :
      ( ( ( insert_o @ A @ A4 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_6899_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A4: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A4 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_6900_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A4: set_int,B: int] :
      ( ( ( insert_int @ A @ A4 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_6901_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_6902_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_6903_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A4: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_6904_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A4: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_6905_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A4: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_6906_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ B5 ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B @ A4 )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6907_disjoint__insert_I2_J,axiom,
    ! [A4: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( bot_bo1488462491386950373nt_int
        = ( inf_in8396524679539076455nt_int @ A4 @ ( insert8897473484851387113nt_int @ B @ B5 ) ) )
      = ( ~ ( member2340774599025711042nt_int @ B @ A4 )
        & ( bot_bo1488462491386950373nt_int
          = ( inf_in8396524679539076455nt_int @ A4 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6908_disjoint__insert_I2_J,axiom,
    ! [A4: set_set_nat,B: set_nat,B5: set_set_nat] :
      ( ( bot_bot_set_set_nat
        = ( inf_inf_set_set_nat @ A4 @ ( insert_set_nat @ B @ B5 ) ) )
      = ( ~ ( member_set_nat @ B @ A4 )
        & ( bot_bot_set_set_nat
          = ( inf_inf_set_set_nat @ A4 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6909_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ B5 ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A4 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6910_disjoint__insert_I2_J,axiom,
    ! [A4: set_o,B: $o,B5: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A4 @ ( insert_o @ B @ B5 ) ) )
      = ( ~ ( member_o @ B @ A4 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6911_disjoint__insert_I2_J,axiom,
    ! [A4: set_nat,B: nat,B5: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A4 @ ( insert_nat @ B @ B5 ) ) )
      = ( ~ ( member_nat @ B @ A4 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6912_disjoint__insert_I2_J,axiom,
    ! [A4: set_int,B: int,B5: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ A4 @ ( insert_int @ B @ B5 ) ) )
      = ( ~ ( member_int @ B @ A4 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6913_disjoint__insert_I1_J,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ B5 @ ( insert9069300056098147895at_nat @ A @ A4 ) )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A @ B5 )
        & ( ( inf_in7913087082777306421at_nat @ B5 @ A4 )
          = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_6914_disjoint__insert_I1_J,axiom,
    ! [B5: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( ( inf_in8396524679539076455nt_int @ B5 @ ( insert8897473484851387113nt_int @ A @ A4 ) )
        = bot_bo1488462491386950373nt_int )
      = ( ~ ( member2340774599025711042nt_int @ A @ B5 )
        & ( ( inf_in8396524679539076455nt_int @ B5 @ A4 )
          = bot_bo1488462491386950373nt_int ) ) ) ).

% disjoint_insert(1)
thf(fact_6915_disjoint__insert_I1_J,axiom,
    ! [B5: set_set_nat,A: set_nat,A4: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ B5 @ ( insert_set_nat @ A @ A4 ) )
        = bot_bot_set_set_nat )
      = ( ~ ( member_set_nat @ A @ B5 )
        & ( ( inf_inf_set_set_nat @ B5 @ A4 )
          = bot_bot_set_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_6916_disjoint__insert_I1_J,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ B5 @ ( insert8211810215607154385at_nat @ A @ A4 ) )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B5 )
        & ( ( inf_in2572325071724192079at_nat @ B5 @ A4 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_6917_disjoint__insert_I1_J,axiom,
    ! [B5: set_o,A: $o,A4: set_o] :
      ( ( ( inf_inf_set_o @ B5 @ ( insert_o @ A @ A4 ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B5 )
        & ( ( inf_inf_set_o @ B5 @ A4 )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_6918_disjoint__insert_I1_J,axiom,
    ! [B5: set_nat,A: nat,A4: set_nat] :
      ( ( ( inf_inf_set_nat @ B5 @ ( insert_nat @ A @ A4 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B5 )
        & ( ( inf_inf_set_nat @ B5 @ A4 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_6919_disjoint__insert_I1_J,axiom,
    ! [B5: set_int,A: int,A4: set_int] :
      ( ( ( inf_inf_set_int @ B5 @ ( insert_int @ A @ A4 ) )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A @ B5 )
        & ( ( inf_inf_set_int @ B5 @ A4 )
          = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
thf(fact_6920_insert__disjoint_I2_J,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) @ B5 ) )
      = ( ~ ( member8757157785044589968at_nat @ A @ B5 )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6921_insert__disjoint_I2_J,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( bot_bo1488462491386950373nt_int
        = ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ A4 ) @ B5 ) )
      = ( ~ ( member2340774599025711042nt_int @ A @ B5 )
        & ( bot_bo1488462491386950373nt_int
          = ( inf_in8396524679539076455nt_int @ A4 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6922_insert__disjoint_I2_J,axiom,
    ! [A: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( bot_bot_set_set_nat
        = ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A4 ) @ B5 ) )
      = ( ~ ( member_set_nat @ A @ B5 )
        & ( bot_bot_set_set_nat
          = ( inf_inf_set_set_nat @ A4 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6923_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) @ B5 ) )
      = ( ~ ( member8440522571783428010at_nat @ A @ B5 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6924_insert__disjoint_I2_J,axiom,
    ! [A: $o,A4: set_o,B5: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B5 ) )
      = ( ~ ( member_o @ A @ B5 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6925_insert__disjoint_I2_J,axiom,
    ! [A: nat,A4: set_nat,B5: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B5 ) )
      = ( ~ ( member_nat @ A @ B5 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6926_insert__disjoint_I2_J,axiom,
    ! [A: int,A4: set_int,B5: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ ( insert_int @ A @ A4 ) @ B5 ) )
      = ( ~ ( member_int @ A @ B5 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6927_insert__disjoint_I1_J,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) @ B5 )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A @ B5 )
        & ( ( inf_in7913087082777306421at_nat @ A4 @ B5 )
          = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_6928_insert__disjoint_I1_J,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ A4 ) @ B5 )
        = bot_bo1488462491386950373nt_int )
      = ( ~ ( member2340774599025711042nt_int @ A @ B5 )
        & ( ( inf_in8396524679539076455nt_int @ A4 @ B5 )
          = bot_bo1488462491386950373nt_int ) ) ) ).

% insert_disjoint(1)
thf(fact_6929_insert__disjoint_I1_J,axiom,
    ! [A: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A4 ) @ B5 )
        = bot_bot_set_set_nat )
      = ( ~ ( member_set_nat @ A @ B5 )
        & ( ( inf_inf_set_set_nat @ A4 @ B5 )
          = bot_bot_set_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_6930_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) @ B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B5 )
        & ( ( inf_in2572325071724192079at_nat @ A4 @ B5 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_6931_insert__disjoint_I1_J,axiom,
    ! [A: $o,A4: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B5 )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B5 )
        & ( ( inf_inf_set_o @ A4 @ B5 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_6932_insert__disjoint_I1_J,axiom,
    ! [A: nat,A4: set_nat,B5: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B5 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B5 )
        & ( ( inf_inf_set_nat @ A4 @ B5 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_6933_insert__disjoint_I1_J,axiom,
    ! [A: int,A4: set_int,B5: set_int] :
      ( ( ( inf_inf_set_int @ ( insert_int @ A @ A4 ) @ B5 )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A @ B5 )
        & ( ( inf_inf_set_int @ A4 @ B5 )
          = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
thf(fact_6934_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      = ( insert9069300056098147895at_nat @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_6935_insert__Diff__single,axiom,
    ! [A: $o,A4: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_6936_insert__Diff__single,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_6937_insert__Diff__single,axiom,
    ! [A: int,A4: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_6938_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( insert8211810215607154385at_nat @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_6939_subset__Compl__singleton,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_6940_subset__Compl__singleton,axiom,
    ! [A4: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int] :
      ( ( ord_le483042692224249369nt_int @ A4 @ ( uminus6423885277529793776nt_int @ ( insert8897473484851387113nt_int @ B @ bot_bo1488462491386950373nt_int ) ) )
      = ( ~ ( member2340774599025711042nt_int @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_6941_subset__Compl__singleton,axiom,
    ! [A4: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_6942_subset__Compl__singleton,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_6943_subset__Compl__singleton,axiom,
    ! [A4: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_6944_subset__Compl__singleton,axiom,
    ! [A4: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_6945_subset__Compl__singleton,axiom,
    ! [A4: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_6946_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] :
      ( produc4947309494688390418_int_o
      @ ^ [Uu2: int,Uv2: int] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_6947_mk__disjoint__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ? [B9: set_Pr4329608150637261639at_nat] :
          ( ( A4
            = ( insert9069300056098147895at_nat @ A @ B9 ) )
          & ~ ( member8757157785044589968at_nat @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_6948_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ? [B9: set_Pr1261947904930325089at_nat] :
          ( ( A4
            = ( insert8211810215607154385at_nat @ A @ B9 ) )
          & ~ ( member8440522571783428010at_nat @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_6949_mk__disjoint__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ? [B9: set_o] :
          ( ( A4
            = ( insert_o @ A @ B9 ) )
          & ~ ( member_o @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_6950_mk__disjoint__insert,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ A @ A4 )
     => ? [B9: set_se6260736226359567993nt_int] :
          ( ( A4
            = ( insert8897473484851387113nt_int @ A @ B9 ) )
          & ~ ( member2340774599025711042nt_int @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_6951_mk__disjoint__insert,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ A4 )
     => ? [B9: set_set_nat] :
          ( ( A4
            = ( insert_set_nat @ A @ B9 ) )
          & ~ ( member_set_nat @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_6952_mk__disjoint__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ? [B9: set_nat] :
          ( ( A4
            = ( insert_nat @ A @ B9 ) )
          & ~ ( member_nat @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_6953_mk__disjoint__insert,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ? [B9: set_int] :
          ( ( A4
            = ( insert_int @ A @ B9 ) )
          & ~ ( member_int @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_6954_insert__commute,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ Y @ A4 ) )
      = ( insert9069300056098147895at_nat @ Y @ ( insert9069300056098147895at_nat @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_6955_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ A4 ) )
      = ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_6956_insert__commute,axiom,
    ! [X: $o,Y: $o,A4: set_o] :
      ( ( insert_o @ X @ ( insert_o @ Y @ A4 ) )
      = ( insert_o @ Y @ ( insert_o @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_6957_insert__commute,axiom,
    ! [X: nat,Y: nat,A4: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ Y @ A4 ) )
      = ( insert_nat @ Y @ ( insert_nat @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_6958_insert__commute,axiom,
    ! [X: int,Y: int,A4: set_int] :
      ( ( insert_int @ X @ ( insert_int @ Y @ A4 ) )
      = ( insert_int @ Y @ ( insert_int @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_6959_insert__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ A4 )
     => ( ~ ( member8757157785044589968at_nat @ B @ B5 )
       => ( ( ( insert9069300056098147895at_nat @ A @ A4 )
            = ( insert9069300056098147895at_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A4 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr4329608150637261639at_nat] :
                  ( ( A4
                    = ( insert9069300056098147895at_nat @ B @ C5 ) )
                  & ~ ( member8757157785044589968at_nat @ B @ C5 )
                  & ( B5
                    = ( insert9069300056098147895at_nat @ A @ C5 ) )
                  & ~ ( member8757157785044589968at_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_6960_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ B @ B5 )
       => ( ( ( insert8211810215607154385at_nat @ A @ A4 )
            = ( insert8211810215607154385at_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A4 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr1261947904930325089at_nat] :
                  ( ( A4
                    = ( insert8211810215607154385at_nat @ B @ C5 ) )
                  & ~ ( member8440522571783428010at_nat @ B @ C5 )
                  & ( B5
                    = ( insert8211810215607154385at_nat @ A @ C5 ) )
                  & ~ ( member8440522571783428010at_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_6961_insert__eq__iff,axiom,
    ! [A: $o,A4: set_o,B: $o,B5: set_o] :
      ( ~ ( member_o @ A @ A4 )
     => ( ~ ( member_o @ B @ B5 )
       => ( ( ( insert_o @ A @ A4 )
            = ( insert_o @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A4 = B5 ) )
            & ( ( A = ~ B )
             => ? [C5: set_o] :
                  ( ( A4
                    = ( insert_o @ B @ C5 ) )
                  & ~ ( member_o @ B @ C5 )
                  & ( B5
                    = ( insert_o @ A @ C5 ) )
                  & ~ ( member_o @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_6962_insert__eq__iff,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ A @ A4 )
     => ( ~ ( member2340774599025711042nt_int @ B @ B5 )
       => ( ( ( insert8897473484851387113nt_int @ A @ A4 )
            = ( insert8897473484851387113nt_int @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A4 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_se6260736226359567993nt_int] :
                  ( ( A4
                    = ( insert8897473484851387113nt_int @ B @ C5 ) )
                  & ~ ( member2340774599025711042nt_int @ B @ C5 )
                  & ( B5
                    = ( insert8897473484851387113nt_int @ A @ C5 ) )
                  & ~ ( member2340774599025711042nt_int @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_6963_insert__eq__iff,axiom,
    ! [A: set_nat,A4: set_set_nat,B: set_nat,B5: set_set_nat] :
      ( ~ ( member_set_nat @ A @ A4 )
     => ( ~ ( member_set_nat @ B @ B5 )
       => ( ( ( insert_set_nat @ A @ A4 )
            = ( insert_set_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A4 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_set_nat] :
                  ( ( A4
                    = ( insert_set_nat @ B @ C5 ) )
                  & ~ ( member_set_nat @ B @ C5 )
                  & ( B5
                    = ( insert_set_nat @ A @ C5 ) )
                  & ~ ( member_set_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_6964_insert__eq__iff,axiom,
    ! [A: nat,A4: set_nat,B: nat,B5: set_nat] :
      ( ~ ( member_nat @ A @ A4 )
     => ( ~ ( member_nat @ B @ B5 )
       => ( ( ( insert_nat @ A @ A4 )
            = ( insert_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A4 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_nat] :
                  ( ( A4
                    = ( insert_nat @ B @ C5 ) )
                  & ~ ( member_nat @ B @ C5 )
                  & ( B5
                    = ( insert_nat @ A @ C5 ) )
                  & ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_6965_insert__eq__iff,axiom,
    ! [A: int,A4: set_int,B: int,B5: set_int] :
      ( ~ ( member_int @ A @ A4 )
     => ( ~ ( member_int @ B @ B5 )
       => ( ( ( insert_int @ A @ A4 )
            = ( insert_int @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A4 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_int] :
                  ( ( A4
                    = ( insert_int @ B @ C5 ) )
                  & ~ ( member_int @ B @ C5 )
                  & ( B5
                    = ( insert_int @ A @ C5 ) )
                  & ~ ( member_int @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_6966_insert__absorb,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( insert9069300056098147895at_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_6967_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( insert8211810215607154385at_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_6968_insert__absorb,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( insert_o @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_6969_insert__absorb,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ A @ A4 )
     => ( ( insert8897473484851387113nt_int @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_6970_insert__absorb,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ A4 )
     => ( ( insert_set_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_6971_insert__absorb,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_6972_insert__absorb,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ( insert_int @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_6973_insert__ident,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A4 )
     => ( ~ ( member8757157785044589968at_nat @ X @ B5 )
       => ( ( ( insert9069300056098147895at_nat @ X @ A4 )
            = ( insert9069300056098147895at_nat @ X @ B5 ) )
          = ( A4 = B5 ) ) ) ) ).

% insert_ident
thf(fact_6974_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B5 )
       => ( ( ( insert8211810215607154385at_nat @ X @ A4 )
            = ( insert8211810215607154385at_nat @ X @ B5 ) )
          = ( A4 = B5 ) ) ) ) ).

% insert_ident
thf(fact_6975_insert__ident,axiom,
    ! [X: $o,A4: set_o,B5: set_o] :
      ( ~ ( member_o @ X @ A4 )
     => ( ~ ( member_o @ X @ B5 )
       => ( ( ( insert_o @ X @ A4 )
            = ( insert_o @ X @ B5 ) )
          = ( A4 = B5 ) ) ) ) ).

% insert_ident
thf(fact_6976_insert__ident,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ X @ A4 )
     => ( ~ ( member2340774599025711042nt_int @ X @ B5 )
       => ( ( ( insert8897473484851387113nt_int @ X @ A4 )
            = ( insert8897473484851387113nt_int @ X @ B5 ) )
          = ( A4 = B5 ) ) ) ) ).

% insert_ident
thf(fact_6977_insert__ident,axiom,
    ! [X: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A4 )
     => ( ~ ( member_set_nat @ X @ B5 )
       => ( ( ( insert_set_nat @ X @ A4 )
            = ( insert_set_nat @ X @ B5 ) )
          = ( A4 = B5 ) ) ) ) ).

% insert_ident
thf(fact_6978_insert__ident,axiom,
    ! [X: nat,A4: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ~ ( member_nat @ X @ B5 )
       => ( ( ( insert_nat @ X @ A4 )
            = ( insert_nat @ X @ B5 ) )
          = ( A4 = B5 ) ) ) ) ).

% insert_ident
thf(fact_6979_insert__ident,axiom,
    ! [X: int,A4: set_int,B5: set_int] :
      ( ~ ( member_int @ X @ A4 )
     => ( ~ ( member_int @ X @ B5 )
       => ( ( ( insert_int @ X @ A4 )
            = ( insert_int @ X @ B5 ) )
          = ( A4 = B5 ) ) ) ) ).

% insert_ident
thf(fact_6980_Set_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A4 )
     => ~ ! [B9: set_Pr4329608150637261639at_nat] :
            ( ( A4
              = ( insert9069300056098147895at_nat @ X @ B9 ) )
           => ( member8757157785044589968at_nat @ X @ B9 ) ) ) ).

% Set.set_insert
thf(fact_6981_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ~ ! [B9: set_Pr1261947904930325089at_nat] :
            ( ( A4
              = ( insert8211810215607154385at_nat @ X @ B9 ) )
           => ( member8440522571783428010at_nat @ X @ B9 ) ) ) ).

% Set.set_insert
thf(fact_6982_Set_Oset__insert,axiom,
    ! [X: $o,A4: set_o] :
      ( ( member_o @ X @ A4 )
     => ~ ! [B9: set_o] :
            ( ( A4
              = ( insert_o @ X @ B9 ) )
           => ( member_o @ X @ B9 ) ) ) ).

% Set.set_insert
thf(fact_6983_Set_Oset__insert,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ X @ A4 )
     => ~ ! [B9: set_se6260736226359567993nt_int] :
            ( ( A4
              = ( insert8897473484851387113nt_int @ X @ B9 ) )
           => ( member2340774599025711042nt_int @ X @ B9 ) ) ) ).

% Set.set_insert
thf(fact_6984_Set_Oset__insert,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ X @ A4 )
     => ~ ! [B9: set_set_nat] :
            ( ( A4
              = ( insert_set_nat @ X @ B9 ) )
           => ( member_set_nat @ X @ B9 ) ) ) ).

% Set.set_insert
thf(fact_6985_Set_Oset__insert,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( member_nat @ X @ A4 )
     => ~ ! [B9: set_nat] :
            ( ( A4
              = ( insert_nat @ X @ B9 ) )
           => ( member_nat @ X @ B9 ) ) ) ).

% Set.set_insert
thf(fact_6986_Set_Oset__insert,axiom,
    ! [X: int,A4: set_int] :
      ( ( member_int @ X @ A4 )
     => ~ ! [B9: set_int] :
            ( ( A4
              = ( insert_int @ X @ B9 ) )
           => ( member_int @ X @ B9 ) ) ) ).

% Set.set_insert
thf(fact_6987_insertI2,axiom,
    ! [A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ A @ B5 )
     => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B5 ) ) ) ).

% insertI2
thf(fact_6988_insertI2,axiom,
    ! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ A @ B5 )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B5 ) ) ) ).

% insertI2
thf(fact_6989_insertI2,axiom,
    ! [A: $o,B5: set_o,B: $o] :
      ( ( member_o @ A @ B5 )
     => ( member_o @ A @ ( insert_o @ B @ B5 ) ) ) ).

% insertI2
thf(fact_6990_insertI2,axiom,
    ! [A: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int,B: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ A @ B5 )
     => ( member2340774599025711042nt_int @ A @ ( insert8897473484851387113nt_int @ B @ B5 ) ) ) ).

% insertI2
thf(fact_6991_insertI2,axiom,
    ! [A: set_nat,B5: set_set_nat,B: set_nat] :
      ( ( member_set_nat @ A @ B5 )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B5 ) ) ) ).

% insertI2
thf(fact_6992_insertI2,axiom,
    ! [A: nat,B5: set_nat,B: nat] :
      ( ( member_nat @ A @ B5 )
     => ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).

% insertI2
thf(fact_6993_insertI2,axiom,
    ! [A: int,B5: set_int,B: int] :
      ( ( member_int @ A @ B5 )
     => ( member_int @ A @ ( insert_int @ B @ B5 ) ) ) ).

% insertI2
thf(fact_6994_insertI1,axiom,
    ! [A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ B5 ) ) ).

% insertI1
thf(fact_6995_insertI1,axiom,
    ! [A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B5 ) ) ).

% insertI1
thf(fact_6996_insertI1,axiom,
    ! [A: $o,B5: set_o] : ( member_o @ A @ ( insert_o @ A @ B5 ) ) ).

% insertI1
thf(fact_6997_insertI1,axiom,
    ! [A: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int] : ( member2340774599025711042nt_int @ A @ ( insert8897473484851387113nt_int @ A @ B5 ) ) ).

% insertI1
thf(fact_6998_insertI1,axiom,
    ! [A: set_nat,B5: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B5 ) ) ).

% insertI1
thf(fact_6999_insertI1,axiom,
    ! [A: nat,B5: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B5 ) ) ).

% insertI1
thf(fact_7000_insertI1,axiom,
    ! [A: int,B5: set_int] : ( member_int @ A @ ( insert_int @ A @ B5 ) ) ).

% insertI1
thf(fact_7001_insertE,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8757157785044589968at_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_7002_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_7003_insertE,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A4 ) )
     => ( ( A = ~ B )
       => ( member_o @ A @ A4 ) ) ) ).

% insertE
thf(fact_7004_insertE,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ A @ ( insert8897473484851387113nt_int @ B @ A4 ) )
     => ( ( A != B )
       => ( member2340774599025711042nt_int @ A @ A4 ) ) ) ).

% insertE
thf(fact_7005_insertE,axiom,
    ! [A: set_nat,B: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member_set_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_7006_insertE,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_7007_insertE,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A4 ) )
     => ( ( A != B )
       => ( member_int @ A @ A4 ) ) ) ).

% insertE
thf(fact_7008_insert__compr,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A5: produc3843707927480180839at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [X4: produc3843707927480180839at_nat] :
              ( ( X4 = A5 )
              | ( member8757157785044589968at_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7009_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A5: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( X4 = A5 )
              | ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7010_insert__compr,axiom,
    ( insert_o
    = ( ^ [A5: $o,B6: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( X4 = A5 )
              | ( member_o @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7011_insert__compr,axiom,
    ( insert_list_nat
    = ( ^ [A5: list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( X4 = A5 )
              | ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7012_insert__compr,axiom,
    ( insert8897473484851387113nt_int
    = ( ^ [A5: set_Pr958786334691620121nt_int,B6: set_se6260736226359567993nt_int] :
          ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] :
              ( ( X4 = A5 )
              | ( member2340774599025711042nt_int @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7013_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A5: set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( X4 = A5 )
              | ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7014_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A5: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( X4 = A5 )
              | ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7015_insert__compr,axiom,
    ( insert_int
    = ( ^ [A5: int,B6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( X4 = A5 )
              | ( member_int @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7016_insert__Collect,axiom,
    ! [A: produc3843707927480180839at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P2 ) )
      = ( collec6321179662152712658at_nat
        @ ^ [U3: produc3843707927480180839at_nat] :
            ( ( U3 != A )
           => ( P2 @ U3 ) ) ) ) ).

% insert_Collect
thf(fact_7017_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U3: product_prod_nat_nat] :
            ( ( U3 != A )
           => ( P2 @ U3 ) ) ) ) ).

% insert_Collect
thf(fact_7018_insert__Collect,axiom,
    ! [A: $o,P2: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P2 ) )
      = ( collect_o
        @ ^ [U3: $o] :
            ( ( U3 != A )
           => ( P2 @ U3 ) ) ) ) ).

% insert_Collect
thf(fact_7019_insert__Collect,axiom,
    ! [A: list_nat,P2: list_nat > $o] :
      ( ( insert_list_nat @ A @ ( collect_list_nat @ P2 ) )
      = ( collect_list_nat
        @ ^ [U3: list_nat] :
            ( ( U3 != A )
           => ( P2 @ U3 ) ) ) ) ).

% insert_Collect
thf(fact_7020_insert__Collect,axiom,
    ! [A: set_Pr958786334691620121nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( insert8897473484851387113nt_int @ A @ ( collec5210948495886036740nt_int @ P2 ) )
      = ( collec5210948495886036740nt_int
        @ ^ [U3: set_Pr958786334691620121nt_int] :
            ( ( U3 != A )
           => ( P2 @ U3 ) ) ) ) ).

% insert_Collect
thf(fact_7021_insert__Collect,axiom,
    ! [A: set_nat,P2: set_nat > $o] :
      ( ( insert_set_nat @ A @ ( collect_set_nat @ P2 ) )
      = ( collect_set_nat
        @ ^ [U3: set_nat] :
            ( ( U3 != A )
           => ( P2 @ U3 ) ) ) ) ).

% insert_Collect
thf(fact_7022_insert__Collect,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P2 ) )
      = ( collect_nat
        @ ^ [U3: nat] :
            ( ( U3 != A )
           => ( P2 @ U3 ) ) ) ) ).

% insert_Collect
thf(fact_7023_insert__Collect,axiom,
    ! [A: int,P2: int > $o] :
      ( ( insert_int @ A @ ( collect_int @ P2 ) )
      = ( collect_int
        @ ^ [U3: int] :
            ( ( U3 != A )
           => ( P2 @ U3 ) ) ) ) ).

% insert_Collect
thf(fact_7024_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_7025_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_7026_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_7027_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_7028_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( ( insert_int @ A @ bot_bot_set_int )
        = ( insert_int @ B @ bot_bot_set_int ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_7029_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A @ A4 )
     != bot_bo228742789529271731at_nat ) ).

% insert_not_empty
thf(fact_7030_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ A4 )
     != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
thf(fact_7031_insert__not__empty,axiom,
    ! [A: $o,A4: set_o] :
      ( ( insert_o @ A @ A4 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_7032_insert__not__empty,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( insert_nat @ A @ A4 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_7033_insert__not__empty,axiom,
    ! [A: int,A4: set_int] :
      ( ( insert_int @ A @ A4 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_7034_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C2: produc3843707927480180839at_nat,D2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ C2 @ ( insert9069300056098147895at_nat @ D2 @ bot_bo228742789529271731at_nat ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_7035_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C2: product_prod_nat_nat,D2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C2 @ ( insert8211810215607154385at_nat @ D2 @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_7036_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C2: $o,D2: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C2 @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_7037_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_7038_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
        = ( insert_int @ C2 @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_7039_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_7040_singleton__iff,axiom,
    ! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ B @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_7041_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_7042_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_7043_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_7044_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_7045_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_7046_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_7047_singletonD,axiom,
    ! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ B @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_7048_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_7049_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_7050_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_7051_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_7052_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_7053_insert__mono,axiom,
    ! [C4: set_Pr4329608150637261639at_nat,D4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C4 @ D4 )
     => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ C4 ) @ ( insert9069300056098147895at_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_7054_insert__mono,axiom,
    ! [C4: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ C4 @ D4 )
     => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C4 ) @ ( insert8211810215607154385at_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_7055_insert__mono,axiom,
    ! [C4: set_o,D4: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C4 @ D4 )
     => ( ord_less_eq_set_o @ ( insert_o @ A @ C4 ) @ ( insert_o @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_7056_insert__mono,axiom,
    ! [C4: set_nat,D4: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C4 @ D4 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C4 ) @ ( insert_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_7057_insert__mono,axiom,
    ! [C4: set_int,D4: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C4 @ D4 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_7058_subset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A4 )
     => ( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B5 ) )
        = ( ord_le1268244103169919719at_nat @ A4 @ B5 ) ) ) ).

% subset_insert
thf(fact_7059_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) )
        = ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ) ).

% subset_insert
thf(fact_7060_subset__insert,axiom,
    ! [X: $o,A4: set_o,B5: set_o] :
      ( ~ ( member_o @ X @ A4 )
     => ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B5 ) )
        = ( ord_less_eq_set_o @ A4 @ B5 ) ) ) ).

% subset_insert
thf(fact_7061_subset__insert,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ X @ A4 )
     => ( ( ord_le483042692224249369nt_int @ A4 @ ( insert8897473484851387113nt_int @ X @ B5 ) )
        = ( ord_le483042692224249369nt_int @ A4 @ B5 ) ) ) ).

% subset_insert
thf(fact_7062_subset__insert,axiom,
    ! [X: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A4 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ ( insert_set_nat @ X @ B5 ) )
        = ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ) ).

% subset_insert
thf(fact_7063_subset__insert,axiom,
    ! [X: nat,A4: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B5 ) )
        = ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).

% subset_insert
thf(fact_7064_subset__insert,axiom,
    ! [X: int,A4: set_int,B5: set_int] :
      ( ~ ( member_int @ X @ A4 )
     => ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ B5 ) )
        = ( ord_less_eq_set_int @ A4 @ B5 ) ) ) ).

% subset_insert
thf(fact_7065_subset__insertI,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B5 @ ( insert9069300056098147895at_nat @ A @ B5 ) ) ).

% subset_insertI
thf(fact_7066_subset__insertI,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B5 @ ( insert8211810215607154385at_nat @ A @ B5 ) ) ).

% subset_insertI
thf(fact_7067_subset__insertI,axiom,
    ! [B5: set_o,A: $o] : ( ord_less_eq_set_o @ B5 @ ( insert_o @ A @ B5 ) ) ).

% subset_insertI
thf(fact_7068_subset__insertI,axiom,
    ! [B5: set_nat,A: nat] : ( ord_less_eq_set_nat @ B5 @ ( insert_nat @ A @ B5 ) ) ).

% subset_insertI
thf(fact_7069_subset__insertI,axiom,
    ! [B5: set_int,A: int] : ( ord_less_eq_set_int @ B5 @ ( insert_int @ A @ B5 ) ) ).

% subset_insertI
thf(fact_7070_subset__insertI2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B5 )
     => ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_7071_subset__insertI2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
     => ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_7072_subset__insertI2,axiom,
    ! [A4: set_o,B5: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A4 @ B5 )
     => ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_7073_subset__insertI2,axiom,
    ! [A4: set_nat,B5: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_7074_subset__insertI2,axiom,
    ! [A4: set_int,B5: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_7075_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,X7: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A4 )
     => ( ( ord_le1268244103169919719at_nat @ X7 @ A4 )
       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ X7 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_7076_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ X7 @ A4 )
       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X7 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_7077_insert__subsetI,axiom,
    ! [X: $o,A4: set_o,X7: set_o] :
      ( ( member_o @ X @ A4 )
     => ( ( ord_less_eq_set_o @ X7 @ A4 )
       => ( ord_less_eq_set_o @ ( insert_o @ X @ X7 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_7078_insert__subsetI,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,X7: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ X @ A4 )
     => ( ( ord_le483042692224249369nt_int @ X7 @ A4 )
       => ( ord_le483042692224249369nt_int @ ( insert8897473484851387113nt_int @ X @ X7 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_7079_insert__subsetI,axiom,
    ! [X: set_nat,A4: set_set_nat,X7: set_set_nat] :
      ( ( member_set_nat @ X @ A4 )
     => ( ( ord_le6893508408891458716et_nat @ X7 @ A4 )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ X7 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_7080_insert__subsetI,axiom,
    ! [X: nat,A4: set_nat,X7: set_nat] :
      ( ( member_nat @ X @ A4 )
     => ( ( ord_less_eq_set_nat @ X7 @ A4 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X7 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_7081_insert__subsetI,axiom,
    ! [X: int,A4: set_int,X7: set_int] :
      ( ( member_int @ X @ A4 )
     => ( ( ord_less_eq_set_int @ X7 @ A4 )
       => ( ord_less_eq_set_int @ ( insert_int @ X @ X7 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_7082_Int__insert__right,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ A @ A4 )
       => ( ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
          = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ A @ A4 )
       => ( ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
          = ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_7083_Int__insert__right,axiom,
    ! [A: $o,A4: set_o,B5: set_o] :
      ( ( ( member_o @ A @ A4 )
       => ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B5 ) )
          = ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B5 ) ) ) )
      & ( ~ ( member_o @ A @ A4 )
       => ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B5 ) )
          = ( inf_inf_set_o @ A4 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_7084_Int__insert__right,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( ( member2340774599025711042nt_int @ A @ A4 )
       => ( ( inf_in8396524679539076455nt_int @ A4 @ ( insert8897473484851387113nt_int @ A @ B5 ) )
          = ( insert8897473484851387113nt_int @ A @ ( inf_in8396524679539076455nt_int @ A4 @ B5 ) ) ) )
      & ( ~ ( member2340774599025711042nt_int @ A @ A4 )
       => ( ( inf_in8396524679539076455nt_int @ A4 @ ( insert8897473484851387113nt_int @ A @ B5 ) )
          = ( inf_in8396524679539076455nt_int @ A4 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_7085_Int__insert__right,axiom,
    ! [A: set_nat,A4: set_set_nat,B5: set_set_nat] :
      ( ( ( member_set_nat @ A @ A4 )
       => ( ( inf_inf_set_set_nat @ A4 @ ( insert_set_nat @ A @ B5 ) )
          = ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A4 @ B5 ) ) ) )
      & ( ~ ( member_set_nat @ A @ A4 )
       => ( ( inf_inf_set_set_nat @ A4 @ ( insert_set_nat @ A @ B5 ) )
          = ( inf_inf_set_set_nat @ A4 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_7086_Int__insert__right,axiom,
    ! [A: int,A4: set_int,B5: set_int] :
      ( ( ( member_int @ A @ A4 )
       => ( ( inf_inf_set_int @ A4 @ ( insert_int @ A @ B5 ) )
          = ( insert_int @ A @ ( inf_inf_set_int @ A4 @ B5 ) ) ) )
      & ( ~ ( member_int @ A @ A4 )
       => ( ( inf_inf_set_int @ A4 @ ( insert_int @ A @ B5 ) )
          = ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_7087_Int__insert__right,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A @ A4 )
       => ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
          = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ A4 )
       => ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
          = ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_7088_Int__insert__right,axiom,
    ! [A: nat,A4: set_nat,B5: set_nat] :
      ( ( ( member_nat @ A @ A4 )
       => ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
          = ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) )
      & ( ~ ( member_nat @ A @ A4 )
       => ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
          = ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_7089_Int__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ A @ C4 )
       => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B5 ) @ C4 )
          = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ B5 @ C4 ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ A @ C4 )
       => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B5 ) @ C4 )
          = ( inf_in7913087082777306421at_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_7090_Int__insert__left,axiom,
    ! [A: $o,C4: set_o,B5: set_o] :
      ( ( ( member_o @ A @ C4 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C4 )
          = ( insert_o @ A @ ( inf_inf_set_o @ B5 @ C4 ) ) ) )
      & ( ~ ( member_o @ A @ C4 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C4 )
          = ( inf_inf_set_o @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_7091_Int__insert__left,axiom,
    ! [A: set_Pr958786334691620121nt_int,C4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( ( member2340774599025711042nt_int @ A @ C4 )
       => ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ B5 ) @ C4 )
          = ( insert8897473484851387113nt_int @ A @ ( inf_in8396524679539076455nt_int @ B5 @ C4 ) ) ) )
      & ( ~ ( member2340774599025711042nt_int @ A @ C4 )
       => ( ( inf_in8396524679539076455nt_int @ ( insert8897473484851387113nt_int @ A @ B5 ) @ C4 )
          = ( inf_in8396524679539076455nt_int @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_7092_Int__insert__left,axiom,
    ! [A: set_nat,C4: set_set_nat,B5: set_set_nat] :
      ( ( ( member_set_nat @ A @ C4 )
       => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B5 ) @ C4 )
          = ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B5 @ C4 ) ) ) )
      & ( ~ ( member_set_nat @ A @ C4 )
       => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B5 ) @ C4 )
          = ( inf_inf_set_set_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_7093_Int__insert__left,axiom,
    ! [A: int,C4: set_int,B5: set_int] :
      ( ( ( member_int @ A @ C4 )
       => ( ( inf_inf_set_int @ ( insert_int @ A @ B5 ) @ C4 )
          = ( insert_int @ A @ ( inf_inf_set_int @ B5 @ C4 ) ) ) )
      & ( ~ ( member_int @ A @ C4 )
       => ( ( inf_inf_set_int @ ( insert_int @ A @ B5 ) @ C4 )
          = ( inf_inf_set_int @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_7094_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A @ C4 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C4 )
          = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B5 @ C4 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ C4 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B5 ) @ C4 )
          = ( inf_in2572325071724192079at_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_7095_Int__insert__left,axiom,
    ! [A: nat,C4: set_nat,B5: set_nat] :
      ( ( ( member_nat @ A @ C4 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C4 )
          = ( insert_nat @ A @ ( inf_inf_set_nat @ B5 @ C4 ) ) ) )
      & ( ~ ( member_nat @ A @ C4 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B5 ) @ C4 )
          = ( inf_inf_set_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_7096_insert__Diff__if,axiom,
    ! [X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ B5 )
       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) @ B5 )
          = ( minus_3314409938677909166at_nat @ A4 @ B5 ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ B5 )
       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) @ B5 )
          = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_7097_insert__Diff__if,axiom,
    ! [X: $o,B5: set_o,A4: set_o] :
      ( ( ( member_o @ X @ B5 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B5 )
          = ( minus_minus_set_o @ A4 @ B5 ) ) )
      & ( ~ ( member_o @ X @ B5 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B5 )
          = ( insert_o @ X @ ( minus_minus_set_o @ A4 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_7098_insert__Diff__if,axiom,
    ! [X: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( ( member2340774599025711042nt_int @ X @ B5 )
       => ( ( minus_2612819937483484256nt_int @ ( insert8897473484851387113nt_int @ X @ A4 ) @ B5 )
          = ( minus_2612819937483484256nt_int @ A4 @ B5 ) ) )
      & ( ~ ( member2340774599025711042nt_int @ X @ B5 )
       => ( ( minus_2612819937483484256nt_int @ ( insert8897473484851387113nt_int @ X @ A4 ) @ B5 )
          = ( insert8897473484851387113nt_int @ X @ ( minus_2612819937483484256nt_int @ A4 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_7099_insert__Diff__if,axiom,
    ! [X: set_nat,B5: set_set_nat,A4: set_set_nat] :
      ( ( ( member_set_nat @ X @ B5 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ B5 )
          = ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) )
      & ( ~ ( member_set_nat @ X @ B5 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ B5 )
          = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_7100_insert__Diff__if,axiom,
    ! [X: nat,B5: set_nat,A4: set_nat] :
      ( ( ( member_nat @ X @ B5 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B5 )
          = ( minus_minus_set_nat @ A4 @ B5 ) ) )
      & ( ~ ( member_nat @ X @ B5 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B5 )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_7101_insert__Diff__if,axiom,
    ! [X: int,B5: set_int,A4: set_int] :
      ( ( ( member_int @ X @ B5 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ B5 )
          = ( minus_minus_set_int @ A4 @ B5 ) ) )
      & ( ~ ( member_int @ X @ B5 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ B5 )
          = ( insert_int @ X @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_7102_insert__Diff__if,axiom,
    ! [X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ B5 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ B5 )
          = ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ B5 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ B5 )
          = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_7103_Collect__conv__if,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P2 @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if
thf(fact_7104_Collect__conv__if,axiom,
    ! [P2: list_nat > $o,A: list_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_7105_Collect__conv__if,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,A: set_Pr958786334691620121nt_int] :
      ( ( ( P2 @ A )
       => ( ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bo1488462491386950373nt_int ) ) ) ).

% Collect_conv_if
thf(fact_7106_Collect__conv__if,axiom,
    ! [P2: set_nat > $o,A: set_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_7107_Collect__conv__if,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P2 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_7108_Collect__conv__if,axiom,
    ! [P2: $o > $o,A: $o] :
      ( ( ( P2 @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_7109_Collect__conv__if,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_7110_Collect__conv__if,axiom,
    ! [P2: int > $o,A: int] :
      ( ( ( P2 @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( X4 = A )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_7111_Collect__conv__if2,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P2 @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7112_Collect__conv__if2,axiom,
    ! [P2: list_nat > $o,A: list_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7113_Collect__conv__if2,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,A: set_Pr958786334691620121nt_int] :
      ( ( ( P2 @ A )
       => ( ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bo1488462491386950373nt_int ) ) ) ).

% Collect_conv_if2
thf(fact_7114_Collect__conv__if2,axiom,
    ! [P2: set_nat > $o,A: set_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7115_Collect__conv__if2,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P2 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7116_Collect__conv__if2,axiom,
    ! [P2: $o > $o,A: $o] :
      ( ( ( P2 @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_7117_Collect__conv__if2,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7118_Collect__conv__if2,axiom,
    ! [P2: int > $o,A: int] :
      ( ( ( P2 @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( A = X4 )
                & ( P2 @ X4 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_7119_insert__def,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A5: product_prod_nat_nat] :
          ( sup_su6327502436637775413at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7120_insert__def,axiom,
    ( insert_o
    = ( ^ [A5: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X4: $o] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7121_insert__def,axiom,
    ( insert_list_nat
    = ( ^ [A5: list_nat] :
          ( sup_sup_set_list_nat
          @ ( collect_list_nat
            @ ^ [X4: list_nat] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7122_insert__def,axiom,
    ( insert8897473484851387113nt_int
    = ( ^ [A5: set_Pr958786334691620121nt_int] :
          ( sup_su2047564715030645325nt_int
          @ ( collec5210948495886036740nt_int
            @ ^ [X4: set_Pr958786334691620121nt_int] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7123_insert__def,axiom,
    ( insert_set_nat
    = ( ^ [A5: set_nat] :
          ( sup_sup_set_set_nat
          @ ( collect_set_nat
            @ ^ [X4: set_nat] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7124_insert__def,axiom,
    ( insert_int
    = ( ^ [A5: int] :
          ( sup_sup_set_int
          @ ( collect_int
            @ ^ [X4: int] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7125_insert__def,axiom,
    ( insert6337962749363155127at_nat
    = ( ^ [A5: produc4166570645942440679at_nat] :
          ( sup_su3035147773818789531at_nat
          @ ( collec5204685387357076818at_nat
            @ ^ [X4: produc4166570645942440679at_nat] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7126_insert__def,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A5: produc3843707927480180839at_nat] :
          ( sup_su5525570899277871387at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7127_insert__def,axiom,
    ( insert_nat
    = ( ^ [A5: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] : X4 = A5 ) ) ) ) ).

% insert_def
thf(fact_7128_subset__singleton__iff,axiom,
    ! [X7: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X7 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( ( X7 = bot_bo228742789529271731at_nat )
        | ( X7
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_7129_subset__singleton__iff,axiom,
    ! [X7: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X7 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( ( X7 = bot_bo2099793752762293965at_nat )
        | ( X7
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_7130_subset__singleton__iff,axiom,
    ! [X7: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X7 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( ( X7 = bot_bot_set_o )
        | ( X7
          = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_7131_subset__singleton__iff,axiom,
    ! [X7: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X7 = bot_bot_set_nat )
        | ( X7
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_7132_subset__singleton__iff,axiom,
    ! [X7: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X7 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X7 = bot_bot_set_int )
        | ( X7
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_7133_subset__singletonD,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
     => ( ( A4 = bot_bo228742789529271731at_nat )
        | ( A4
          = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_7134_subset__singletonD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
     => ( ( A4 = bot_bo2099793752762293965at_nat )
        | ( A4
          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_7135_subset__singletonD,axiom,
    ! [A4: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) )
     => ( ( A4 = bot_bot_set_o )
        | ( A4
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_7136_subset__singletonD,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A4 = bot_bot_set_nat )
        | ( A4
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_7137_subset__singletonD,axiom,
    ! [A4: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A4 = bot_bot_set_int )
        | ( A4
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_7138_insert__is__Un,axiom,
    ( insert6337962749363155127at_nat
    = ( ^ [A5: produc4166570645942440679at_nat] : ( sup_su3035147773818789531at_nat @ ( insert6337962749363155127at_nat @ A5 @ bot_bo8422036546324065075at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_7139_insert__is__Un,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A5: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A5 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_7140_insert__is__Un,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A5: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A5 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_7141_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A5: $o] : ( sup_sup_set_o @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_7142_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_7143_insert__is__Un,axiom,
    ( insert_int
    = ( ^ [A5: int] : ( sup_sup_set_int @ ( insert_int @ A5 @ bot_bot_set_int ) ) ) ) ).

% insert_is_Un
thf(fact_7144_Un__singleton__iff,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,X: produc4166570645942440679at_nat] :
      ( ( ( sup_su3035147773818789531at_nat @ A4 @ B5 )
        = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
      = ( ( ( A4 = bot_bo8422036546324065075at_nat )
          & ( B5
            = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) ) )
        | ( ( A4
            = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
          & ( B5 = bot_bo8422036546324065075at_nat ) )
        | ( ( A4
            = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
          & ( B5
            = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_7145_Un__singleton__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A4 @ B5 )
        = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B5
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B5 = bot_bo228742789529271731at_nat ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B5
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_7146_Un__singleton__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A4 @ B5 )
        = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B5
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B5 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B5
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_7147_Un__singleton__iff,axiom,
    ! [A4: set_o,B5: set_o,X: $o] :
      ( ( ( sup_sup_set_o @ A4 @ B5 )
        = ( insert_o @ X @ bot_bot_set_o ) )
      = ( ( ( A4 = bot_bot_set_o )
          & ( B5
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A4
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B5 = bot_bot_set_o ) )
        | ( ( A4
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B5
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_7148_Un__singleton__iff,axiom,
    ! [A4: set_nat,B5: set_nat,X: nat] :
      ( ( ( sup_sup_set_nat @ A4 @ B5 )
        = ( insert_nat @ X @ bot_bot_set_nat ) )
      = ( ( ( A4 = bot_bot_set_nat )
          & ( B5
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A4
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B5 = bot_bot_set_nat ) )
        | ( ( A4
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B5
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_7149_Un__singleton__iff,axiom,
    ! [A4: set_int,B5: set_int,X: int] :
      ( ( ( sup_sup_set_int @ A4 @ B5 )
        = ( insert_int @ X @ bot_bot_set_int ) )
      = ( ( ( A4 = bot_bot_set_int )
          & ( B5
            = ( insert_int @ X @ bot_bot_set_int ) ) )
        | ( ( A4
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B5 = bot_bot_set_int ) )
        | ( ( A4
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B5
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_7150_singleton__Un__iff,axiom,
    ! [X: produc4166570645942440679at_nat,A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] :
      ( ( ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat )
        = ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
      = ( ( ( A4 = bot_bo8422036546324065075at_nat )
          & ( B5
            = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) ) )
        | ( ( A4
            = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
          & ( B5 = bot_bo8422036546324065075at_nat ) )
        | ( ( A4
            = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) )
          & ( B5
            = ( insert6337962749363155127at_nat @ X @ bot_bo8422036546324065075at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_7151_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat )
        = ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
      = ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B5
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B5 = bot_bo228742789529271731at_nat ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B5
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_7152_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat )
        = ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
      = ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B5
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B5 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B5
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_7153_singleton__Un__iff,axiom,
    ! [X: $o,A4: set_o,B5: set_o] :
      ( ( ( insert_o @ X @ bot_bot_set_o )
        = ( sup_sup_set_o @ A4 @ B5 ) )
      = ( ( ( A4 = bot_bot_set_o )
          & ( B5
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A4
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B5 = bot_bot_set_o ) )
        | ( ( A4
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B5
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_7154_singleton__Un__iff,axiom,
    ! [X: nat,A4: set_nat,B5: set_nat] :
      ( ( ( insert_nat @ X @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A4 @ B5 ) )
      = ( ( ( A4 = bot_bot_set_nat )
          & ( B5
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A4
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B5 = bot_bot_set_nat ) )
        | ( ( A4
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B5
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_7155_singleton__Un__iff,axiom,
    ! [X: int,A4: set_int,B5: set_int] :
      ( ( ( insert_int @ X @ bot_bot_set_int )
        = ( sup_sup_set_int @ A4 @ B5 ) )
      = ( ( ( A4 = bot_bot_set_int )
          & ( B5
            = ( insert_int @ X @ bot_bot_set_int ) ) )
        | ( ( A4
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B5 = bot_bot_set_int ) )
        | ( ( A4
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B5
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_7156_set__minus__singleton__eq,axiom,
    ! [X: produc3843707927480180839at_nat,X7: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ X7 )
     => ( ( minus_3314409938677909166at_nat @ X7 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_7157_set__minus__singleton__eq,axiom,
    ! [X: set_Pr958786334691620121nt_int,X7: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ X @ X7 )
     => ( ( minus_2612819937483484256nt_int @ X7 @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_7158_set__minus__singleton__eq,axiom,
    ! [X: set_nat,X7: set_set_nat] :
      ( ~ ( member_set_nat @ X @ X7 )
     => ( ( minus_2163939370556025621et_nat @ X7 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_7159_set__minus__singleton__eq,axiom,
    ! [X: $o,X7: set_o] :
      ( ~ ( member_o @ X @ X7 )
     => ( ( minus_minus_set_o @ X7 @ ( insert_o @ X @ bot_bot_set_o ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_7160_set__minus__singleton__eq,axiom,
    ! [X: nat,X7: set_nat] :
      ( ~ ( member_nat @ X @ X7 )
     => ( ( minus_minus_set_nat @ X7 @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_7161_set__minus__singleton__eq,axiom,
    ! [X: int,X7: set_int] :
      ( ~ ( member_int @ X @ X7 )
     => ( ( minus_minus_set_int @ X7 @ ( insert_int @ X @ bot_bot_set_int ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_7162_set__minus__singleton__eq,axiom,
    ! [X: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ X7 )
     => ( ( minus_1356011639430497352at_nat @ X7 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = X7 ) ) ).

% set_minus_singleton_eq
thf(fact_7163_insert__minus__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( X != Y )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_7164_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A4: set_o] :
      ( ( X != Y )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ ( insert_o @ Y @ bot_bot_set_o ) )
        = ( insert_o @ X @ ( minus_minus_set_o @ A4 @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).

% insert_minus_eq
thf(fact_7165_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A4: set_nat] :
      ( ( X != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
        = ( insert_nat @ X @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_7166_insert__minus__eq,axiom,
    ! [X: int,Y: int,A4: set_int] :
      ( ( X != Y )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ ( insert_int @ Y @ bot_bot_set_int ) )
        = ( insert_int @ X @ ( minus_minus_set_int @ A4 @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_7167_insert__minus__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( X != Y )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_7168_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A4 )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_7169_Diff__insert__absorb,axiom,
    ! [X: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ~ ( member2340774599025711042nt_int @ X @ A4 )
     => ( ( minus_2612819937483484256nt_int @ ( insert8897473484851387113nt_int @ X @ A4 ) @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_7170_Diff__insert__absorb,axiom,
    ! [X: set_nat,A4: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A4 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A4 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_7171_Diff__insert__absorb,axiom,
    ! [X: $o,A4: set_o] :
      ( ~ ( member_o @ X @ A4 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ ( insert_o @ X @ bot_bot_set_o ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_7172_Diff__insert__absorb,axiom,
    ! [X: nat,A4: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_7173_Diff__insert__absorb,axiom,
    ! [X: int,A4: set_int] :
      ( ~ ( member_int @ X @ A4 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A4 ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_7174_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A4 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_7175_Diff__insert2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_7176_Diff__insert2,axiom,
    ! [A4: set_o,A: $o,B5: set_o] :
      ( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B5 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_7177_Diff__insert2,axiom,
    ! [A4: set_nat,A: nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_7178_Diff__insert2,axiom,
    ! [A4: set_int,A: int,B5: set_int] :
      ( ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B5 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_7179_Diff__insert2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_7180_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_7181_insert__Diff,axiom,
    ! [A: set_Pr958786334691620121nt_int,A4: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ A @ A4 )
     => ( ( insert8897473484851387113nt_int @ A @ ( minus_2612819937483484256nt_int @ A4 @ ( insert8897473484851387113nt_int @ A @ bot_bo1488462491386950373nt_int ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_7182_insert__Diff,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( ( member_set_nat @ A @ A4 )
     => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_7183_insert__Diff,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_7184_insert__Diff,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_7185_insert__Diff,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_7186_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_7187_Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B5 ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).

% Diff_insert
thf(fact_7188_Diff__insert,axiom,
    ! [A4: set_o,A: $o,B5: set_o] :
      ( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B5 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ B5 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_7189_Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B5 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_7190_Diff__insert,axiom,
    ! [A4: set_int,A: int,B5: set_int] :
      ( ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B5 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_7191_Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B5 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).

% Diff_insert
thf(fact_7192_subset__Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( minus_3314409938677909166at_nat @ B5 @ ( insert9069300056098147895at_nat @ X @ C4 ) ) )
      = ( ( ord_le1268244103169919719at_nat @ A4 @ ( minus_3314409938677909166at_nat @ B5 @ C4 ) )
        & ~ ( member8757157785044589968at_nat @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_7193_subset__Diff__insert,axiom,
    ! [A4: set_o,B5: set_o,X: $o,C4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B5 @ ( insert_o @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B5 @ C4 ) )
        & ~ ( member_o @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_7194_subset__Diff__insert,axiom,
    ! [A4: set_se6260736226359567993nt_int,B5: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,C4: set_se6260736226359567993nt_int] :
      ( ( ord_le483042692224249369nt_int @ A4 @ ( minus_2612819937483484256nt_int @ B5 @ ( insert8897473484851387113nt_int @ X @ C4 ) ) )
      = ( ( ord_le483042692224249369nt_int @ A4 @ ( minus_2612819937483484256nt_int @ B5 @ C4 ) )
        & ~ ( member2340774599025711042nt_int @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_7195_subset__Diff__insert,axiom,
    ! [A4: set_set_nat,B5: set_set_nat,X: set_nat,C4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( minus_2163939370556025621et_nat @ B5 @ ( insert_set_nat @ X @ C4 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A4 @ ( minus_2163939370556025621et_nat @ B5 @ C4 ) )
        & ~ ( member_set_nat @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_7196_subset__Diff__insert,axiom,
    ! [A4: set_nat,B5: set_nat,X: nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B5 @ C4 ) )
        & ~ ( member_nat @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_7197_subset__Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B5 @ ( insert8211810215607154385at_nat @ X @ C4 ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B5 @ C4 ) )
        & ~ ( member8440522571783428010at_nat @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_7198_subset__Diff__insert,axiom,
    ! [A4: set_int,B5: set_int,X: int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( minus_minus_set_int @ B5 @ ( insert_int @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_int @ A4 @ ( minus_minus_set_int @ B5 @ C4 ) )
        & ~ ( member_int @ X @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_7199_subset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B5 ) )
      = ( ( ( member8757157785044589968at_nat @ X @ A4 )
         => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B5 ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ A4 )
         => ( ord_le1268244103169919719at_nat @ A4 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_7200_subset__insert__iff,axiom,
    ! [A4: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( ord_le483042692224249369nt_int @ A4 @ ( insert8897473484851387113nt_int @ X @ B5 ) )
      = ( ( ( member2340774599025711042nt_int @ X @ A4 )
         => ( ord_le483042692224249369nt_int @ ( minus_2612819937483484256nt_int @ A4 @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) ) @ B5 ) )
        & ( ~ ( member2340774599025711042nt_int @ X @ A4 )
         => ( ord_le483042692224249369nt_int @ A4 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_7201_subset__insert__iff,axiom,
    ! [A4: set_set_nat,X: set_nat,B5: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( insert_set_nat @ X @ B5 ) )
      = ( ( ( member_set_nat @ X @ A4 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B5 ) )
        & ( ~ ( member_set_nat @ X @ A4 )
         => ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_7202_subset__insert__iff,axiom,
    ! [A4: set_o,X: $o,B5: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B5 ) )
      = ( ( ( member_o @ X @ A4 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B5 ) )
        & ( ~ ( member_o @ X @ A4 )
         => ( ord_less_eq_set_o @ A4 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_7203_subset__insert__iff,axiom,
    ! [A4: set_nat,X: nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B5 ) )
      = ( ( ( member_nat @ X @ A4 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_7204_subset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) )
      = ( ( ( member8440522571783428010at_nat @ X @ A4 )
         => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B5 ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ A4 )
         => ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_7205_subset__insert__iff,axiom,
    ! [A4: set_int,X: int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ B5 ) )
      = ( ( ( member_int @ X @ A4 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ord_less_eq_set_int @ A4 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_7206_Diff__single__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B5 )
     => ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_7207_Diff__single__insert,axiom,
    ! [A4: set_o,X: $o,B5: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B5 )
     => ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_7208_Diff__single__insert,axiom,
    ! [A4: set_nat,X: nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 )
     => ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_7209_Diff__single__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B5 )
     => ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_7210_Diff__single__insert,axiom,
    ! [A4: set_int,X: int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 )
     => ( ord_less_eq_set_int @ A4 @ ( insert_int @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_7211_remove__subset,axiom,
    ! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ S )
     => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ S @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_7212_remove__subset,axiom,
    ! [X: set_Pr958786334691620121nt_int,S: set_se6260736226359567993nt_int] :
      ( ( member2340774599025711042nt_int @ X @ S )
     => ( ord_le1924305788584680229nt_int @ ( minus_2612819937483484256nt_int @ S @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) ) @ S ) ) ).

% remove_subset
thf(fact_7213_remove__subset,axiom,
    ! [X: set_nat,S: set_set_nat] :
      ( ( member_set_nat @ X @ S )
     => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_7214_remove__subset,axiom,
    ! [X: $o,S: set_o] :
      ( ( member_o @ X @ S )
     => ( ord_less_set_o @ ( minus_minus_set_o @ S @ ( insert_o @ X @ bot_bot_set_o ) ) @ S ) ) ).

% remove_subset
thf(fact_7215_remove__subset,axiom,
    ! [X: nat,S: set_nat] :
      ( ( member_nat @ X @ S )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_7216_remove__subset,axiom,
    ! [X: int,S: set_int] :
      ( ( member_int @ X @ S )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S @ ( insert_int @ X @ bot_bot_set_int ) ) @ S ) ) ).

% remove_subset
thf(fact_7217_remove__subset,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ S )
     => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_7218_Compl__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) )
      = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).

% Compl_insert
thf(fact_7219_Compl__insert,axiom,
    ! [X: $o,A4: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X @ A4 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A4 ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_7220_Compl__insert,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A4 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_7221_Compl__insert,axiom,
    ! [X: int,A4: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A4 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_7222_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) )
      = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% Compl_insert
thf(fact_7223_psubset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B5 ) )
      = ( ( ( member8757157785044589968at_nat @ X @ B5 )
         => ( ord_le2604355607129572851at_nat @ A4 @ B5 ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ B5 )
         => ( ( ( member8757157785044589968at_nat @ X @ A4 )
             => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B5 ) )
            & ( ~ ( member8757157785044589968at_nat @ X @ A4 )
             => ( ord_le1268244103169919719at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7224_psubset__insert__iff,axiom,
    ! [A4: set_se6260736226359567993nt_int,X: set_Pr958786334691620121nt_int,B5: set_se6260736226359567993nt_int] :
      ( ( ord_le1924305788584680229nt_int @ A4 @ ( insert8897473484851387113nt_int @ X @ B5 ) )
      = ( ( ( member2340774599025711042nt_int @ X @ B5 )
         => ( ord_le1924305788584680229nt_int @ A4 @ B5 ) )
        & ( ~ ( member2340774599025711042nt_int @ X @ B5 )
         => ( ( ( member2340774599025711042nt_int @ X @ A4 )
             => ( ord_le1924305788584680229nt_int @ ( minus_2612819937483484256nt_int @ A4 @ ( insert8897473484851387113nt_int @ X @ bot_bo1488462491386950373nt_int ) ) @ B5 ) )
            & ( ~ ( member2340774599025711042nt_int @ X @ A4 )
             => ( ord_le483042692224249369nt_int @ A4 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7225_psubset__insert__iff,axiom,
    ! [A4: set_set_nat,X: set_nat,B5: set_set_nat] :
      ( ( ord_less_set_set_nat @ A4 @ ( insert_set_nat @ X @ B5 ) )
      = ( ( ( member_set_nat @ X @ B5 )
         => ( ord_less_set_set_nat @ A4 @ B5 ) )
        & ( ~ ( member_set_nat @ X @ B5 )
         => ( ( ( member_set_nat @ X @ A4 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B5 ) )
            & ( ~ ( member_set_nat @ X @ A4 )
             => ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7226_psubset__insert__iff,axiom,
    ! [A4: set_o,X: $o,B5: set_o] :
      ( ( ord_less_set_o @ A4 @ ( insert_o @ X @ B5 ) )
      = ( ( ( member_o @ X @ B5 )
         => ( ord_less_set_o @ A4 @ B5 ) )
        & ( ~ ( member_o @ X @ B5 )
         => ( ( ( member_o @ X @ A4 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B5 ) )
            & ( ~ ( member_o @ X @ A4 )
             => ( ord_less_eq_set_o @ A4 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7227_psubset__insert__iff,axiom,
    ! [A4: set_nat,X: nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A4 @ ( insert_nat @ X @ B5 ) )
      = ( ( ( member_nat @ X @ B5 )
         => ( ord_less_set_nat @ A4 @ B5 ) )
        & ( ~ ( member_nat @ X @ B5 )
         => ( ( ( member_nat @ X @ A4 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
            & ( ~ ( member_nat @ X @ A4 )
             => ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7228_psubset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B5 ) )
      = ( ( ( member8440522571783428010at_nat @ X @ B5 )
         => ( ord_le7866589430770878221at_nat @ A4 @ B5 ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ B5 )
         => ( ( ( member8440522571783428010at_nat @ X @ A4 )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B5 ) )
            & ( ~ ( member8440522571783428010at_nat @ X @ A4 )
             => ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7229_psubset__insert__iff,axiom,
    ! [A4: set_int,X: int,B5: set_int] :
      ( ( ord_less_set_int @ A4 @ ( insert_int @ X @ B5 ) )
      = ( ( ( member_int @ X @ B5 )
         => ( ord_less_set_int @ A4 @ B5 ) )
        & ( ~ ( member_int @ X @ B5 )
         => ( ( ( member_int @ X @ A4 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 ) )
            & ( ~ ( member_int @ X @ A4 )
             => ( ord_less_eq_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_7230_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 )
        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
        @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(8)
thf(fact_7231_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q8: int,R5: int] : ( plus_plus_int @ Q8 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_7232_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_7233_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N: nat] :
          ( if_nat
          @ ( ( M3 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_7234_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_7235_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L ) )
     => ( ( ( ( member_int @ K2 @ ( 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 @ K2 @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K2 @ ( 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 @ K2 @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_7236_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_7237_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_7238_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z: int] :
                ( ( ord_less_eq_int @ D5 @ Z )
                & ( ord_less_int @ Z8 @ Z ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_7239_trancl__single,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( transi2703068831062848130at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ bot_bo228742789529271731at_nat ) )
      = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ bot_bo228742789529271731at_nat ) ) ).

% trancl_single
thf(fact_7240_trancl__single,axiom,
    ! [A: int,B: int] :
      ( ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B ) @ bot_bo1796632182523588997nt_int ) )
      = ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B ) @ bot_bo1796632182523588997nt_int ) ) ).

% trancl_single
thf(fact_7241_trancl__single,axiom,
    ! [A: nat,B: nat] :
      ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ bot_bo2099793752762293965at_nat ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ bot_bo2099793752762293965at_nat ) ) ).

% trancl_single
thf(fact_7242_trancl__sub__insert__trancl,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( transi2703068831062848130at_nat @ R3 ) @ ( transi2703068831062848130at_nat @ ( insert9069300056098147895at_nat @ X @ R3 ) ) ) ).

% trancl_sub_insert_trancl
thf(fact_7243_trancl__sub__insert__trancl,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( transi6264000038957366511cl_nat @ R3 ) @ ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ X @ R3 ) ) ) ).

% trancl_sub_insert_trancl
thf(fact_7244_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_7245_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_7246_trancl__insert2,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,R2: set_Pr8551490117392284871at_nat] :
      ( ( transi5749863603491437800at_nat @ ( insert6337962749363155127at_nat @ ( produc4348348721325984599at_nat @ A @ B ) @ R2 ) )
      = ( sup_su3035147773818789531at_nat @ ( transi5749863603491437800at_nat @ R2 )
        @ ( collec5204685387357076818at_nat
          @ ( produc5410597840814988588_nat_o
            @ ^ [X4: multis2468970476368604999at_nat,Y4: multis2468970476368604999at_nat] :
                ( ( ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ X4 @ A ) @ ( transi5749863603491437800at_nat @ R2 ) )
                  | ( X4 = A ) )
                & ( ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ B @ Y4 ) @ ( transi5749863603491437800at_nat @ R2 ) )
                  | ( Y4 = B ) ) ) ) ) ) ) ).

% trancl_insert2
thf(fact_7247_trancl__insert2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( transi2703068831062848130at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) )
      = ( sup_su5525570899277871387at_nat @ ( transi2703068831062848130at_nat @ R2 )
        @ ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
                ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ A ) @ ( transi2703068831062848130at_nat @ R2 ) )
                  | ( X4 = A ) )
                & ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ Y4 ) @ ( transi2703068831062848130at_nat @ R2 ) )
                  | ( Y4 = B ) ) ) ) ) ) ) ).

% trancl_insert2
thf(fact_7248_trancl__insert2,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) )
      = ( sup_su6327502436637775413at_nat @ ( transi6264000038957366511cl_nat @ R2 )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] :
                ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ A ) @ ( transi6264000038957366511cl_nat @ R2 ) )
                  | ( X4 = A ) )
                & ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ Y4 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
                  | ( Y4 = B ) ) ) ) ) ) ) ).

% trancl_insert2
thf(fact_7249_trancl__insert2,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 ) )
      = ( sup_su6024340866399070445nt_int @ ( transi6261509568448316235cl_int @ R2 )
        @ ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [X4: int,Y4: int] :
                ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ A ) @ ( transi6261509568448316235cl_int @ R2 ) )
                  | ( X4 = A ) )
                & ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ Y4 ) @ ( transi6261509568448316235cl_int @ R2 ) )
                  | ( Y4 = B ) ) ) ) ) ) ) ).

% trancl_insert2
thf(fact_7250_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z: int] :
                ( ( ord_less_eq_int @ D5 @ Z8 )
                & ( ord_less_int @ Z8 @ Z ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_7251_log_Osimps,axiom,
    ( log
    = ( ^ [B4: code_natural,I: code_natural] :
          ( if_Code_natural
          @ ( ( ord_le1926595141338095240atural @ B4 @ one_one_Code_natural )
            | ( ord_le5570908160329646204atural @ I @ B4 ) )
          @ one_one_Code_natural
          @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B4 @ ( divide5121882707175180666atural @ I @ B4 ) ) ) ) ) ) ).

% log.simps
thf(fact_7252_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_7253_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_7254_minus__shift__def,axiom,
    ( minus_shift
    = ( ^ [R5: code_natural,K3: code_natural,L2: code_natural] : ( if_Code_natural @ ( ord_le5570908160329646204atural @ K3 @ L2 ) @ ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ R5 @ K3 ) @ L2 ) @ ( minus_7197305767214868737atural @ K3 @ L2 ) ) ) ) ).

% minus_shift_def
thf(fact_7255_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_7256_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_7257_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z3: nat] :
      ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z3 ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z3 ) )
        = ( insert_nat @ N2 @ ( nat_set_decode @ Z3 ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_7258_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_7259_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K2: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K2 )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) ) ) ).

% take_bit_Suc_from_most
thf(fact_7260_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_7261_xor__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_7262_Suc__0__xor__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_7263_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_7264_scomp__apply,axiom,
    ( produc5538323210962509403atural
    = ( ^ [F3: produc7822875418678951345atural > produc5835291356934675326atural,G3: code_natural > produc7822875418678951345atural > produc5835291356934675326atural,X4: produc7822875418678951345atural] : ( produc8638916746724166107atural @ G3 @ ( F3 @ X4 ) ) ) ) ).

% scomp_apply
thf(fact_7265_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_7266_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K2 ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ).

% signed_take_bit_negative_iff
thf(fact_7267_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_7268_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_7269_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_7270_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_7271_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_7272_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_7273_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_7274_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_7275_scomp__scomp,axiom,
    ! [F: produc7822875418678951345atural > produc5835291356934675326atural,G: code_natural > produc7822875418678951345atural > produc5835291356934675326atural,H: code_natural > produc7822875418678951345atural > produc5835291356934675326atural] :
      ( ( produc5538323210962509403atural @ ( produc5538323210962509403atural @ F @ G ) @ H )
      = ( produc5538323210962509403atural @ F
        @ ^ [X4: code_natural] : ( produc5538323210962509403atural @ ( G @ X4 ) @ H ) ) ) ).

% scomp_scomp
thf(fact_7276_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [N5: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N5 )
          | ~ ( bit_se1146084159140164899it_int @ B @ N5 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_7277_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [N5: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N5 )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N5 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_7278_scomp__Pair,axiom,
    ! [X: produc7822875418678951345atural > produc5835291356934675326atural] :
      ( ( produc5538323210962509403atural @ X @ produc6639722614265839536atural )
      = X ) ).

% scomp_Pair
thf(fact_7279_Pair__scomp,axiom,
    ! [X: code_natural,F: code_natural > produc7822875418678951345atural > produc5835291356934675326atural] :
      ( ( produc5538323210962509403atural @ ( produc6639722614265839536atural @ X ) @ F )
      = ( F @ X ) ) ).

% Pair_scomp
thf(fact_7280_bit__take__bit__iff,axiom,
    ! [M: nat,A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N2 )
      = ( ( ord_less_nat @ N2 @ M )
        & ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).

% bit_take_bit_iff
thf(fact_7281_bit__take__bit__iff,axiom,
    ! [M: nat,A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N2 )
      = ( ( ord_less_nat @ N2 @ M )
        & ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).

% bit_take_bit_iff
thf(fact_7282_scomp__def,axiom,
    ( produc5538323210962509403atural
    = ( ^ [F3: produc7822875418678951345atural > produc5835291356934675326atural,G3: code_natural > produc7822875418678951345atural > produc5835291356934675326atural,X4: produc7822875418678951345atural] : ( produc8638916746724166107atural @ G3 @ ( F3 @ X4 ) ) ) ) ).

% scomp_def
thf(fact_7283_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se8040316288895769887atural @ ( groups2241214984954800037atural @ zero_n8403883297036319079atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7284_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7285_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7286_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7287_iterate_Osimps,axiom,
    ( iterate_int_int
    = ( ^ [K3: code_natural,F3: int > int > product_prod_int_int,X4: int] : ( if_int2409958687428134794nt_int @ ( K3 = zero_z2226904508553997617atural ) @ ( product_Pair_int_int @ X4 ) @ ( produc6963641260316735276nt_int @ ( F3 @ X4 ) @ ( iterate_int_int @ ( minus_7197305767214868737atural @ K3 @ one_one_Code_natural ) @ F3 ) ) ) ) ) ).

% iterate.simps
thf(fact_7288_iterate_Osimps,axiom,
    ( iterat1233435593421607556et_nat
    = ( ^ [K3: code_natural,F3: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > produc3925858234332021118et_nat,X4: produc3658429121746597890et_nat > $o] : ( if_Pro6440128116348121305et_nat @ ( K3 = zero_z2226904508553997617atural ) @ ( produc5001842942810119800et_nat @ X4 ) @ ( produc2318911012118706502et_nat @ ( F3 @ X4 ) @ ( iterat1233435593421607556et_nat @ ( minus_7197305767214868737atural @ K3 @ one_one_Code_natural ) @ F3 ) ) ) ) ) ).

% iterate.simps
thf(fact_7289_iterate_Osimps,axiom,
    ( iterat2702368289246022656et_nat
    = ( ^ [K3: code_natural,F3: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat,X4: produc3658429121746597890et_nat > $o] : ( if_Pro5796790085669187537et_nat @ ( K3 = zero_z2226904508553997617atural ) @ ( produc2245416461498447860et_nat @ X4 ) @ ( produc7537777246833756098et_nat @ ( F3 @ X4 ) @ ( iterat2702368289246022656et_nat @ ( minus_7197305767214868737atural @ K3 @ one_one_Code_natural ) @ F3 ) ) ) ) ) ).

% iterate.simps
thf(fact_7290_iterate_Osimps,axiom,
    ( iterat9120928542346325760it_nat
    = ( ^ [K3: code_natural,F3: b > produc6653097349344004940it_nat > produc7388388658123137530it_nat,X4: b] : ( if_Pro4080778131203054751it_nat @ ( K3 = zero_z2226904508553997617atural ) @ ( produc4082563078715348724it_nat @ X4 ) @ ( produc8824551317683396724it_nat @ ( F3 @ X4 ) @ ( iterat9120928542346325760it_nat @ ( minus_7197305767214868737atural @ K3 @ one_one_Code_natural ) @ F3 ) ) ) ) ) ).

% iterate.simps
thf(fact_7291_iterate_Osimps,axiom,
    ( iterat4993027441371875583it_nat
    = ( ^ [K3: code_natural,F3: a > produc6653097349344004940it_nat > produc3260487557148687353it_nat,X4: a] : ( if_Pro3578090444564244254it_nat @ ( K3 = zero_z2226904508553997617atural ) @ ( produc9178034014595674355it_nat @ X4 ) @ ( produc3714732129343117170it_nat @ ( F3 @ X4 ) @ ( iterat4993027441371875583it_nat @ ( minus_7197305767214868737atural @ K3 @ one_one_Code_natural ) @ F3 ) ) ) ) ) ).

% iterate.simps
thf(fact_7292_iterate_Osimps,axiom,
    ( iterat8892046348760725948atural
    = ( ^ [K3: code_natural,F3: code_natural > produc7822875418678951345atural > produc5835291356934675326atural,X4: code_natural] : ( if_Pro2760142802149639654atural @ ( K3 = zero_z2226904508553997617atural ) @ ( produc6639722614265839536atural @ X4 ) @ ( produc5538323210962509403atural @ ( F3 @ X4 ) @ ( iterat8892046348760725948atural @ ( minus_7197305767214868737atural @ K3 @ one_one_Code_natural ) @ F3 ) ) ) ) ) ).

% iterate.simps
thf(fact_7293_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: int > int > product_prod_int_int,Xb: int,Y: int > product_prod_int_int] :
      ( ( ( iterate_int_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( X = zero_z2226904508553997617atural )
         => ( Y
            = ( product_Pair_int_int @ Xb ) ) )
        & ( ( X != zero_z2226904508553997617atural )
         => ( Y
            = ( produc6963641260316735276nt_int @ ( Xa @ Xb ) @ ( iterate_int_int @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) ) ) ).

% iterate.elims
thf(fact_7294_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > produc3925858234332021118et_nat,Xb: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > produc3925858234332021118et_nat] :
      ( ( ( iterat1233435593421607556et_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( X = zero_z2226904508553997617atural )
         => ( Y
            = ( produc5001842942810119800et_nat @ Xb ) ) )
        & ( ( X != zero_z2226904508553997617atural )
         => ( Y
            = ( produc2318911012118706502et_nat @ ( Xa @ Xb ) @ ( iterat1233435593421607556et_nat @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) ) ) ).

% iterate.elims
thf(fact_7295_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat,Xb: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat > produc2732055786443039994et_nat] :
      ( ( ( iterat2702368289246022656et_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( X = zero_z2226904508553997617atural )
         => ( Y
            = ( produc2245416461498447860et_nat @ Xb ) ) )
        & ( ( X != zero_z2226904508553997617atural )
         => ( Y
            = ( produc7537777246833756098et_nat @ ( Xa @ Xb ) @ ( iterat2702368289246022656et_nat @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) ) ) ).

% iterate.elims
thf(fact_7296_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: b > produc6653097349344004940it_nat > produc7388388658123137530it_nat,Xb: b,Y: produc6653097349344004940it_nat > produc7388388658123137530it_nat] :
      ( ( ( iterat9120928542346325760it_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( X = zero_z2226904508553997617atural )
         => ( Y
            = ( produc4082563078715348724it_nat @ Xb ) ) )
        & ( ( X != zero_z2226904508553997617atural )
         => ( Y
            = ( produc8824551317683396724it_nat @ ( Xa @ Xb ) @ ( iterat9120928542346325760it_nat @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) ) ) ).

% iterate.elims
thf(fact_7297_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: a > produc6653097349344004940it_nat > produc3260487557148687353it_nat,Xb: a,Y: produc6653097349344004940it_nat > produc3260487557148687353it_nat] :
      ( ( ( iterat4993027441371875583it_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( X = zero_z2226904508553997617atural )
         => ( Y
            = ( produc9178034014595674355it_nat @ Xb ) ) )
        & ( ( X != zero_z2226904508553997617atural )
         => ( Y
            = ( produc3714732129343117170it_nat @ ( Xa @ Xb ) @ ( iterat4993027441371875583it_nat @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) ) ) ).

% iterate.elims
thf(fact_7298_iterate_Oelims,axiom,
    ! [X: code_natural,Xa: code_natural > produc7822875418678951345atural > produc5835291356934675326atural,Xb: code_natural,Y: produc7822875418678951345atural > produc5835291356934675326atural] :
      ( ( ( iterat8892046348760725948atural @ X @ Xa @ Xb )
        = Y )
     => ( ( ( X = zero_z2226904508553997617atural )
         => ( Y
            = ( produc6639722614265839536atural @ Xb ) ) )
        & ( ( X != zero_z2226904508553997617atural )
         => ( Y
            = ( produc5538323210962509403atural @ ( Xa @ Xb ) @ ( iterat8892046348760725948atural @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) ) ) ).

% iterate.elims
thf(fact_7299_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K2: int] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( bit_se1146084159140164899it_int @ K2 @ N2 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K2 ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_7300_bit__concat__bit__iff,axiom,
    ! [M: nat,K2: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K2 @ L ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K2 @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_7301_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N5: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ N5 @ M4 )
             => ( ( bit_se1146084159140164899it_int @ K2 @ M4 )
                = ( bit_se1146084159140164899it_int @ K2 @ N5 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N5 )
             => ( ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N5 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K2 @ N5 ) ) ) ) ) ).

% int_bit_bound
thf(fact_7302_even__bit__succ__iff,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ N2 )
        = ( ( bit_se8040316288895769887atural @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_7303_even__bit__succ__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N2 )
        = ( ( bit_se9216721137139052372nteger @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_7304_even__bit__succ__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_7305_even__bit__succ__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_7306_bit__sum__mult__2__cases,axiom,
    ! [A: code_natural,B: code_natural,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se8040316288895769887atural @ A @ ( suc @ J3 ) )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se8040316288895769887atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_7307_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J3 ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_7308_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J3 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_7309_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J3 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_7310_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_7311_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_7312_xor__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ A @ one_one_Code_natural )
      = ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n8403883297036319079atural
          @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7313_xor__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7314_xor__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7315_xor__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7316_one__xor__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ A )
      = ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n8403883297036319079atural
          @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7317_one__xor__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7318_one__xor__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7319_one__xor__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7320_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% set_bit_eq
thf(fact_7321_horner__sum__simps_I2_J,axiom,
    ! [F: int > rat,A: rat,X: int,Xs: list_int] :
      ( ( groups7852591826665563233nt_rat @ F @ A @ ( cons_int @ X @ Xs ) )
      = ( plus_plus_rat @ ( F @ X ) @ ( times_times_rat @ A @ ( groups7852591826665563233nt_rat @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7322_horner__sum__simps_I2_J,axiom,
    ! [F: nat > rat,A: rat,X: nat,Xs: list_nat] :
      ( ( groups6853238114764508677at_rat @ F @ A @ ( cons_nat @ X @ Xs ) )
      = ( plus_plus_rat @ ( F @ X ) @ ( times_times_rat @ A @ ( groups6853238114764508677at_rat @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7323_horner__sum__simps_I2_J,axiom,
    ! [F: int > nat,A: nat,X: int,Xs: list_int] :
      ( ( groups8487721886752058969nt_nat @ F @ A @ ( cons_int @ X @ Xs ) )
      = ( plus_plus_nat @ ( F @ X ) @ ( times_times_nat @ A @ ( groups8487721886752058969nt_nat @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7324_horner__sum__simps_I2_J,axiom,
    ! [F: nat > nat,A: nat,X: nat,Xs: list_nat] :
      ( ( groups7488368174851004413at_nat @ F @ A @ ( cons_nat @ X @ Xs ) )
      = ( plus_plus_nat @ ( F @ X ) @ ( times_times_nat @ A @ ( groups7488368174851004413at_nat @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7325_horner__sum__simps_I2_J,axiom,
    ! [F: int > int,A: int,X: int,Xs: list_int] :
      ( ( groups8485231416243008693nt_int @ F @ A @ ( cons_int @ X @ Xs ) )
      = ( plus_plus_int @ ( F @ X ) @ ( times_times_int @ A @ ( groups8485231416243008693nt_int @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7326_horner__sum__simps_I2_J,axiom,
    ! [F: nat > int,A: int,X: nat,Xs: list_nat] :
      ( ( groups7485877704341954137at_int @ F @ A @ ( cons_nat @ X @ Xs ) )
      = ( plus_plus_int @ ( F @ X ) @ ( times_times_int @ A @ ( groups7485877704341954137at_int @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7327_horner__sum__simps_I2_J,axiom,
    ! [F: $o > int,A: int,X: $o,Xs: list_o] :
      ( ( groups9116527308978886569_o_int @ F @ A @ ( cons_o @ X @ Xs ) )
      = ( plus_plus_int @ ( F @ X ) @ ( times_times_int @ A @ ( groups9116527308978886569_o_int @ F @ A @ Xs ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_7328_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: int > int > product_prod_int_int,Xb: int,Y: int > product_prod_int_int] :
      ( ( ( iterate_int_int @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P5309764456724190780nt_int @ iterate_rel_int_int @ ( produc6480503542405096427nt_int @ X @ ( produc8811497915304161060nt_int @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( product_Pair_int_int @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc6963641260316735276nt_int @ ( Xa @ Xb ) @ ( iterate_int_int @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P5309764456724190780nt_int @ iterate_rel_int_int @ ( produc6480503542405096427nt_int @ X @ ( produc8811497915304161060nt_int @ Xa @ Xb ) ) ) ) ) ) ).

% iterate.pelims
thf(fact_7329_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > produc3925858234332021118et_nat,Xb: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > produc3925858234332021118et_nat] :
      ( ( ( iterat1233435593421607556et_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P2863604582003348699_nat_o @ iterat2000369294365210145et_nat @ ( produc1685672994636548566_nat_o @ X @ ( produc417064122970982599_nat_o @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc5001842942810119800et_nat @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc2318911012118706502et_nat @ ( Xa @ Xb ) @ ( iterat1233435593421607556et_nat @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P2863604582003348699_nat_o @ iterat2000369294365210145et_nat @ ( produc1685672994636548566_nat_o @ X @ ( produc417064122970982599_nat_o @ Xa @ Xb ) ) ) ) ) ) ).

% iterate.pelims
thf(fact_7330_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat,Xb: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat > produc2732055786443039994et_nat] :
      ( ( ( iterat2702368289246022656et_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P5472185023545635811_nat_o @ iterat7486693702726674333et_nat @ ( produc722914403947760862_nat_o @ X @ ( produc5247183594425207759_nat_o @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc2245416461498447860et_nat @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc7537777246833756098et_nat @ ( Xa @ Xb ) @ ( iterat2702368289246022656et_nat @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P5472185023545635811_nat_o @ iterat7486693702726674333et_nat @ ( produc722914403947760862_nat_o @ X @ ( produc5247183594425207759_nat_o @ Xa @ Xb ) ) ) ) ) ) ).

% iterate.pelims
thf(fact_7331_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: b > produc6653097349344004940it_nat > produc7388388658123137530it_nat,Xb: b,Y: produc6653097349344004940it_nat > produc7388388658123137530it_nat] :
      ( ( ( iterat9120928542346325760it_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8256022550677833493_nat_b @ iterat7179843830775474589it_nat @ ( produc7509976829882447632_nat_b @ X @ ( produc5258162843807585025_nat_b @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc4082563078715348724it_nat @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc8824551317683396724it_nat @ ( Xa @ Xb ) @ ( iterat9120928542346325760it_nat @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P8256022550677833493_nat_b @ iterat7179843830775474589it_nat @ ( produc7509976829882447632_nat_b @ X @ ( produc5258162843807585025_nat_b @ Xa @ Xb ) ) ) ) ) ) ).

% iterate.pelims
thf(fact_7332_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: a > produc6653097349344004940it_nat > produc3260487557148687353it_nat,Xb: a,Y: produc6653097349344004940it_nat > produc3260487557148687353it_nat] :
      ( ( ( iterat4993027441371875583it_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7857408453303007510_nat_a @ iterat3051942729801024412it_nat @ ( produc1823638919063574929_nat_a @ X @ ( produc8795196974146716930_nat_a @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc9178034014595674355it_nat @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc3714732129343117170it_nat @ ( Xa @ Xb ) @ ( iterat4993027441371875583it_nat @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P7857408453303007510_nat_a @ iterat3051942729801024412it_nat @ ( produc1823638919063574929_nat_a @ X @ ( produc8795196974146716930_nat_a @ Xa @ Xb ) ) ) ) ) ) ).

% iterate.pelims
thf(fact_7333_iterate_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural > produc7822875418678951345atural > produc5835291356934675326atural,Xb: code_natural,Y: produc7822875418678951345atural > produc5835291356934675326atural] :
      ( ( ( iterat8892046348760725948atural @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P9117636446167716760atural @ iterat8136814461032266713atural @ ( produc7296465590736685127atural @ X @ ( produc2044891599319335296atural @ Xa @ Xb ) ) )
       => ~ ( ( ( ( X = zero_z2226904508553997617atural )
               => ( Y
                  = ( produc6639722614265839536atural @ Xb ) ) )
              & ( ( X != zero_z2226904508553997617atural )
               => ( Y
                  = ( produc5538323210962509403atural @ ( Xa @ Xb ) @ ( iterat8892046348760725948atural @ ( minus_7197305767214868737atural @ X @ one_one_Code_natural ) @ Xa ) ) ) ) )
           => ~ ( accp_P9117636446167716760atural @ iterat8136814461032266713atural @ ( produc7296465590736685127atural @ X @ ( produc2044891599319335296atural @ Xa @ Xb ) ) ) ) ) ) ).

% iterate.pelims
thf(fact_7334_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] :
      ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_7335_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_7336_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_7337_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_7338_the__elem__eq,axiom,
    ! [X: int] :
      ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% the_elem_eq
thf(fact_7339_inc__shift__def,axiom,
    ( inc_shift
    = ( ^ [V2: code_natural,K3: code_natural] : ( if_Code_natural @ ( V2 = K3 ) @ one_one_Code_natural @ ( plus_p4538020629002901425atural @ K3 @ one_one_Code_natural ) ) ) ) ).

% inc_shift_def
thf(fact_7340_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q8: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q8 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q8 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_7341_xor__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K2 @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_7342_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K2 @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K2 @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_7343_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_7344_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_7345_plus__integer__code_I1_J,axiom,
    ! [K2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K2 @ zero_z3403309356797280102nteger )
      = K2 ) ).

% plus_integer_code(1)
thf(fact_7346_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_7347_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_7348_bit__nat__iff,axiom,
    ! [K2: int,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K2 ) @ N2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        & ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ).

% bit_nat_iff
thf(fact_7349_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_7350_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( 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_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_7351_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_7352_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_7353_is__singleton__the__elem,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat] :
          ( A6
          = ( insert9069300056098147895at_nat @ ( the_el221668144340439132at_nat @ A6 ) @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7354_is__singleton__the__elem,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7355_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
          ( A6
          = ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7356_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
          ( A6
          = ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7357_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
          ( A6
          = ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7358_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_7359_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
                      @ ^ [Q8: num] : ( some_num @ ( bit0 @ Q8 ) )
                      @ ( 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_7360_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).

% is_singletonI
thf(fact_7361_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).

% is_singletonI
thf(fact_7362_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_7363_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_7364_is__singletonI,axiom,
    ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_7365_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_7366_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_7367_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_7368_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K2 ) )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_7369_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% not_negative_int_iff
thf(fact_7370_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_7371_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_7372_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_7373_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_7374_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_7375_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_7376_nat_Ocase__distrib,axiom,
    ! [H: option_num > option_num,F1: option_num,F2: nat > option_num,Nat: nat] :
      ( ( H @ ( case_nat_option_num @ F1 @ F2 @ Nat ) )
      = ( case_nat_option_num @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7377_nat_Ocase__distrib,axiom,
    ! [H: option_num > nat,F1: option_num,F2: nat > option_num,Nat: nat] :
      ( ( H @ ( case_nat_option_num @ F1 @ F2 @ Nat ) )
      = ( case_nat_nat @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7378_nat_Ocase__distrib,axiom,
    ! [H: option_num > $o,F1: option_num,F2: nat > option_num,Nat: nat] :
      ( ( H @ ( case_nat_option_num @ F1 @ F2 @ Nat ) )
      = ( case_nat_o @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7379_nat_Ocase__distrib,axiom,
    ! [H: nat > option_num,F1: nat,F2: nat > nat,Nat: nat] :
      ( ( H @ ( case_nat_nat @ F1 @ F2 @ Nat ) )
      = ( case_nat_option_num @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7380_nat_Ocase__distrib,axiom,
    ! [H: nat > nat,F1: nat,F2: nat > nat,Nat: nat] :
      ( ( H @ ( case_nat_nat @ F1 @ F2 @ Nat ) )
      = ( case_nat_nat @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7381_nat_Ocase__distrib,axiom,
    ! [H: nat > $o,F1: nat,F2: nat > nat,Nat: nat] :
      ( ( H @ ( case_nat_nat @ F1 @ F2 @ Nat ) )
      = ( case_nat_o @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7382_nat_Ocase__distrib,axiom,
    ! [H: $o > option_num,F1: $o,F2: nat > $o,Nat: nat] :
      ( ( H @ ( case_nat_o @ F1 @ F2 @ Nat ) )
      = ( case_nat_option_num @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7383_nat_Ocase__distrib,axiom,
    ! [H: $o > nat,F1: $o,F2: nat > $o,Nat: nat] :
      ( ( H @ ( case_nat_o @ F1 @ F2 @ Nat ) )
      = ( case_nat_nat @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7384_nat_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: $o,F2: nat > $o,Nat: nat] :
      ( ( H @ ( case_nat_o @ F1 @ F2 @ Nat ) )
      = ( case_nat_o @ ( H @ F1 )
        @ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_7385_num_Ocase__distrib,axiom,
    ! [H: option_num > option_num,F1: option_num,F2: num > option_num,F32: num > option_num,Num: num] :
      ( ( H @ ( case_num_option_num @ F1 @ F2 @ F32 @ Num ) )
      = ( case_num_option_num @ ( H @ F1 )
        @ ^ [X4: num] : ( H @ ( F2 @ X4 ) )
        @ ^ [X4: num] : ( H @ ( F32 @ X4 ) )
        @ Num ) ) ).

% num.case_distrib
thf(fact_7386_old_Onat_Osimps_I4_J,axiom,
    ! [F1: option_num,F2: nat > option_num] :
      ( ( case_nat_option_num @ F1 @ F2 @ zero_zero_nat )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_7387_old_Onat_Osimps_I4_J,axiom,
    ! [F1: nat,F2: nat > nat] :
      ( ( case_nat_nat @ F1 @ F2 @ zero_zero_nat )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_7388_old_Onat_Osimps_I4_J,axiom,
    ! [F1: $o,F2: nat > $o] :
      ( ( case_nat_o @ F1 @ F2 @ zero_zero_nat )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_7389_old_Onat_Osimps_I5_J,axiom,
    ! [F1: option_num,F2: nat > option_num,X2: nat] :
      ( ( case_nat_option_num @ F1 @ F2 @ ( suc @ X2 ) )
      = ( F2 @ X2 ) ) ).

% old.nat.simps(5)
thf(fact_7390_old_Onat_Osimps_I5_J,axiom,
    ! [F1: nat,F2: nat > nat,X2: nat] :
      ( ( case_nat_nat @ F1 @ F2 @ ( suc @ X2 ) )
      = ( F2 @ X2 ) ) ).

% old.nat.simps(5)
thf(fact_7391_old_Onat_Osimps_I5_J,axiom,
    ! [F1: $o,F2: nat > $o,X2: nat] :
      ( ( case_nat_o @ F1 @ F2 @ ( suc @ X2 ) )
      = ( F2 @ X2 ) ) ).

% old.nat.simps(5)
thf(fact_7392_not__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
      = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).

% not_add_distrib
thf(fact_7393_not__diff__distrib,axiom,
    ! [A: int,B: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
      = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).

% not_diff_distrib
thf(fact_7394_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).

% plus_integer.abs_eq
thf(fact_7395_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_7396_verit__eq__simplify_I17_J,axiom,
    ! [F1: option_num,F2: num > option_num,F32: num > option_num,X2: num] :
      ( ( case_num_option_num @ F1 @ F2 @ F32 @ ( bit0 @ X2 ) )
      = ( F2 @ X2 ) ) ).

% verit_eq_simplify(17)
thf(fact_7397_verit__eq__simplify_I16_J,axiom,
    ! [F1: option_num,F2: num > option_num,F32: num > option_num] :
      ( ( case_num_option_num @ F1 @ F2 @ F32 @ one )
      = F1 ) ).

% verit_eq_simplify(16)
thf(fact_7398_verit__eq__simplify_I18_J,axiom,
    ! [F1: option_num,F2: num > option_num,F32: num > option_num,X32: num] :
      ( ( case_num_option_num @ F1 @ F2 @ F32 @ ( bit1 @ X32 ) )
      = ( F32 @ X32 ) ) ).

% verit_eq_simplify(18)
thf(fact_7399_is__singletonI_H,axiom,
    ! [A4: set_se6260736226359567993nt_int] :
      ( ( A4 != bot_bo1488462491386950373nt_int )
     => ( ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X3 @ A4 )
           => ( ( member2340774599025711042nt_int @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_sin6299389887212142093nt_int @ A4 ) ) ) ).

% is_singletonI'
thf(fact_7400_is__singletonI_H,axiom,
    ! [A4: set_set_nat] :
      ( ( A4 != bot_bot_set_set_nat )
     => ( ! [X3: set_nat,Y3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
           => ( ( member_set_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_set_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_7401_is__singletonI_H,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
           => ( ( member8440522571783428010at_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_sin2850979758926227957at_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_7402_is__singletonI_H,axiom,
    ! [A4: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X3: $o,Y3: $o] :
            ( ( member_o @ X3 @ A4 )
           => ( ( member_o @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_o @ A4 ) ) ) ).

% is_singletonI'
thf(fact_7403_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_7404_is__singletonI_H,axiom,
    ! [A4: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X3: int,Y3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ( member_int @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_int @ A4 ) ) ) ).

% is_singletonI'
thf(fact_7405_minus__eq__not__plus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A5: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A5 ) @ one_one_Code_integer ) ) ) ).

% minus_eq_not_plus_1
thf(fact_7406_minus__eq__not__plus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A5: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A5 ) @ one_one_int ) ) ) ).

% minus_eq_not_plus_1
thf(fact_7407_nth__Cons,axiom,
    ! [X: int,Xs: list_int,N2: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ N2 )
      = ( case_nat_int @ X @ ( nth_int @ Xs ) @ N2 ) ) ).

% nth_Cons
thf(fact_7408_nth__Cons,axiom,
    ! [X: option_num,Xs: list_option_num,N2: nat] :
      ( ( nth_option_num @ ( cons_option_num @ X @ Xs ) @ N2 )
      = ( case_nat_option_num @ X @ ( nth_option_num @ Xs ) @ N2 ) ) ).

% nth_Cons
thf(fact_7409_nth__Cons,axiom,
    ! [X: nat,Xs: list_nat,N2: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N2 )
      = ( case_nat_nat @ X @ ( nth_nat @ Xs ) @ N2 ) ) ).

% nth_Cons
thf(fact_7410_nth__Cons,axiom,
    ! [X: $o,Xs: list_o,N2: nat] :
      ( ( nth_o @ ( cons_o @ X @ Xs ) @ N2 )
      = ( case_nat_o @ X @ ( nth_o @ Xs ) @ N2 ) ) ).

% nth_Cons
thf(fact_7411_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( bit_se1745604003318907178nteger @ M @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N2 ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% take_bit_not_mask_eq_0
thf(fact_7412_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) )
        = zero_zero_int ) ) ).

% take_bit_not_mask_eq_0
thf(fact_7413_is__singleton__def,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat] :
        ? [X4: produc3843707927480180839at_nat] :
          ( A6
          = ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_7414_is__singleton__def,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
        ? [X4: product_prod_nat_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_7415_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
        ? [X4: $o] :
          ( A6
          = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_7416_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
        ? [X4: nat] :
          ( A6
          = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_7417_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
        ? [X4: int] :
          ( A6
          = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_7418_is__singletonE,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( is_sin2937591304547752795at_nat @ A4 )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( A4
           != ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% is_singletonE
thf(fact_7419_is__singletonE,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( is_sin2850979758926227957at_nat @ A4 )
     => ~ ! [X3: product_prod_nat_nat] :
            ( A4
           != ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% is_singletonE
thf(fact_7420_is__singletonE,axiom,
    ! [A4: set_o] :
      ( ( is_singleton_o @ A4 )
     => ~ ! [X3: $o] :
            ( A4
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_7421_is__singletonE,axiom,
    ! [A4: set_nat] :
      ( ( is_singleton_nat @ A4 )
     => ~ ! [X3: nat] :
            ( A4
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_7422_is__singletonE,axiom,
    ! [A4: set_int] :
      ( ( is_singleton_int @ A4 )
     => ~ ! [X3: int] :
            ( A4
           != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_7423_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_7424_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_7425_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_7426_and__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_7427_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_7428_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,J: code_integer] : ( if_num @ ( J = 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_7429_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,J: code_integer] : ( if_int @ ( J = 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_7430_nat_Osplit__sels_I2_J,axiom,
    ! [P2: option_num > $o,F1: option_num,F2: nat > option_num,Nat: nat] :
      ( ( P2 @ ( case_nat_option_num @ F1 @ F2 @ Nat ) )
      = ( ~ ( ( ( Nat = zero_zero_nat )
              & ~ ( P2 @ F1 ) )
            | ( ( Nat
                = ( suc @ ( pred @ Nat ) ) )
              & ~ ( P2 @ ( F2 @ ( pred @ Nat ) ) ) ) ) ) ) ).

% nat.split_sels(2)
thf(fact_7431_nat_Osplit__sels_I2_J,axiom,
    ! [P2: nat > $o,F1: nat,F2: nat > nat,Nat: nat] :
      ( ( P2 @ ( case_nat_nat @ F1 @ F2 @ Nat ) )
      = ( ~ ( ( ( Nat = zero_zero_nat )
              & ~ ( P2 @ F1 ) )
            | ( ( Nat
                = ( suc @ ( pred @ Nat ) ) )
              & ~ ( P2 @ ( F2 @ ( pred @ Nat ) ) ) ) ) ) ) ).

% nat.split_sels(2)
thf(fact_7432_nat_Osplit__sels_I2_J,axiom,
    ! [P2: $o > $o,F1: $o,F2: nat > $o,Nat: nat] :
      ( ( P2 @ ( case_nat_o @ F1 @ F2 @ Nat ) )
      = ( ~ ( ( ( Nat = zero_zero_nat )
              & ~ ( P2 @ F1 ) )
            | ( ( Nat
                = ( suc @ ( pred @ Nat ) ) )
              & ~ ( P2 @ ( F2 @ ( pred @ Nat ) ) ) ) ) ) ) ).

% nat.split_sels(2)
thf(fact_7433_nat_Osplit__sels_I1_J,axiom,
    ! [P2: option_num > $o,F1: option_num,F2: nat > option_num,Nat: nat] :
      ( ( P2 @ ( case_nat_option_num @ F1 @ F2 @ Nat ) )
      = ( ( ( Nat = zero_zero_nat )
         => ( P2 @ F1 ) )
        & ( ( Nat
            = ( suc @ ( pred @ Nat ) ) )
         => ( P2 @ ( F2 @ ( pred @ Nat ) ) ) ) ) ) ).

% nat.split_sels(1)
thf(fact_7434_nat_Osplit__sels_I1_J,axiom,
    ! [P2: nat > $o,F1: nat,F2: nat > nat,Nat: nat] :
      ( ( P2 @ ( case_nat_nat @ F1 @ F2 @ Nat ) )
      = ( ( ( Nat = zero_zero_nat )
         => ( P2 @ F1 ) )
        & ( ( Nat
            = ( suc @ ( pred @ Nat ) ) )
         => ( P2 @ ( F2 @ ( pred @ Nat ) ) ) ) ) ) ).

% nat.split_sels(1)
thf(fact_7435_nat_Osplit__sels_I1_J,axiom,
    ! [P2: $o > $o,F1: $o,F2: nat > $o,Nat: nat] :
      ( ( P2 @ ( case_nat_o @ F1 @ F2 @ Nat ) )
      = ( ( ( Nat = zero_zero_nat )
         => ( P2 @ F1 ) )
        & ( ( Nat
            = ( suc @ ( pred @ Nat ) ) )
         => ( P2 @ ( F2 @ ( pred @ Nat ) ) ) ) ) ) ).

% nat.split_sels(1)
thf(fact_7436_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
          @ ^ [Q8: nat] : ( ord_less_nat @ Q8 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_7437_of__int__sum,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups3906332499630173760nt_rat
        @ ^ [X4: int] : ( ring_1_of_int_rat @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_int_sum
thf(fact_7438_of__int__sum,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_int_sum
thf(fact_7439_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% plus_integer.rep_eq
thf(fact_7440_of__nat__sum,axiom,
    ! [F: int > nat,A4: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_7441_of__nat__sum,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_7442_of__nat__sum,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri4939895301339042750nteger @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups7501900531339628137nteger
        @ ^ [X4: nat] : ( semiri4939895301339042750nteger @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_7443_of__nat__sum,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( semiri1316708129612266289at_nat @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_7444_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X23: nat] : X23 ) ) ).

% pred_def
thf(fact_7445_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A4: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( modulo_modulo_nat @ ( F @ I ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ A ) ) ).

% mod_sum_eq
thf(fact_7446_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A4: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I: int] : ( modulo_modulo_int @ ( F @ I ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ A ) ) ).

% mod_sum_eq
thf(fact_7447_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_7448_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_7449_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_7450_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X4: code_integer,Xa2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa2 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_7451_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_7452_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_7453_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X4: code_integer,Xa2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa2 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_7454_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7455_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7456_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_7457_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_7458_bit__numeral__rec_I1_J,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se8040316288895769887atural @ ( numera5444537566228673987atural @ ( bit0 @ W2 ) ) @ N2 )
      = ( case_nat_o @ $false @ ( bit_se8040316288895769887atural @ ( numera5444537566228673987atural @ W2 ) ) @ N2 ) ) ).

% bit_numeral_rec(1)
thf(fact_7459_bit__numeral__rec_I1_J,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se9216721137139052372nteger @ ( numera6620942414471956472nteger @ ( bit0 @ W2 ) ) @ N2 )
      = ( case_nat_o @ $false @ ( bit_se9216721137139052372nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ N2 ) ) ).

% bit_numeral_rec(1)
thf(fact_7460_bit__numeral__rec_I1_J,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) @ N2 )
      = ( case_nat_o @ $false @ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) ) @ N2 ) ) ).

% bit_numeral_rec(1)
thf(fact_7461_bit__numeral__rec_I1_J,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) @ N2 )
      = ( case_nat_o @ $false @ ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W2 ) ) @ N2 ) ) ).

% bit_numeral_rec(1)
thf(fact_7462_bit__numeral__rec_I2_J,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se8040316288895769887atural @ ( numera5444537566228673987atural @ ( bit1 @ W2 ) ) @ N2 )
      = ( case_nat_o @ $true @ ( bit_se8040316288895769887atural @ ( numera5444537566228673987atural @ W2 ) ) @ N2 ) ) ).

% bit_numeral_rec(2)
thf(fact_7463_bit__numeral__rec_I2_J,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se9216721137139052372nteger @ ( numera6620942414471956472nteger @ ( bit1 @ W2 ) ) @ N2 )
      = ( case_nat_o @ $true @ ( bit_se9216721137139052372nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ N2 ) ) ).

% bit_numeral_rec(2)
thf(fact_7464_bit__numeral__rec_I2_J,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) @ N2 )
      = ( case_nat_o @ $true @ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) ) @ N2 ) ) ).

% bit_numeral_rec(2)
thf(fact_7465_bit__numeral__rec_I2_J,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) @ N2 )
      = ( case_nat_o @ $true @ ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W2 ) ) @ N2 ) ) ).

% bit_numeral_rec(2)
thf(fact_7466_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
          @ ^ [Q8: nat] : ( ord_less_nat @ Q8 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7467_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
          @ ^ [Q8: nat] : ( ord_less_nat @ Q8 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7468_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
          @ ^ [Q8: nat] : ( ord_less_nat @ Q8 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7469_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
          @ ^ [Q8: nat] : ( ord_less_nat @ Q8 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7470_sum__abs__ge__zero,axiom,
    ! [F: int > int,A4: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I: int] : ( abs_abs_int @ ( F @ I ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_7471_sum__abs,axiom,
    ! [F: int > int,A4: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I: int] : ( abs_abs_int @ ( F @ I ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_7472_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
                  @ ^ [I: $o] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7473_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
                  @ ^ [I: nat] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7474_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
                  @ ^ [I: int] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7475_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
                  @ ^ [I: $o] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7476_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
                  @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7477_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
                  @ ^ [I: int] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7478_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
                  @ ^ [I: $o] : ( times_times_int @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7479_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
                  @ ^ [I: nat] : ( times_times_int @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7480_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
                  @ ^ [I: int] : ( times_times_int @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7481_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
                  @ ^ [I: set_nat] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7482_abs__sum__abs,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( abs_abs_int
        @ ( groups4538972089207619220nt_int
          @ ^ [A5: int] : ( abs_abs_int @ ( F @ A5 ) )
          @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [A5: int] : ( abs_abs_int @ ( F @ A5 ) )
        @ A4 ) ) ).

% abs_sum_abs
thf(fact_7483_sum_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu2: nat] : zero_zero_nat
        @ A4 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_7484_sum_Oneutral__const,axiom,
    ! [A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu2: int] : zero_zero_int
        @ A4 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_7485_int__sum,axiom,
    ! [F: int > nat,A4: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% int_sum
thf(fact_7486_int__sum,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% int_sum
thf(fact_7487_sum_Oswap,axiom,
    ! [G: nat > nat > nat,B5: set_nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( G @ I ) @ B5 )
        @ A4 )
      = ( groups3542108847815614940at_nat
        @ ^ [J: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( G @ I @ J )
            @ A4 )
        @ B5 ) ) ).

% sum.swap
thf(fact_7488_sum_Oswap,axiom,
    ! [G: int > int > int,B5: set_int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [I: int] : ( groups4538972089207619220nt_int @ ( G @ I ) @ B5 )
        @ A4 )
      = ( groups4538972089207619220nt_int
        @ ^ [J: int] :
            ( groups4538972089207619220nt_int
            @ ^ [I: int] : ( G @ I @ J )
            @ A4 )
        @ B5 ) ) ).

% sum.swap
thf(fact_7489_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_7490_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_7491_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_7492_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_7493_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_7494_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_7495_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_7496_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_7497_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_7498_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_7499_sum_Odistrib,axiom,
    ! [G: nat > nat,H: nat > nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( plus_plus_nat @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H @ A4 ) ) ) ).

% sum.distrib
thf(fact_7500_sum_Odistrib,axiom,
    ! [G: int > int,H: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( plus_plus_int @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H @ A4 ) ) ) ).

% sum.distrib
thf(fact_7501_sum__product,axiom,
    ! [F: nat > nat,A4: set_nat,G: nat > nat,B5: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_7502_sum__product,axiom,
    ! [F: int > int,A4: set_int,G: int > int,B5: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J: int] : ( times_times_int @ ( F @ I ) @ ( G @ J ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_7503_sum__distrib__right,axiom,
    ! [F: nat > nat,A4: set_nat,R2: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ R2 )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R2 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_7504_sum__distrib__right,axiom,
    ! [F: int > int,A4: set_int,R2: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ R2 )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ ( F @ N ) @ R2 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_7505_sum__distrib__left,axiom,
    ! [R2: nat,F: nat > nat,A4: set_nat] :
      ( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ R2 @ ( F @ N ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_7506_sum__distrib__left,axiom,
    ! [R2: int,F: int > int,A4: set_int] :
      ( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ R2 @ ( F @ N ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_7507_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( minus_minus_int @ ( F @ X4 ) @ ( G @ X4 ) )
        @ A4 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_7508_sum__negf,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( uminus_uminus_int @ ( F @ X4 ) )
        @ A4 )
      = ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) ) ) ).

% sum_negf
thf(fact_7509_sum__nonpos,axiom,
    ! [A4: set_o,F: $o > code_integer] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups4406642042086082107nteger @ F @ A4 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_7510_sum__nonpos,axiom,
    ! [A4: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A4 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_7511_sum__nonpos,axiom,
    ! [A4: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A4 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_7512_sum__nonpos,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7513_sum__nonpos,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7514_sum__nonpos,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7515_sum__nonpos,axiom,
    ! [A4: set_o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7516_sum__nonpos,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7517_sum__nonpos,axiom,
    ! [A4: set_o,F: $o > int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A4 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_7518_sum__nonpos,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_int @ ( F @ X3 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_7519_sum__nonneg,axiom,
    ! [A4: set_o,F: $o > code_integer] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups4406642042086082107nteger @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7520_sum__nonneg,axiom,
    ! [A4: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups7501900531339628137nteger @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7521_sum__nonneg,axiom,
    ! [A4: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups7873554091576472773nteger @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7522_sum__nonneg,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7523_sum__nonneg,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7524_sum__nonneg,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7525_sum__nonneg,axiom,
    ! [A4: set_o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7526_sum__nonneg,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7527_sum__nonneg,axiom,
    ! [A4: set_o,F: $o > int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7528_sum__nonneg,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7529_sum__subtractf__nat,axiom,
    ! [A4: set_o,G: $o > nat,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups8507830703676809646_o_nat
          @ ^ [X4: $o] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( groups8507830703676809646_o_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7530_sum__subtractf__nat,axiom,
    ! [A4: set_se6260736226359567993nt_int,G: set_Pr958786334691620121nt_int > nat,F: set_Pr958786334691620121nt_int > nat] :
      ( ! [X3: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups185207323561075247nt_nat
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups185207323561075247nt_nat @ F @ A4 ) @ ( groups185207323561075247nt_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7531_sum__subtractf__nat,axiom,
    ! [A4: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X4: set_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( groups8294997508430121362at_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7532_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: int > nat,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7533_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7534_sum__comp__morphism,axiom,
    ! [H: nat > code_integer,G: nat > nat,A4: set_nat] :
      ( ( ( H @ zero_zero_nat )
        = zero_z3403309356797280102nteger )
     => ( ! [X3: nat,Y3: nat] :
            ( ( H @ ( plus_plus_nat @ X3 @ Y3 ) )
            = ( plus_p5714425477246183910nteger @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups7501900531339628137nteger @ ( comp_n3898172953802868194er_nat @ H @ G ) @ A4 )
          = ( H @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7535_sum__comp__morphism,axiom,
    ! [H: nat > rat,G: nat > nat,A4: set_nat] :
      ( ( ( H @ zero_zero_nat )
        = zero_zero_rat )
     => ( ! [X3: nat,Y3: nat] :
            ( ( H @ ( plus_plus_nat @ X3 @ Y3 ) )
            = ( plus_plus_rat @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups2906978787729119204at_rat @ ( comp_nat_rat_nat @ H @ G ) @ A4 )
          = ( H @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7536_sum__comp__morphism,axiom,
    ! [H: nat > int,G: nat > nat,A4: set_nat] :
      ( ( ( H @ zero_zero_nat )
        = zero_zero_int )
     => ( ! [X3: nat,Y3: nat] :
            ( ( H @ ( plus_plus_nat @ X3 @ Y3 ) )
            = ( plus_plus_int @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups3539618377306564664at_int @ ( comp_nat_int_nat @ H @ G ) @ A4 )
          = ( H @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7537_sum__comp__morphism,axiom,
    ! [H: int > code_integer,G: int > int,A4: set_int] :
      ( ( ( H @ zero_zero_int )
        = zero_z3403309356797280102nteger )
     => ( ! [X3: int,Y3: int] :
            ( ( H @ ( plus_plus_int @ X3 @ Y3 ) )
            = ( plus_p5714425477246183910nteger @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups7873554091576472773nteger @ ( comp_i1585864551200866970er_int @ H @ G ) @ A4 )
          = ( H @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7538_sum__comp__morphism,axiom,
    ! [H: int > rat,G: int > int,A4: set_int] :
      ( ( ( H @ zero_zero_int )
        = zero_zero_rat )
     => ( ! [X3: int,Y3: int] :
            ( ( H @ ( plus_plus_int @ X3 @ Y3 ) )
            = ( plus_plus_rat @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups3906332499630173760nt_rat @ ( comp_int_rat_int @ H @ G ) @ A4 )
          = ( H @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7539_sum__comp__morphism,axiom,
    ! [H: int > nat,G: int > int,A4: set_int] :
      ( ( ( H @ zero_zero_int )
        = zero_zero_nat )
     => ( ! [X3: int,Y3: int] :
            ( ( H @ ( plus_plus_int @ X3 @ Y3 ) )
            = ( plus_plus_nat @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups4541462559716669496nt_nat @ ( comp_int_nat_int @ H @ G ) @ A4 )
          = ( H @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7540_sum__comp__morphism,axiom,
    ! [H: code_integer > nat,G: nat > code_integer,A4: set_nat] :
      ( ( ( H @ zero_z3403309356797280102nteger )
        = zero_zero_nat )
     => ( ! [X3: code_integer,Y3: code_integer] :
            ( ( H @ ( plus_p5714425477246183910nteger @ X3 @ Y3 ) )
            = ( plus_plus_nat @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups3542108847815614940at_nat @ ( comp_C4049556595663050210at_nat @ H @ G ) @ A4 )
          = ( H @ ( groups7501900531339628137nteger @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7541_sum__comp__morphism,axiom,
    ! [H: rat > nat,G: nat > rat,A4: set_nat] :
      ( ( ( H @ zero_zero_rat )
        = zero_zero_nat )
     => ( ! [X3: rat,Y3: rat] :
            ( ( H @ ( plus_plus_rat @ X3 @ Y3 ) )
            = ( plus_plus_nat @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups3542108847815614940at_nat @ ( comp_rat_nat_nat @ H @ G ) @ A4 )
          = ( H @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7542_sum__comp__morphism,axiom,
    ! [H: int > nat,G: nat > int,A4: set_nat] :
      ( ( ( H @ zero_zero_int )
        = zero_zero_nat )
     => ( ! [X3: int,Y3: int] :
            ( ( H @ ( plus_plus_int @ X3 @ Y3 ) )
            = ( plus_plus_nat @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups3542108847815614940at_nat @ ( comp_int_nat_nat @ H @ G ) @ A4 )
          = ( H @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7543_sum__comp__morphism,axiom,
    ! [H: nat > nat,G: nat > nat,A4: set_nat] :
      ( ( ( H @ zero_zero_nat )
        = zero_zero_nat )
     => ( ! [X3: nat,Y3: nat] :
            ( ( H @ ( plus_plus_nat @ X3 @ Y3 ) )
            = ( plus_plus_nat @ ( H @ X3 ) @ ( H @ Y3 ) ) )
       => ( ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ H @ G ) @ A4 )
          = ( H @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7544_sum__SucD,axiom,
    ! [F: nat > nat,A4: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A4 )
        = ( suc @ N2 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_7545_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,J: code_integer] : ( if_nat @ ( J = 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_7546_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_7547_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_7548_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
          @ ( groups7501900531339628137nteger
            @ ^ [I: nat] :
                ( if_Code_integer
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I )
                @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I ) )
                @ zero_z3403309356797280102nteger )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_7549_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_7550_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_7551_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
          @ ( groups7501900531339628137nteger
            @ ^ [I: nat] : ( if_Code_integer @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I ) ) @ zero_z3403309356797280102nteger )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_7552_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_7553_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_7554_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L4: set_int,H5: set_int] :
      ( ( ( set_or370866239135849197et_int @ L @ H )
        = ( set_or370866239135849197et_int @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H = H5 ) )
        | ( ~ ( ord_less_eq_set_int @ L @ H )
          & ~ ( ord_less_eq_set_int @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_7555_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L4: rat,H5: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H )
        = ( set_or633870826150836451st_rat @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H = H5 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H )
          & ~ ( ord_less_eq_rat @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_7556_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L4: num,H5: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H )
        = ( set_or7049704709247886629st_num @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H = H5 ) )
        | ( ~ ( ord_less_eq_num @ L @ H )
          & ~ ( ord_less_eq_num @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_7557_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L4: nat,H5: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H )
        = ( set_or1269000886237332187st_nat @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H = H5 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H )
          & ~ ( ord_less_eq_nat @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_7558_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L4: int,H5: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H )
        = ( set_or1266510415728281911st_int @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H = H5 ) )
        | ( ~ ( ord_less_eq_int @ L @ H )
          & ~ ( ord_less_eq_int @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_7559_Icc__eq__Icc,axiom,
    ! [L: code_integer,H: code_integer,L4: code_integer,H5: code_integer] :
      ( ( ( set_or189985376899183464nteger @ L @ H )
        = ( set_or189985376899183464nteger @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H = H5 ) )
        | ( ~ ( ord_le3102999989581377725nteger @ L @ H )
          & ~ ( ord_le3102999989581377725nteger @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_7560_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_7561_atLeastAtMost__iff,axiom,
    ! [I2: set_Pr958786334691620121nt_int,L: set_Pr958786334691620121nt_int,U: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ I2 @ ( set_or2481441762145802318nt_int @ L @ U ) )
      = ( ( ord_le2843351958646193337nt_int @ L @ I2 )
        & ( ord_le2843351958646193337nt_int @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_7562_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_7563_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_7564_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_7565_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_7566_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_7567_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_7568_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_7569_atMost__iff,axiom,
    ! [I2: $o,K2: $o] :
      ( ( member_o @ I2 @ ( set_ord_atMost_o @ K2 ) )
      = ( ord_less_eq_o @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_7570_atMost__iff,axiom,
    ! [I2: set_Pr958786334691620121nt_int,K2: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ I2 @ ( set_or2459421552957432928nt_int @ K2 ) )
      = ( ord_le2843351958646193337nt_int @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_7571_atMost__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K2 ) )
      = ( ord_less_eq_set_nat @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_7572_atMost__iff,axiom,
    ! [I2: set_int,K2: set_int] :
      ( ( member_set_int @ I2 @ ( set_or58775011639299419et_int @ K2 ) )
      = ( ord_less_eq_set_int @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_7573_atMost__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K2 ) )
      = ( ord_less_eq_rat @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_7574_atMost__iff,axiom,
    ! [I2: num,K2: num] :
      ( ( member_num @ I2 @ ( set_ord_atMost_num @ K2 ) )
      = ( ord_less_eq_num @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_7575_atMost__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K2 ) )
      = ( ord_less_eq_nat @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_7576_atMost__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( member_int @ I2 @ ( set_ord_atMost_int @ K2 ) )
      = ( ord_less_eq_int @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_7577_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_7578_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_7579_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_7580_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_7581_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_7582_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_7583_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_7584_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_7585_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_7586_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_7587_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_7588_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_7589_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_7590_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_7591_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_7592_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_7593_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_7594_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_7595_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_7596_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_7597_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_7598_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D2: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( ( ord_less_eq_set_int @ C2 @ A )
          & ( ord_less_eq_set_int @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_7599_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C2 @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_7600_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C2 @ A )
          & ( ord_less_eq_num @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_7601_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C2 @ A )
          & ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_7602_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C2 @ A )
          & ( ord_less_eq_int @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_7603_atLeastatMost__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C2 @ D2 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B )
        | ( ( ord_le3102999989581377725nteger @ C2 @ A )
          & ( ord_le3102999989581377725nteger @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_7604_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_7605_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_7606_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_7607_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_7608_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_7609_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H5: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H ) @ ( set_or58775011639299419et_int @ H5 ) )
      = ( ~ ( ord_less_eq_set_int @ L @ H )
        | ( ord_less_eq_set_int @ H @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7610_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H5: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H ) @ ( set_ord_atMost_rat @ H5 ) )
      = ( ~ ( ord_less_eq_rat @ L @ H )
        | ( ord_less_eq_rat @ H @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7611_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H5: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H ) @ ( set_ord_atMost_num @ H5 ) )
      = ( ~ ( ord_less_eq_num @ L @ H )
        | ( ord_less_eq_num @ H @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7612_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H5: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H5 ) )
      = ( ~ ( ord_less_eq_nat @ L @ H )
        | ( ord_less_eq_nat @ H @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7613_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H5: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atMost_int @ H5 ) )
      = ( ~ ( ord_less_eq_int @ L @ H )
        | ( ord_less_eq_int @ H @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7614_Icc__subset__Iic__iff,axiom,
    ! [L: code_integer,H: code_integer,H5: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ L @ H ) @ ( set_or9101266186257409494nteger @ H5 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ L @ H )
        | ( ord_le3102999989581377725nteger @ H @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7615_sum_OatMost__Suc,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7616_sum_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7617_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7618_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7619_nat__of__integer__non__positive,axiom,
    ! [K2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K2 )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_7620_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_7621_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_7622_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_7623_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_7624_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_7625_not__Iic__le__Icc,axiom,
    ! [H: int,L4: int,H5: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H ) @ ( set_or1266510415728281911st_int @ L4 @ H5 ) ) ).

% not_Iic_le_Icc
thf(fact_7626_atMost__def,axiom,
    ( set_or2459421552957432928nt_int
    = ( ^ [U3: set_Pr958786334691620121nt_int] :
          ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X4 @ U3 ) ) ) ) ).

% atMost_def
thf(fact_7627_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U3: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U3 ) ) ) ) ).

% atMost_def
thf(fact_7628_atMost__def,axiom,
    ( set_or58775011639299419et_int
    = ( ^ [U3: set_int] :
          ( collect_set_int
          @ ^ [X4: set_int] : ( ord_less_eq_set_int @ X4 @ U3 ) ) ) ) ).

% atMost_def
thf(fact_7629_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U3: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_eq_rat @ X4 @ U3 ) ) ) ) ).

% atMost_def
thf(fact_7630_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U3: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_eq_num @ X4 @ U3 ) ) ) ) ).

% atMost_def
thf(fact_7631_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U3: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U3 ) ) ) ) ).

% atMost_def
thf(fact_7632_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U3: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_eq_int @ X4 @ U3 ) ) ) ) ).

% atMost_def
thf(fact_7633_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_7634_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_7635_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_7636_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_7637_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_7638_ex__nat__less,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
            & ( P2 @ M3 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P2 @ X4 ) ) ) ) ).

% ex_nat_less
thf(fact_7639_all__nat__less,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
           => ( P2 @ M3 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P2 @ X4 ) ) ) ) ).

% all_nat_less
thf(fact_7640_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [G: nat > nat,M: nat,K2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ K2 ) ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.atLeastAtMost_shift_bounds
thf(fact_7641_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
        @ ^ [I: nat] : ( G @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7642_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7643_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_7644_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_7645_atLeastatMost__psubset__iff,axiom,
    ! [A: assn,B: assn,C2: assn,D2: assn] :
      ( ( ord_less_set_assn @ ( set_or7959216805967363635t_assn @ A @ B ) @ ( set_or7959216805967363635t_assn @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_assn @ A @ B )
          | ( ( ord_less_eq_assn @ C2 @ A )
            & ( ord_less_eq_assn @ B @ D2 )
            & ( ( ord_less_assn @ C2 @ A )
              | ( ord_less_assn @ B @ D2 ) ) ) )
        & ( ord_less_eq_assn @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_7646_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D2: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B )
          | ( ( ord_less_eq_set_int @ C2 @ A )
            & ( ord_less_eq_set_int @ B @ D2 )
            & ( ( ord_less_set_int @ C2 @ A )
              | ( ord_less_set_int @ B @ D2 ) ) ) )
        & ( ord_less_eq_set_int @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_7647_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C2 @ A )
            & ( ord_less_eq_rat @ B @ D2 )
            & ( ( ord_less_rat @ C2 @ A )
              | ( ord_less_rat @ B @ D2 ) ) ) )
        & ( ord_less_eq_rat @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_7648_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C2 @ A )
            & ( ord_less_eq_num @ B @ D2 )
            & ( ( ord_less_num @ C2 @ A )
              | ( ord_less_num @ B @ D2 ) ) ) )
        & ( ord_less_eq_num @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_7649_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C2 @ A )
            & ( ord_less_eq_nat @ B @ D2 )
            & ( ( ord_less_nat @ C2 @ A )
              | ( ord_less_nat @ B @ D2 ) ) ) )
        & ( ord_less_eq_nat @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_7650_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C2 @ A )
            & ( ord_less_eq_int @ B @ D2 )
            & ( ( ord_less_int @ C2 @ A )
              | ( ord_less_int @ B @ D2 ) ) ) )
        & ( ord_less_eq_int @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_7651_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C2 @ D2 ) )
      = ( ( ~ ( ord_le3102999989581377725nteger @ A @ B )
          | ( ( ord_le3102999989581377725nteger @ C2 @ A )
            & ( ord_le3102999989581377725nteger @ B @ D2 )
            & ( ( ord_le6747313008572928689nteger @ C2 @ A )
              | ( ord_le6747313008572928689nteger @ B @ D2 ) ) ) )
        & ( ord_le3102999989581377725nteger @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_7652_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_7653_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_7654_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_7655_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7656_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_7657_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat ) @ ( groups6857163185585827899at_nat @ ( comp_n8698576032424989604at_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_7658_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat ) @ ( groups2906978787729119204at_rat @ ( comp_nat_rat_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_7659_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat ) @ ( groups3539618377306564664at_int @ ( comp_nat_int_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_7660_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat ) @ ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_7661_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7662_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7663_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7664_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7665_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_7666_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_7667_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_7668_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_7669_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_7670_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_7671_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_7672_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_7673_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7674_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7675_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7676_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7677_sum__telescope,axiom,
    ! [F: nat > rat,I2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( minus_minus_rat @ ( F @ I ) @ ( F @ ( suc @ I ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_7678_sum__telescope,axiom,
    ! [F: nat > int,I2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( minus_minus_int @ ( F @ I ) @ ( F @ ( suc @ I ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_7679_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
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7680_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
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7681_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
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7682_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
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7683_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( minus_minus_rat @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7684_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7685_sum_OatLeast__atMost__pred__shift,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ( comp_nat_nat_nat @ G
          @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.atLeast_atMost_pred_shift
thf(fact_7686_sum__choose__lower,axiom,
    ! [R2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_7687_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J ) @ 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_7688_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J ) @ 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_7689_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_7690_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_7691_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_7692_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_7693_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_7694_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_7695_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_7696_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_7697_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_7698_sum_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ M ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% sum.atLeastAtMost_shift_0
thf(fact_7699_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_7700_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I: nat,J: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I @ J ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( G @ I @ ( minus_minus_nat @ K3 @ I ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_7701_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_7702_vandermonde,axiom,
    ! [M: nat,N2: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R2 ) ) ).

% vandermonde
thf(fact_7703_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_7704_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_7705_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_7706_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_7707_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_7708_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_7709_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_7710_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_7711_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_7712_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_7713_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
        @ ^ [I: nat] : ( plus_p7104986032573967614at_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_7714_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
        @ ^ [I: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_7715_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
        @ ^ [I: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_7716_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
        @ ^ [I: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_7717_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_7718_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_7719_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_7720_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_7721_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_7722_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_7723_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_7724_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_7725_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_7726_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_7727_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_7728_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_7729_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
        @ ^ [I: nat] : ( plus_p7104986032573967614at_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7730_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
        @ ^ [I: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7731_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
        @ ^ [I: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7732_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
        @ ^ [I: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7733_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_7734_sum_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups7501900531339628137nteger
            @ ^ [J: nat] : ( if_Code_integer @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_Code_integer @ ( J = K2 ) @ zero_z3403309356797280102nteger @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups7501900531339628137nteger
            @ ^ [J: nat] : ( if_Code_integer @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7735_sum_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > multis2468970476368604999at_nat,H: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups6857163185585827899at_nat
            @ ^ [J: nat] : ( if_mul8430962117462786573at_nat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_mul8430962117462786573at_nat @ ( J = K2 ) @ zero_z1048942125864253310at_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups6857163185585827899at_nat
            @ ^ [J: nat] : ( if_mul8430962117462786573at_nat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7736_sum_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > rat,H: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J: nat] : ( if_rat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_rat @ ( J = K2 ) @ zero_zero_rat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J: nat] : ( if_rat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7737_sum_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J: nat] : ( if_int @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_int @ ( J = K2 ) @ zero_zero_int @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J: nat] : ( if_int @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7738_sum_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_nat @ ( J = K2 ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7739_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_7740_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J ) @ K2 )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_7741_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_7742_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_7743_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_7744_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I ) @ ( semiri681578069525770553at_rat @ I ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_7745_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I ) @ ( semiri1314217659103216013at_int @ I ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_7746_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I ) @ ( semiri4939895301339042750nteger @ I ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_7747_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_7748_double__arith__series,axiom,
    ! [A: rat,D2: rat,N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I ) @ 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_7749_double__arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) )
        @ ( groups6325495683096345652atural
          @ ^ [I: nat] : ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ I ) @ 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_7750_double__arith__series,axiom,
    ! [A: int,D2: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I ) @ 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_7751_double__arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
        @ ( groups7501900531339628137nteger
          @ ^ [I: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I ) @ 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_7752_double__arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I ) @ 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_7753_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_7754_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_7755_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_7756_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_7757_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_7758_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_7759_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( times_times_nat @ I @ ( binomial @ N2 @ I ) )
        @ ( 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_7760_arith__series__nat,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I @ 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_7761_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_7762_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_7763_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_7764_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups7501900531339628137nteger
          @ ^ [I: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_7765_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_7766_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_7767_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_7768_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_7769_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_7770_arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] :
      ( ( groups6325495683096345652atural
        @ ^ [I: nat] : ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ I ) @ 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_7771_arith__series,axiom,
    ! [A: int,D2: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I ) @ 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_7772_arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I ) @ 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_7773_arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I ) @ 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_7774_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_7775_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_7776_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_7777_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_7778_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_7779_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_7780_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_7781_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_7782_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_7783_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( 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 @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7784_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_7785_pochhammer__times__pochhammer__half,axiom,
    ! [Z3: rat,N2: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z3 @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z3 @ ( 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_7786_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
                  @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S5 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_7787_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_7788_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_7789_sum__power2,axiom,
    ! [K2: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_7790_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_7791_apsnd__conv,axiom,
    ! [F: int > int,X: int,Y: int] :
      ( ( produc4463282112584876420nt_int @ F @ ( product_Pair_int_int @ X @ Y ) )
      = ( product_Pair_int_int @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_7792_apsnd__conv,axiom,
    ! [F: produc3658429121746597890et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( produc8750854537940449737_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
      = ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_7793_apsnd__conv,axiom,
    ! [F: produc3658429121746597890et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( produc7714581247149323085_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
      = ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_7794_apsnd__conv,axiom,
    ! [F: produc3925858234332021118et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( produc1515462096303866701_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
      = ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_7795_apsnd__conv,axiom,
    ! [F: produc3925858234332021118et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( produc969530845752564945_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
      = ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_7796_apsnd__conv,axiom,
    ! [F: produc6653097349344004940it_nat > produc6653097349344004940it_nat,X: b,Y: produc6653097349344004940it_nat] :
      ( ( produc701890989253350759_nat_b @ F @ ( produc4082563078715348724it_nat @ X @ Y ) )
      = ( produc4082563078715348724it_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_7797_apsnd__conv,axiom,
    ! [F: produc6653097349344004940it_nat > produc6653097349344004940it_nat,X: a,Y: produc6653097349344004940it_nat] :
      ( ( produc701890989253350758_nat_a @ F @ ( produc9178034014595674355it_nat @ X @ Y ) )
      = ( produc9178034014595674355it_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_7798_apsnd__conv,axiom,
    ! [F: code_integer > code_integer,X: code_integer,Y: code_integer] :
      ( ( produc6499014454317279255nteger @ F @ ( produc1086072967326762835nteger @ X @ Y ) )
      = ( produc1086072967326762835nteger @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_7799_prod_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu2: nat] : one_one_nat
        @ A4 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_7800_prod_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu2: nat] : one_one_int
        @ A4 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7801_prod_Oneutral__const,axiom,
    ! [A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu2: int] : one_one_int
        @ A4 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7802_of__nat__prod,axiom,
    ! [F: int > nat,A4: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_nat_prod
thf(fact_7803_of__nat__prod,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ F @ A4 ) )
      = ( groups3455450783089532116nteger
        @ ^ [X4: nat] : ( semiri4939895301339042750nteger @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_nat_prod
thf(fact_7804_of__nat__prod,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
      = ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( semiri1316708129612266289at_nat @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_nat_prod
thf(fact_7805_of__nat__prod,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A4 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_nat_prod
thf(fact_7806_of__int__prod,axiom,
    ! [F: nat > int,A4: set_nat] :
      ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A4 ) )
      = ( groups73079841787564623at_rat
        @ ^ [X4: nat] : ( ring_1_of_int_rat @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_int_prod
thf(fact_7807_of__int__prod,axiom,
    ! [F: nat > int,A4: set_nat] :
      ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A4 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_int_prod
thf(fact_7808_of__int__prod,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A4 ) )
      = ( groups1072433553688619179nt_rat
        @ ^ [X4: int] : ( ring_1_of_int_rat @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_int_prod
thf(fact_7809_of__int__prod,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A4 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( ring_1_of_int_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% of_int_prod
thf(fact_7810_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_7811_atLeastLessThan__iff,axiom,
    ! [I2: set_Pr958786334691620121nt_int,L: set_Pr958786334691620121nt_int,U: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ I2 @ ( set_or2826562449856406570nt_int @ L @ U ) )
      = ( ( ord_le2843351958646193337nt_int @ L @ I2 )
        & ( ord_le7563427860532173253nt_int @ I2 @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_7812_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_7813_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_7814_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_7815_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_7816_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_7817_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_7818_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_7819_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_7820_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_7821_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_7822_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_7823_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_7824_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_7825_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
thf(fact_7826_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_7827_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_7828_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_7829_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_7830_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_7831_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_7832_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_7833_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_7834_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_7835_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_7836_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_7837_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_7838_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_7839_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_7840_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_7841_ivl__subset,axiom,
    ! [I2: rat,J2: rat,M: rat,N2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ I2 @ J2 ) @ ( set_or4029947393144176647an_rat @ M @ N2 ) )
      = ( ( ord_less_eq_rat @ J2 @ I2 )
        | ( ( ord_less_eq_rat @ M @ I2 )
          & ( ord_less_eq_rat @ J2 @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_7842_ivl__subset,axiom,
    ! [I2: num,J2: num,M: num,N2: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I2 @ J2 ) @ ( set_or1222409239386451017an_num @ M @ N2 ) )
      = ( ( ord_less_eq_num @ J2 @ I2 )
        | ( ( ord_less_eq_num @ M @ I2 )
          & ( ord_less_eq_num @ J2 @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_7843_ivl__subset,axiom,
    ! [I2: nat,J2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J2 ) @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ J2 @ I2 )
        | ( ( ord_less_eq_nat @ M @ I2 )
          & ( ord_less_eq_nat @ J2 @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_7844_ivl__subset,axiom,
    ! [I2: code_integer,J2: code_integer,M: code_integer,N2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ I2 @ J2 ) @ ( set_or8404916559141939852nteger @ M @ N2 ) )
      = ( ( ord_le3102999989581377725nteger @ J2 @ I2 )
        | ( ( ord_le3102999989581377725nteger @ M @ I2 )
          & ( ord_le3102999989581377725nteger @ J2 @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_7845_ivl__subset,axiom,
    ! [I2: int,J2: int,M: int,N2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I2 @ J2 ) @ ( set_or4662586982721622107an_int @ M @ N2 ) )
      = ( ( ord_less_eq_int @ J2 @ I2 )
        | ( ( ord_less_eq_int @ M @ I2 )
          & ( ord_less_eq_int @ J2 @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_7846_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_7847_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_7848_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_7849_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_7850_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_7851_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_7852_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_7853_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_7854_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_7855_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_7856_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_7857_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_7858_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_7859_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_7860_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_7861_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_7862_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_7863_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_7864_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_7865_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_7866_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_7867_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_7868_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [G: nat > nat,M: nat,K2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ K2 ) ) @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.atLeastLessThan_shift_bounds
thf(fact_7869_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [G: nat > int,M: nat,K2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ ( plus_plus_nat @ K2 ) ) @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.atLeastLessThan_shift_bounds
thf(fact_7870_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_7871_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_7872_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_7873_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_7874_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
        @ ^ [I: nat] : ( G @ ( suc @ I ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_7875_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
        @ ^ [I: nat] : ( G @ ( suc @ I ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_7876_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ( set_or4029947393144176647an_rat @ A @ B )
            = ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
          = ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7877_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ C2 @ D2 )
       => ( ( ( set_or1222409239386451017an_num @ A @ B )
            = ( set_or1222409239386451017an_num @ C2 @ D2 ) )
          = ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7878_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
          = ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7879_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ( ( set_or8404916559141939852nteger @ A @ B )
            = ( set_or8404916559141939852nteger @ C2 @ D2 ) )
          = ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7880_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ( set_or4662586982721622107an_int @ A @ B )
            = ( set_or4662586982721622107an_int @ C2 @ D2 ) )
          = ( ( A = C2 )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7881_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C2 @ D2 )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7882_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C2 @ D2 ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C2 @ D2 )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7883_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C2 @ D2 )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7884_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C2 @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7885_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C2 @ D2 ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C2 @ D2 )
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7886_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C2 @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7887_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C2 @ D2 ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C2 @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7888_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C2 @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7889_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C2 @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7890_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C2 @ D2 ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C2 @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7891_prod_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat,G: nat > nat,H: nat > nat] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C2 @ X3 )
             => ( ( ord_less_nat @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups708209901874060359at_nat @ H @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_7892_prod_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat,G: nat > int,H: nat > int] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C2 @ X3 )
             => ( ( ord_less_nat @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups705719431365010083at_int @ H @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_7893_prod_Oivl__cong,axiom,
    ! [A: int,C2: int,B: int,D2: int,G: int > int,H: int > int] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( ( ord_less_eq_int @ C2 @ X3 )
             => ( ( ord_less_int @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups1705073143266064639nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups1705073143266064639nt_int @ H @ ( set_or4662586982721622107an_int @ C2 @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_7894_prod_Oswap,axiom,
    ! [G: nat > nat > nat,B5: set_nat,A4: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( groups708209901874060359at_nat @ ( G @ I ) @ B5 )
        @ A4 )
      = ( groups708209901874060359at_nat
        @ ^ [J: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I: nat] : ( G @ I @ J )
            @ A4 )
        @ B5 ) ) ).

% prod.swap
thf(fact_7895_prod_Oswap,axiom,
    ! [G: nat > nat > int,B5: set_nat,A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I: nat] : ( groups705719431365010083at_int @ ( G @ I ) @ B5 )
        @ A4 )
      = ( groups705719431365010083at_int
        @ ^ [J: nat] :
            ( groups705719431365010083at_int
            @ ^ [I: nat] : ( G @ I @ J )
            @ A4 )
        @ B5 ) ) ).

% prod.swap
thf(fact_7896_prod_Oswap,axiom,
    ! [G: nat > int > int,B5: set_int,A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I: nat] : ( groups1705073143266064639nt_int @ ( G @ I ) @ B5 )
        @ A4 )
      = ( groups1705073143266064639nt_int
        @ ^ [J: int] :
            ( groups705719431365010083at_int
            @ ^ [I: nat] : ( G @ I @ J )
            @ A4 )
        @ B5 ) ) ).

% prod.swap
thf(fact_7897_prod_Oswap,axiom,
    ! [G: int > nat > int,B5: set_nat,A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [I: int] : ( groups705719431365010083at_int @ ( G @ I ) @ B5 )
        @ A4 )
      = ( groups705719431365010083at_int
        @ ^ [J: nat] :
            ( groups1705073143266064639nt_int
            @ ^ [I: int] : ( G @ I @ J )
            @ A4 )
        @ B5 ) ) ).

% prod.swap
thf(fact_7898_prod_Oswap,axiom,
    ! [G: int > int > int,B5: set_int,A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [I: int] : ( groups1705073143266064639nt_int @ ( G @ I ) @ B5 )
        @ A4 )
      = ( groups1705073143266064639nt_int
        @ ^ [J: int] :
            ( groups1705073143266064639nt_int
            @ ^ [I: int] : ( G @ I @ J )
            @ A4 )
        @ B5 ) ) ).

% prod.swap
thf(fact_7899_prod_Odistrib,axiom,
    ! [G: nat > nat,H: nat > nat,A4: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( times_times_nat @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A4 ) @ ( groups708209901874060359at_nat @ H @ A4 ) ) ) ).

% prod.distrib
thf(fact_7900_prod_Odistrib,axiom,
    ! [G: nat > int,H: nat > int,A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( times_times_int @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A4 ) @ ( groups705719431365010083at_int @ H @ A4 ) ) ) ).

% prod.distrib
thf(fact_7901_prod_Odistrib,axiom,
    ! [G: int > int,H: int > int,A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( times_times_int @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A4 ) @ ( groups1705073143266064639nt_int @ H @ A4 ) ) ) ).

% prod.distrib
thf(fact_7902_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_7903_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_7904_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_7905_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_7906_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_7907_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_7908_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_7909_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_7910_prod__power__distrib,axiom,
    ! [F: nat > nat,A4: set_nat,N2: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ N2 )
      = ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( power_power_nat @ ( F @ X4 ) @ N2 )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_7911_prod__power__distrib,axiom,
    ! [F: nat > int,A4: set_nat,N2: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A4 ) @ N2 )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( power_power_int @ ( F @ X4 ) @ N2 )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_7912_prod__power__distrib,axiom,
    ! [F: int > int,A4: set_int,N2: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ N2 )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( power_power_int @ ( F @ X4 ) @ N2 )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_7913_prod_OatLeastLessThan__shift__0,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ M ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% prod.atLeastLessThan_shift_0
thf(fact_7914_prod_OatLeastLessThan__shift__0,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ ( plus_plus_nat @ M ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% prod.atLeastLessThan_shift_0
thf(fact_7915_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A4: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I: nat] : ( modulo_modulo_nat @ ( F @ I ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_7916_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A4: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I: nat] : ( modulo_modulo_int @ ( F @ I ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_7917_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A4: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I: int] : ( modulo_modulo_int @ ( F @ I ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_7918_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_7919_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_7920_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_7921_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_7922_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ( comp_nat_nat_nat @ G
          @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.atLeast_lessThan_pred_shift
thf(fact_7923_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ( comp_nat_int_nat @ G
          @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.atLeast_lessThan_pred_shift
thf(fact_7924_prod_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_7925_prod_OatLeastLessThan__rev,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_7926_prod_Onested__swap,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( groups708209901874060359at_nat @ ( A @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% prod.nested_swap
thf(fact_7927_prod_Onested__swap,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I: nat] : ( groups705719431365010083at_int @ ( A @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J: nat] :
            ( groups705719431365010083at_int
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% prod.nested_swap
thf(fact_7928_apsnd__compose,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc9020447175693601247nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
      = ( produc9020447175693601247nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ X ) ) ).

% apsnd_compose
thf(fact_7929_apsnd__compose,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc6499014454317279255nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
      = ( produc6499014454317279255nteger @ ( comp_C7449957260575251196nteger @ F @ G ) @ X ) ) ).

% apsnd_compose
thf(fact_7930_prod__nonneg,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_7931_prod__nonneg,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_7932_prod__nonneg,axiom,
    ! [A4: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_7933_prod__mono,axiom,
    ! [A4: set_o,F: $o > code_integer,G: $o > code_integer] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A4 ) @ ( groups7694694392188491536nteger @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7934_prod__mono,axiom,
    ! [A4: set_nat,F: nat > code_integer,G: nat > code_integer] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7935_prod__mono,axiom,
    ! [A4: set_int,F: int > code_integer,G: int > code_integer] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7936_prod__mono,axiom,
    ! [A4: set_o,F: $o > rat,G: $o > rat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A4 )
         => ( ( 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 @ A4 ) @ ( groups2869687844427037835_o_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7937_prod__mono,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7938_prod__mono,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7939_prod__mono,axiom,
    ! [A4: set_o,F: $o > nat,G: $o > nat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A4 )
         => ( ( 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 @ A4 ) @ ( groups3504817904513533571_o_nat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7940_prod__mono,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7941_prod__mono,axiom,
    ! [A4: set_o,F: $o > int,G: $o > int] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A4 )
         => ( ( 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 @ A4 ) @ ( groups3502327434004483295_o_int @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7942_prod__mono,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7943_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_7944_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_7945_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_7946_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_7947_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_7948_prod__pos,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_7949_prod__pos,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_7950_prod__pos,axiom,
    ! [A4: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_7951_prod__ge__1,axiom,
    ! [A4: set_o,F: $o > code_integer] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7952_prod__ge__1,axiom,
    ! [A4: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7953_prod__ge__1,axiom,
    ! [A4: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7954_prod__ge__1,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7955_prod__ge__1,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7956_prod__ge__1,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7957_prod__ge__1,axiom,
    ! [A4: set_o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups3504817904513533571_o_nat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7958_prod__ge__1,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7959_prod__ge__1,axiom,
    ! [A4: set_o,F: $o > int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7960_prod__ge__1,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7961_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
     => ( ( ord_less_eq_rat @ B @ A )
        | ( ( ord_less_eq_rat @ C2 @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_7962_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C2 @ D2 ) )
     => ( ( ord_less_eq_num @ B @ A )
        | ( ( ord_less_eq_num @ C2 @ A )
          & ( ord_less_eq_num @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_7963_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C2 @ A )
          & ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_7964_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C2 @ D2 ) )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
        | ( ( ord_le3102999989581377725nteger @ C2 @ A )
          & ( ord_le3102999989581377725nteger @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_7965_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C2 @ D2 ) )
     => ( ( ord_less_eq_int @ B @ A )
        | ( ( ord_less_eq_int @ C2 @ A )
          & ( ord_less_eq_int @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_7966_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [G: nat > nat,M: nat,K2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ K2 ) ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.atLeastAtMost_shift_bounds
thf(fact_7967_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [G: nat > int,M: nat,K2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ ( plus_plus_nat @ K2 ) ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.atLeastAtMost_shift_bounds
thf(fact_7968_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_7969_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_7970_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_7971_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_7972_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_7973_all__nat__less__eq,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N2 )
           => ( P2 @ M3 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P2 @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_7974_ex__nat__less__eq,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N2 )
            & ( P2 @ M3 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P2 @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_7975_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
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7976_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
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7977_pochhammer__prod,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ I ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_7978_pochhammer__prod,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N: nat] :
          ( groups3455450783089532116nteger
          @ ^ [I: nat] : ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ I ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_7979_pochhammer__prod,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ I ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_7980_pochhammer__prod,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N: nat] :
          ( groups705719431365010083at_int
          @ ^ [I: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ I ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_7981_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_7982_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_7983_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_7984_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [G: nat > nat,M: nat,K2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ K2 ) ) @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.atLeastLessThan_shift_bounds
thf(fact_7985_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
        @ ^ [I: nat] : ( G @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_7986_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
        @ ^ [I: nat] : ( G @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_7987_power__sum,axiom,
    ! [C2: nat,F: nat > nat,A4: set_nat] :
      ( ( power_power_nat @ C2 @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A5: nat] : ( power_power_nat @ C2 @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_7988_power__sum,axiom,
    ! [C2: int,F: nat > nat,A4: set_nat] :
      ( ( power_power_int @ C2 @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups705719431365010083at_int
        @ ^ [A5: nat] : ( power_power_int @ C2 @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_7989_power__sum,axiom,
    ! [C2: int,F: int > nat,A4: set_int] :
      ( ( power_power_int @ C2 @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A5: int] : ( power_power_int @ C2 @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_7990_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_7991_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_7992_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
        @ ^ [I: nat] : ( G @ ( suc @ I ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_7993_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_7994_prod__le__1,axiom,
    ! [A4: set_o,F: $o > code_integer] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A4 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_7995_prod__le__1,axiom,
    ! [A4: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A4 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_7996_prod__le__1,axiom,
    ! [A4: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A4 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_7997_prod__le__1,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ( 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 @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7998_prod__le__1,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7999_prod__le__1,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8000_prod__le__1,axiom,
    ! [A4: set_o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ( 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 @ A4 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_8001_prod__le__1,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_8002_prod__le__1,axiom,
    ! [A4: set_o,F: $o > int] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A4 )
         => ( ( 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 @ A4 ) @ one_one_int ) ) ).

% prod_le_1
thf(fact_8003_prod__le__1,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( 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 @ A4 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_8004_fact__split,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( semiri773545260158071498ct_rat @ N2 )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K2 ) @ N2 ) ) ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ) ).

% fact_split
thf(fact_8005_fact__split,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K2 ) @ N2 ) ) ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ) ).

% fact_split
thf(fact_8006_fact__split,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( semiri3624122377584611663nteger @ N2 )
        = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K2 ) @ N2 ) ) ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ) ).

% fact_split
thf(fact_8007_fact__split,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K2 ) @ N2 ) ) ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K2 ) ) ) ) ) ).

% fact_split
thf(fact_8008_sum_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat,G: nat > nat,H: nat > nat] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C2 @ X3 )
             => ( ( ord_less_nat @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups3542108847815614940at_nat @ H @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_8009_sum_Oivl__cong,axiom,
    ! [A: int,C2: int,B: int,D2: int,G: int > int,H: int > int] :
      ( ( A = C2 )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( ( ord_less_eq_int @ C2 @ X3 )
             => ( ( ord_less_int @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) ) )
         => ( ( groups4538972089207619220nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups4538972089207619220nt_int @ H @ ( set_or4662586982721622107an_int @ C2 @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_8010_gbinomial__altdef__of__nat,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( divide_divide_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ I ) ) @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ K3 @ I ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) ) ) ) ).

% gbinomial_altdef_of_nat
thf(fact_8011_binomial__altdef__of__nat,axiom,
    ! [K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ N2 )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K2 ) )
        = ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I ) ) @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ K2 @ I ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% binomial_altdef_of_nat
thf(fact_8012_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_8013_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_8014_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_8015_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_8016_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_8017_gbinomial__mult__fact,axiom,
    ! [K2: nat,A: rat] :
      ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ).

% gbinomial_mult_fact
thf(fact_8018_gbinomial__mult__fact_H,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ).

% gbinomial_mult_fact'
thf(fact_8019_gbinomial__prod__rev,axiom,
    ( gbinom7368847122466276068atural
    = ( ^ [A5: code_natural,K3: nat] :
          ( divide5121882707175180666atural
          @ ( groups2279045934846249631atural
            @ ^ [I: nat] : ( minus_7197305767214868737atural @ A5 @ ( semiri3763490453095760265atural @ I ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri2447717529341329178atural @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_8020_gbinomial__prod__rev,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] :
          ( divide_divide_rat
          @ ( groups73079841787564623at_rat
            @ ^ [I: nat] : ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ I ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_8021_gbinomial__prod__rev,axiom,
    ( gbinom8545251970709558553nteger
    = ( ^ [A5: code_integer,K3: nat] :
          ( divide6298287555418463151nteger
          @ ( groups3455450783089532116nteger
            @ ^ [I: nat] : ( minus_8373710615458151222nteger @ A5 @ ( semiri4939895301339042750nteger @ I ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri3624122377584611663nteger @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_8022_gbinomial__prod__rev,axiom,
    ( gbinomial_nat
    = ( ^ [A5: nat,K3: nat] :
          ( divide_divide_nat
          @ ( groups708209901874060359at_nat
            @ ^ [I: nat] : ( minus_minus_nat @ A5 @ ( semiri1316708129612266289at_nat @ I ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_8023_gbinomial__prod__rev,axiom,
    ( gbinomial_int
    = ( ^ [A5: int,K3: nat] :
          ( divide_divide_int
          @ ( groups705719431365010083at_int
            @ ^ [I: nat] : ( minus_minus_int @ A5 @ ( semiri1314217659103216013at_int @ I ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K3 ) )
          @ ( semiri1406184849735516958ct_int @ K3 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_8024_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_8025_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_8026_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_8027_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_8028_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_8029_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_8030_bset_I1_J,axiom,
    ! [D4: int,B5: set_int,P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( P2 @ X3 )
           => ( P2 @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( Q2 @ X3 )
             => ( Q2 @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
             => ( ( P2 @ ( minus_minus_int @ X6 @ D4 ) )
                & ( Q2 @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_8031_bset_I2_J,axiom,
    ! [D4: int,B5: set_int,P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( P2 @ X3 )
           => ( P2 @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( Q2 @ X3 )
             => ( Q2 @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
             => ( ( P2 @ ( minus_minus_int @ X6 @ D4 ) )
                | ( Q2 @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_8032_aset_I1_J,axiom,
    ! [D4: int,A4: set_int,P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A4 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( P2 @ X3 )
           => ( P2 @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( Q2 @ X3 )
             => ( Q2 @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( ( P2 @ X6 )
                & ( Q2 @ X6 ) )
             => ( ( P2 @ ( plus_plus_int @ X6 @ D4 ) )
                & ( Q2 @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_8033_aset_I2_J,axiom,
    ! [D4: int,A4: set_int,P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A4 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( P2 @ X3 )
           => ( P2 @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( Q2 @ X3 )
             => ( Q2 @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( ( P2 @ X6 )
                | ( Q2 @ X6 ) )
             => ( ( P2 @ ( plus_plus_int @ X6 @ D4 ) )
                | ( Q2 @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_8034_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J2 @ K2 ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J2 ) @ ( set_or4665077453230672383an_nat @ J2 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_8035_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_8036_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_8037_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_8038_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_8039_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_8040_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_8041_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_8042_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_8043_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_8044_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_8045_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: assn,B: assn,C2: assn,D2: assn] :
      ( ( ord_less_eq_set_assn @ ( set_or7959216805967363635t_assn @ A @ B ) @ ( set_or5523502641901747543n_assn @ C2 @ D2 ) )
      = ( ( ord_less_eq_assn @ A @ B )
       => ( ( ord_less_eq_assn @ C2 @ A )
          & ( ord_less_assn @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8046_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D2: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or8585797421378605585et_int @ C2 @ D2 ) )
      = ( ( ord_less_eq_set_int @ A @ B )
       => ( ( ord_less_eq_set_int @ C2 @ A )
          & ( ord_less_set_int @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8047_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
      = ( ( ord_less_eq_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C2 @ A )
          & ( ord_less_rat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8048_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: num,B: num,C2: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C2 @ D2 ) )
      = ( ( ord_less_eq_num @ A @ B )
       => ( ( ord_less_eq_num @ C2 @ A )
          & ( ord_less_num @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8049_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
      = ( ( ord_less_eq_nat @ A @ B )
       => ( ( ord_less_eq_nat @ C2 @ A )
          & ( ord_less_nat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8050_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C2 @ D2 ) )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ( ord_le3102999989581377725nteger @ C2 @ A )
          & ( ord_le6747313008572928689nteger @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8051_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C2 @ D2 ) )
      = ( ( ord_less_eq_int @ A @ B )
       => ( ( ord_less_eq_int @ C2 @ A )
          & ( ord_less_int @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8052_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D2 ) )
      = ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C2 @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_8053_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_8054_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_8055_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_8056_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_8057_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_8058_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8059_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8060_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8061_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8062_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat ) @ ( groups6857163185585827899at_nat @ ( comp_n8698576032424989604at_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_8063_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat ) @ ( groups2906978787729119204at_rat @ ( comp_nat_rat_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_8064_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat ) @ ( groups3539618377306564664at_int @ ( comp_nat_int_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_8065_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat ) @ ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_8066_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_8067_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_8068_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_8069_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_8070_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_8071_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_8072_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_8073_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_8074_sum_OatLeastLessThan__shift__0,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ M ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% sum.atLeastLessThan_shift_0
thf(fact_8075_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_8076_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_8077_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_8078_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_8079_aset_I10_J,axiom,
    ! [D2: int,D4: int,A4: set_int,T6: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T6 ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X6 @ D4 ) @ T6 ) ) ) ) ) ).

% aset(10)
thf(fact_8080_aset_I9_J,axiom,
    ! [D2: int,D4: int,A4: set_int,T6: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T6 ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X6 @ D4 ) @ T6 ) ) ) ) ) ).

% aset(9)
thf(fact_8081_bset_I10_J,axiom,
    ! [D2: int,D4: int,B5: set_int,T6: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T6 ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ D4 ) @ T6 ) ) ) ) ) ).

% bset(10)
thf(fact_8082_bset_I9_J,axiom,
    ! [D2: int,D4: int,B5: set_int,T6: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T6 ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ D4 ) @ T6 ) ) ) ) ) ).

% bset(9)
thf(fact_8083_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
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8084_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
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8085_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
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8086_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
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8087_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_8088_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8089_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8090_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8091_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8092_prod_OatLeast__atMost__pred__shift,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ( comp_nat_nat_nat @ G
          @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.atLeast_atMost_pred_shift
thf(fact_8093_prod_OatLeast__atMost__pred__shift,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ( comp_nat_int_nat @ G
          @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.atLeast_atMost_pred_shift
thf(fact_8094_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_8095_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_8096_fact__prod,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [N: nat] :
          ( semiri4939895301339042750nteger
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_8097_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_8098_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( minus_minus_rat @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_8099_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_8100_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ( comp_nat_nat_nat @ G
          @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.atLeast_lessThan_pred_shift
thf(fact_8101_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_8102_sum_Onested__swap,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( A @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% sum.nested_swap
thf(fact_8103_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_8104_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_8105_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_8106_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_8107_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_8108_prod_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ M ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% prod.atLeastAtMost_shift_0
thf(fact_8109_prod_OatLeastAtMost__shift__0,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ ( plus_plus_nat @ M ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% prod.atLeastAtMost_shift_0
thf(fact_8110_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_8111_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_8112_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_8113_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_8114_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: $o,U: $o] :
      ( ( ord_less_eq_o @ L @ U )
     => ( ( sup_sup_set_o @ ( set_or7139685690850216873Than_o @ L @ U ) @ ( insert_o @ U @ bot_bot_set_o ) )
        = ( set_or8904488021354931149Most_o @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_8115_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ U ) @ ( insert_rat @ U @ bot_bot_set_rat ) )
        = ( set_or633870826150836451st_rat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_8116_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_eq_num @ L @ U )
     => ( ( sup_sup_set_num @ ( set_or1222409239386451017an_num @ L @ U ) @ ( insert_num @ U @ bot_bot_set_num ) )
        = ( set_or7049704709247886629st_num @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_8117_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
        = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_8118_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ U ) @ ( insert_Code_integer @ U @ bot_bo3990330152332043303nteger ) )
        = ( set_or189985376899183464nteger @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_8119_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_eq_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ U ) @ ( insert_int @ U @ bot_bot_set_int ) )
        = ( set_or1266510415728281911st_int @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_8120_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_8121_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_8122_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_8123_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_8124_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_8125_periodic__finite__ex,axiom,
    ! [D2: int,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K: int] :
            ( ( P2 @ X3 )
            = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
       => ( ( ? [X11: int] : ( P2 @ X11 ) )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_8126_bset_I3_J,axiom,
    ! [D4: int,T6: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T6 @ one_one_int ) @ B5 )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( X6 = T6 )
             => ( ( minus_minus_int @ X6 @ D4 )
                = T6 ) ) ) ) ) ).

% bset(3)
thf(fact_8127_bset_I4_J,axiom,
    ! [D4: int,T6: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T6 @ B5 )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( X6 != T6 )
             => ( ( minus_minus_int @ X6 @ D4 )
               != T6 ) ) ) ) ) ).

% bset(4)
thf(fact_8128_bset_I5_J,axiom,
    ! [D4: int,B5: set_int,T6: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
         => ( ( ord_less_int @ X6 @ T6 )
           => ( ord_less_int @ ( minus_minus_int @ X6 @ D4 ) @ T6 ) ) ) ) ).

% bset(5)
thf(fact_8129_bset_I7_J,axiom,
    ! [D4: int,T6: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T6 @ B5 )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( ord_less_int @ T6 @ X6 )
             => ( ord_less_int @ T6 @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_8130_aset_I3_J,axiom,
    ! [D4: int,T6: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T6 @ one_one_int ) @ A4 )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( X6 = T6 )
             => ( ( plus_plus_int @ X6 @ D4 )
                = T6 ) ) ) ) ) ).

% aset(3)
thf(fact_8131_aset_I4_J,axiom,
    ! [D4: int,T6: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T6 @ A4 )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( X6 != T6 )
             => ( ( plus_plus_int @ X6 @ D4 )
               != T6 ) ) ) ) ) ).

% aset(4)
thf(fact_8132_aset_I5_J,axiom,
    ! [D4: int,T6: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T6 @ A4 )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( ord_less_int @ X6 @ T6 )
             => ( ord_less_int @ ( plus_plus_int @ X6 @ D4 ) @ T6 ) ) ) ) ) ).

% aset(5)
thf(fact_8133_aset_I7_J,axiom,
    ! [D4: int,A4: set_int,T6: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
         => ( ( ord_less_int @ T6 @ X6 )
           => ( ord_less_int @ T6 @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_8134_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K2: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K2 ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K2 ) )
          = ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K2 ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K2 ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_8135_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_8136_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
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_8137_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I: int,J: int] : ( if_set_int @ ( ord_less_int @ J @ I ) @ bot_bot_set_int @ ( insert_int @ I @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ) ).

% simp_from_to
thf(fact_8138_aset_I8_J,axiom,
    ! [D4: int,A4: set_int,T6: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
         => ( ( ord_less_eq_int @ T6 @ X6 )
           => ( ord_less_eq_int @ T6 @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_8139_aset_I6_J,axiom,
    ! [D4: int,T6: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T6 @ one_one_int ) @ A4 )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( ord_less_eq_int @ X6 @ T6 )
             => ( ord_less_eq_int @ ( plus_plus_int @ X6 @ D4 ) @ T6 ) ) ) ) ) ).

% aset(6)
thf(fact_8140_bset_I8_J,axiom,
    ! [D4: int,T6: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T6 @ one_one_int ) @ B5 )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
           => ( ( ord_less_eq_int @ T6 @ X6 )
             => ( ord_less_eq_int @ T6 @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_8141_bset_I6_J,axiom,
    ! [D4: int,B5: set_int,T6: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( ( member_int @ Xa4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa4 ) ) ) )
         => ( ( ord_less_eq_int @ X6 @ T6 )
           => ( ord_less_eq_int @ ( minus_minus_int @ X6 @ D4 ) @ T6 ) ) ) ) ).

% bset(6)
thf(fact_8142_cpmi,axiom,
    ! [D4: int,P2: int > $o,P6: int > $o,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z6: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z6 )
           => ( ( P2 @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa3: int] :
                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B5 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
             => ( ( P2 @ X3 )
               => ( P2 @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D4 ) ) ) )
           => ( ( ? [X11: int] : ( P2 @ X11 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ B5 )
                        & ( P2 @ ( plus_plus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_8143_cppi,axiom,
    ! [D4: int,P2: int > $o,P6: int > $o,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z6: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z6 @ X3 )
           => ( ( P2 @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa3: int] :
                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A4 )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
             => ( ( P2 @ X3 )
               => ( P2 @ ( plus_plus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D4 ) ) ) )
           => ( ( ? [X11: int] : ( P2 @ X11 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ A4 )
                        & ( P2 @ ( minus_minus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_8144_pochhammer__Suc__prod,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8145_pochhammer__Suc__prod,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( groups3455450783089532116nteger
        @ ^ [I: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8146_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8147_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8148_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8149_pochhammer__prod__rev,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N: nat] :
          ( groups3455450783089532116nteger
          @ ^ [I: nat] : ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N @ I ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8150_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8151_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N: nat] :
          ( groups705719431365010083at_int
          @ ^ [I: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8152_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_8153_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
        @ ^ [I: nat] : ( times_times_assn @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_8154_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
        @ ^ [I: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_8155_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
        @ ^ [I: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_8156_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
        @ ^ [I: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_8157_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
        @ ^ [I: nat] : ( times_times_assn @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8158_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
        @ ^ [I: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8159_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
        @ ^ [I: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8160_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
        @ ^ [I: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8161_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8162_pochhammer__Suc__prod__rev,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( groups3455450783089532116nteger
        @ ^ [I: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N2 @ I ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8163_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8164_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8165_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
                @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S5 ) ) )
                @ ( 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
                    @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S5 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_8166_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_8167_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_8168_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_8169_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_8170_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_8171_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_8172_horner__sum__eq__sum,axiom,
    ( groups9119017779487936845_o_nat
    = ( ^ [F3: $o > nat,A5: nat,Xs3: list_o] :
          ( groups3542108847815614940at_nat
          @ ^ [N: nat] : ( times_times_nat @ ( F3 @ ( nth_o @ Xs3 @ N ) ) @ ( power_power_nat @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8173_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_8174_horner__sum__eq__sum,axiom,
    ( groups9116527308978886569_o_int
    = ( ^ [F3: $o > int,A5: int,Xs3: list_o] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( times_times_int @ ( F3 @ ( nth_o @ Xs3 @ N ) ) @ ( power_power_int @ A5 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs3 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8175_prod_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups3455450783089532116nteger
            @ ^ [J: nat] : ( if_Code_integer @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_Code_integer @ ( J = K2 ) @ one_one_Code_integer @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups3455450783089532116nteger
            @ ^ [J: nat] : ( if_Code_integer @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8176_prod_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > assn,H: nat > assn] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups6906906614972039071t_assn
            @ ^ [J: nat] : ( if_assn @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_assn @ ( J = K2 ) @ one_one_assn @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups6906906614972039071t_assn
            @ ^ [J: nat] : ( if_assn @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8177_prod_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > rat,H: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J: nat] : ( if_rat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_rat @ ( J = K2 ) @ one_one_rat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J: nat] : ( if_rat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8178_prod_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_nat @ ( J = K2 ) @ one_one_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8179_prod_Ozero__middle,axiom,
    ! [P3: nat,K2: nat,G: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P3 )
     => ( ( ord_less_eq_nat @ K2 @ P3 )
       => ( ( groups705719431365010083at_int
            @ ^ [J: nat] : ( if_int @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( if_int @ ( J = K2 ) @ one_one_int @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P3 ) )
          = ( groups705719431365010083at_int
            @ ^ [J: nat] : ( if_int @ ( ord_less_nat @ J @ K2 ) @ ( G @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8180_gbinomial__Suc,axiom,
    ! [A: code_natural,K2: nat] :
      ( ( gbinom7368847122466276068atural @ A @ ( suc @ K2 ) )
      = ( divide5121882707175180666atural
        @ ( groups2279045934846249631atural
          @ ^ [I: nat] : ( minus_7197305767214868737atural @ A @ ( semiri3763490453095760265atural @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri2447717529341329178atural @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8181_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K2 ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8182_gbinomial__Suc,axiom,
    ! [A: code_integer,K2: nat] :
      ( ( gbinom8545251970709558553nteger @ A @ ( suc @ K2 ) )
      = ( divide6298287555418463151nteger
        @ ( groups3455450783089532116nteger
          @ ^ [I: nat] : ( minus_8373710615458151222nteger @ A @ ( semiri4939895301339042750nteger @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri3624122377584611663nteger @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8183_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K2 ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8184_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K2 ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8185_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_8186_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_8187_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I: nat,J: nat] : ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( G @ I @ ( minus_minus_nat @ K3 @ I ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_8188_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_8189_lessThan__iff,axiom,
    ! [I2: $o,K2: $o] :
      ( ( member_o @ I2 @ ( set_ord_lessThan_o @ K2 ) )
      = ( ord_less_o @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_8190_lessThan__iff,axiom,
    ! [I2: set_Pr958786334691620121nt_int,K2: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ I2 @ ( set_or4940836740269066044nt_int @ K2 ) )
      = ( ord_le7563427860532173253nt_int @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_8191_lessThan__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or890127255671739683et_nat @ K2 ) )
      = ( ord_less_set_nat @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_8192_lessThan__iff,axiom,
    ! [I2: assn,K2: assn] :
      ( ( member_assn @ I2 @ ( set_or7637083652282234053n_assn @ K2 ) )
      = ( ord_less_assn @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_8193_lessThan__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K2 ) )
      = ( ord_less_rat @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_8194_lessThan__iff,axiom,
    ! [I2: num,K2: num] :
      ( ( member_num @ I2 @ ( set_ord_lessThan_num @ K2 ) )
      = ( ord_less_num @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_8195_lessThan__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K2 ) )
      = ( ord_less_nat @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_8196_lessThan__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( member_int @ I2 @ ( set_ord_lessThan_int @ K2 ) )
      = ( ord_less_int @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_8197_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_8198_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_8199_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_8200_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_8201_sum_OlessThan__Suc,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_8202_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_8203_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_8204_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_8205_int__prod,axiom,
    ! [F: int > nat,A4: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% int_prod
thf(fact_8206_int__prod,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A4 ) )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( semiri1314217659103216013at_int @ ( F @ X4 ) )
        @ A4 ) ) ).

% int_prod
thf(fact_8207_lessThan__def,axiom,
    ( set_or4940836740269066044nt_int
    = ( ^ [U3: set_Pr958786334691620121nt_int] :
          ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( ord_le7563427860532173253nt_int @ X4 @ U3 ) ) ) ) ).

% lessThan_def
thf(fact_8208_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U3: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_set_nat @ X4 @ U3 ) ) ) ) ).

% lessThan_def
thf(fact_8209_lessThan__def,axiom,
    ( set_or7637083652282234053n_assn
    = ( ^ [U3: assn] :
          ( collect_assn
          @ ^ [X4: assn] : ( ord_less_assn @ X4 @ U3 ) ) ) ) ).

% lessThan_def
thf(fact_8210_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U3: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_rat @ X4 @ U3 ) ) ) ) ).

% lessThan_def
thf(fact_8211_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U3: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_num @ X4 @ U3 ) ) ) ) ).

% lessThan_def
thf(fact_8212_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U3: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ X4 @ U3 ) ) ) ) ).

% lessThan_def
thf(fact_8213_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U3: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_int @ X4 @ U3 ) ) ) ) ).

% lessThan_def
thf(fact_8214_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_8215_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_8216_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_8217_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_8218_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_8219_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_8220_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8221_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_8222_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_8223_prod__int__eq,axiom,
    ! [I2: nat,J2: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ J2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ) ).

% prod_int_eq
thf(fact_8224_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_8225_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_8226_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_8227_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_8228_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_8229_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I: nat,J: nat] : ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I: nat] : ( G @ I @ ( minus_minus_nat @ K3 @ I ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_8230_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I: nat,J: nat] : ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I: nat] : ( G @ I @ ( minus_minus_nat @ K3 @ I ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_8231_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_8232_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_8233_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_8234_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_8235_sum__diff__distrib,axiom,
    ! [Q2: int > nat,P2: int > nat,N2: int] :
      ( ! [X3: int] : ( ord_less_eq_nat @ ( Q2 @ X3 ) @ ( P2 @ X3 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P2 @ ( set_ord_lessThan_int @ N2 ) ) @ ( groups4541462559716669496nt_nat @ Q2 @ ( set_ord_lessThan_int @ N2 ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( P2 @ X4 ) @ ( Q2 @ X4 ) )
          @ ( set_ord_lessThan_int @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8236_sum__diff__distrib,axiom,
    ! [Q2: nat > nat,P2: nat > nat,N2: nat] :
      ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q2 @ X3 ) @ ( P2 @ X3 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P2 @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q2 @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( P2 @ X4 ) @ ( Q2 @ X4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8237_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8238_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8239_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8240_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8241_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_8242_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_8243_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_8244_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_8245_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8246_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8247_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8248_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8249_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_8250_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_8251_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_8252_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_8253_prod__int__plus__eq,axiom,
    ! [I2: nat,J2: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J2 ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J2 ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_8254_sum_Onat__group,axiom,
    ! [G: nat > nat,K2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [M3: nat] : ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M3 @ K2 ) @ ( plus_plus_nat @ ( times_times_nat @ M3 @ K2 ) @ K2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K2 ) ) ) ) ).

% sum.nat_group
thf(fact_8255_prod_Onat__group,axiom,
    ! [G: nat > nat,K2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [M3: nat] : ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M3 @ K2 ) @ ( plus_plus_nat @ ( times_times_nat @ M3 @ K2 ) @ K2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K2 ) ) ) ) ).

% prod.nat_group
thf(fact_8256_prod_Onat__group,axiom,
    ! [G: nat > int,K2: nat,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [M3: nat] : ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M3 @ K2 ) @ ( plus_plus_nat @ ( times_times_nat @ M3 @ K2 ) @ K2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K2 ) ) ) ) ).

% prod.nat_group
thf(fact_8257_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( A @ I ) @ ( set_ord_lessThan_nat @ I ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_8258_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( groups708209901874060359at_nat @ ( A @ I ) @ ( set_ord_lessThan_nat @ I ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_8259_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I: nat] : ( groups705719431365010083at_int @ ( A @ I ) @ ( set_ord_lessThan_nat @ I ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J: nat] :
            ( groups705719431365010083at_int
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_8260_Iio__Int__singleton,axiom,
    ! [X: $o,K2: $o] :
      ( ( ( ord_less_o @ X @ K2 )
       => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
          = ( insert_o @ X @ bot_bot_set_o ) ) )
      & ( ~ ( ord_less_o @ X @ K2 )
       => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
          = bot_bot_set_o ) ) ) ).

% Iio_Int_singleton
thf(fact_8261_Iio__Int__singleton,axiom,
    ! [X: assn,K2: assn] :
      ( ( ( ord_less_assn @ X @ K2 )
       => ( ( inf_inf_set_assn @ ( set_or7637083652282234053n_assn @ K2 ) @ ( insert_assn @ X @ bot_bot_set_assn ) )
          = ( insert_assn @ X @ bot_bot_set_assn ) ) )
      & ( ~ ( ord_less_assn @ X @ K2 )
       => ( ( inf_inf_set_assn @ ( set_or7637083652282234053n_assn @ K2 ) @ ( insert_assn @ X @ bot_bot_set_assn ) )
          = bot_bot_set_assn ) ) ) ).

% Iio_Int_singleton
thf(fact_8262_Iio__Int__singleton,axiom,
    ! [X: rat,K2: rat] :
      ( ( ( ord_less_rat @ X @ K2 )
       => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K2 ) @ ( insert_rat @ X @ bot_bot_set_rat ) )
          = ( insert_rat @ X @ bot_bot_set_rat ) ) )
      & ( ~ ( ord_less_rat @ X @ K2 )
       => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K2 ) @ ( insert_rat @ X @ bot_bot_set_rat ) )
          = bot_bot_set_rat ) ) ) ).

% Iio_Int_singleton
thf(fact_8263_Iio__Int__singleton,axiom,
    ! [X: num,K2: num] :
      ( ( ( ord_less_num @ X @ K2 )
       => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K2 ) @ ( insert_num @ X @ bot_bot_set_num ) )
          = ( insert_num @ X @ bot_bot_set_num ) ) )
      & ( ~ ( ord_less_num @ X @ K2 )
       => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K2 ) @ ( insert_num @ X @ bot_bot_set_num ) )
          = bot_bot_set_num ) ) ) ).

% Iio_Int_singleton
thf(fact_8264_Iio__Int__singleton,axiom,
    ! [X: nat,K2: nat] :
      ( ( ( ord_less_nat @ X @ K2 )
       => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) )
      & ( ~ ( ord_less_nat @ X @ K2 )
       => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = bot_bot_set_nat ) ) ) ).

% Iio_Int_singleton
thf(fact_8265_Iio__Int__singleton,axiom,
    ! [X: int,K2: int] :
      ( ( ( ord_less_int @ X @ K2 )
       => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) )
      & ( ~ ( ord_less_int @ X @ K2 )
       => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
          = bot_bot_set_int ) ) ) ).

% Iio_Int_singleton
thf(fact_8266_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: $o] :
      ( ( sup_sup_set_o @ ( set_ord_lessThan_o @ U ) @ ( insert_o @ U @ bot_bot_set_o ) )
      = ( set_ord_atMost_o @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_8267_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: nat] :
      ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
      = ( set_ord_atMost_nat @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_8268_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: int] :
      ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ U ) @ ( insert_int @ U @ bot_bot_set_int ) )
      = ( set_ord_atMost_int @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_8269_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_8270_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_8271_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_8272_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_8273_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_8274_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_8275_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_8276_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_8277_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_8278_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_8279_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_8280_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_8281_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8282_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8283_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8284_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8285_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8286_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8287_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8288_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
          @ ^ [I: nat] : ( G @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8289_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_8290_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
          @ ^ [I: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) ) @ ( power_power_rat @ X @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8291_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
          @ ^ [I: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) ) @ ( power_power_int @ X @ I ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8292_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_8293_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_8294_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
          @ ^ [I: nat] : ( power_8256067586552552935nteger @ X @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8295_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
          @ ^ [I: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8296_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
          @ ^ [I: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8297_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I: nat,J: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I @ J ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I: nat] : ( G @ I @ ( minus_minus_nat @ K3 @ I ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8298_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I: nat,J: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I @ J ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I: nat] : ( G @ I @ ( minus_minus_nat @ K3 @ I ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8299_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_8300_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_8301_signed__take__bit__code,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N: nat,A5: code_integer] : ( if_Code_integer @ ( bit_se9216721137139052372nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A5 ) @ N ) @ ( plus_p5714425477246183910nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A5 ) @ ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) @ ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A5 ) ) ) ) ).

% signed_take_bit_code
thf(fact_8302_signed__take__bit__code,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,A5: int] : ( if_int @ ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A5 ) @ N ) @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A5 ) @ ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) ) ) @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A5 ) ) ) ) ).

% signed_take_bit_code
thf(fact_8303_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: nat > rat,D2: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) ) @ ( D2 @ I ) )
            @ A4 )
          = ( divide_divide_rat @ ( C2 @ zero_zero_nat ) @ ( D2 @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) ) @ ( D2 @ I ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_8304_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( 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_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_8305_execute__of__list,axiom,
    ! [Xs: list_nat,H: heap_e7401611519738050253t_unit] :
      ( ( heap_T2203789307644377761ay_nat @ ( array_of_list_nat @ Xs ) @ H )
      = ( some_P8206036975937309155it_nat
        @ ( produc8657510640128716596it_nat
          @ ^ [R5: array_nat,H7: heap_e7401611519738050253t_unit] : ( produc4907525368653469954it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_nat @ Xs ) ) ) )
          @ ( array_alloc_nat @ Xs @ H ) ) ) ) ).

% execute_of_list
thf(fact_8306_execute__of__list,axiom,
    ! [Xs: list_o,H: heap_e7401611519738050253t_unit] :
      ( ( heap_T8140152426084121245rray_o @ ( array_of_list_o @ Xs ) @ H )
      = ( some_P3509045262911171395it_nat
        @ ( produc5512802776183222702it_nat
          @ ^ [R5: array_o,H7: heap_e7401611519738050253t_unit] : ( produc2890842502232952598it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_o @ Xs ) ) ) )
          @ ( array_alloc_o @ Xs @ H ) ) ) ) ).

% execute_of_list
thf(fact_8307_execute__of__list,axiom,
    ! [Xs: list_int,H: heap_e7401611519738050253t_unit] :
      ( ( heap_T7249310324989956861ay_int @ ( array_of_list_int @ Xs ) @ H )
      = ( some_P8650484732927216191it_nat
        @ ( produc3016605123915955252it_nat
          @ ^ [R5: array_int,H7: heap_e7401611519738050253t_unit] : ( produc5351973125643376990it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_int @ Xs ) ) ) )
          @ ( array_alloc_int @ Xs @ H ) ) ) ) ).

% execute_of_list
thf(fact_8308_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_8309_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_8310_push__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ A ) )
      = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_8311_push__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
      = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_8312_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_8313_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_8314_or__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K2 @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_8315_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K2 @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K2 @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_8316_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_8317_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_8318_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_8319_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_8320_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_8321_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_8322_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_8323_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_8324_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_8325_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_8326_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_8327_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_8328_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_8329_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_8330_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_8331_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_8332_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_8333_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_8334_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_8335_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_8336_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_8337_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_8338_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_8339_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_8340_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_8341_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_8342_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_8343_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_8344_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_8345_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_8346_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_8347_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_8348_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_8349_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_8350_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_8351_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_8352_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_8353_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_8354_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_8355_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_8356_sum_Oinsert,axiom,
    ! [A4: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X @ A4 )
       => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A4 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8357_sum_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A4 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8358_sum_Oinsert,axiom,
    ! [A4: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A4 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8359_sum_Oinsert,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ~ ( member_Code_integer @ X @ A4 )
       => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A4 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8360_sum_Oinsert,axiom,
    ! [A4: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X @ A4 )
       => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A4 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8361_sum_Oinsert,axiom,
    ! [A4: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A4 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8362_sum_Oinsert,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ~ ( member_Code_integer @ X @ A4 )
       => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A4 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8363_sum_Oinsert,axiom,
    ! [A4: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X @ A4 )
       => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X @ A4 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8364_sum_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A4 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8365_sum_Oinsert,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ~ ( member_Code_integer @ X @ A4 )
       => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A4 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_8366_prod__pos__nat__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8367_prod__pos__nat__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups3190895334310489300er_nat @ F @ A4 ) )
        = ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8368_prod__pos__nat__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) )
        = ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8369_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8370_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: int > $o,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8371_sum__of__bool__mult__eq,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups6602215022474089585er_rat @ F @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8372_sum__of__bool__mult__eq,axiom,
    ! [A4: set_nat,P2: nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8373_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: int > $o,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8374_sum__of__bool__mult__eq,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat
          @ ^ [X4: code_integer] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups7237345082560585321er_nat @ F @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8375_sum__of__bool__mult__eq,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7234854612051535045er_int
          @ ^ [X4: code_integer] : ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups7234854612051535045er_int @ F @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8376_sum__of__bool__mult__eq,axiom,
    ! [A4: set_nat,P2: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3539618377306564664at_int
          @ ^ [X4: nat] : ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8377_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: int > $o,F: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7873554091576472773nteger
          @ ^ [X4: int] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups7873554091576472773nteger @ F @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8378_sum__of__bool__mult__eq,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups879477027807139574nteger
          @ ^ [X4: code_integer] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups879477027807139574nteger @ F @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8379_sum__of__bool__mult__eq,axiom,
    ! [A4: set_nat,P2: nat > $o,F: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups7501900531339628137nteger
          @ ^ [X4: nat] : ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( P2 @ X4 ) ) @ ( F @ X4 ) )
          @ A4 )
        = ( groups7501900531339628137nteger @ F @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_8380_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F: int > rat,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( times_times_rat @ ( F @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8381_sum__mult__of__bool__eq,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( times_times_rat @ ( F @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups6602215022474089585er_rat @ F @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8382_sum__mult__of__bool__eq,axiom,
    ! [A4: set_nat,F: nat > rat,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( times_times_rat @ ( F @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8383_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F: int > nat,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( times_times_nat @ ( F @ X4 ) @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8384_sum__mult__of__bool__eq,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat
          @ ^ [X4: code_integer] : ( times_times_nat @ ( F @ X4 ) @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups7237345082560585321er_nat @ F @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8385_sum__mult__of__bool__eq,axiom,
    ! [A4: set_Code_integer,F: code_integer > int,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7234854612051535045er_int
          @ ^ [X4: code_integer] : ( times_times_int @ ( F @ X4 ) @ ( zero_n2684676970156552555ol_int @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups7234854612051535045er_int @ F @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8386_sum__mult__of__bool__eq,axiom,
    ! [A4: set_nat,F: nat > int,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3539618377306564664at_int
          @ ^ [X4: nat] : ( times_times_int @ ( F @ X4 ) @ ( zero_n2684676970156552555ol_int @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8387_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F: int > code_integer,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7873554091576472773nteger
          @ ^ [X4: int] : ( times_3573771949741848930nteger @ ( F @ X4 ) @ ( zero_n356916108424825756nteger @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups7873554091576472773nteger @ F @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8388_sum__mult__of__bool__eq,axiom,
    ! [A4: set_Code_integer,F: code_integer > code_integer,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups879477027807139574nteger
          @ ^ [X4: code_integer] : ( times_3573771949741848930nteger @ ( F @ X4 ) @ ( zero_n356916108424825756nteger @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups879477027807139574nteger @ F @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8389_sum__mult__of__bool__eq,axiom,
    ! [A4: set_nat,F: nat > code_integer,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups7501900531339628137nteger
          @ ^ [X4: nat] : ( times_3573771949741848930nteger @ ( F @ X4 ) @ ( zero_n356916108424825756nteger @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups7501900531339628137nteger @ F @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_8390_sum__zero__power,axiom,
    ! [A4: set_nat,C2: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) )
            @ A4 )
          = ( C2 @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_8391_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_8392_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_8393_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_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_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_8394_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_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_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_8395_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_8396_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_8397_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_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_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_8398_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_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_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_8399_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_8400_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_8401_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_8402_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_8403_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_8404_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_8405_push__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_8406_push__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_8407_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ S )
     => ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [S5: set_Pr958786334691620121nt_int] :
              ( ( minus_1052850069191792384nt_int @ S5 @ S )
              = ( minus_1052850069191792384nt_int @ S3 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8408_finite__if__eq__beyond__finite,axiom,
    ! [S: set_nat,S3: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S5: set_nat] :
              ( ( minus_minus_set_nat @ S5 @ S )
              = ( minus_minus_set_nat @ S3 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8409_finite__if__eq__beyond__finite,axiom,
    ! [S: set_int,S3: set_int] :
      ( ( finite_finite_int @ S )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S5: set_int] :
              ( ( minus_minus_set_int @ S5 @ S )
              = ( minus_minus_set_int @ S3 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8410_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Code_integer,S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S5: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S5 @ S )
              = ( minus_2355218937544613996nteger @ S3 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8411_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [S5: set_Pr1261947904930325089at_nat] :
              ( ( minus_1356011639430497352at_nat @ S5 @ S )
              = ( minus_1356011639430497352at_nat @ S3 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8412_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_8413_finite__M__bounded__by__nat,axiom,
    ! [P2: nat > $o,I2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P2 @ K3 )
            & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_8414_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_8415_sum_Oswap__restrict,axiom,
    ! [A4: set_o,B5: set_nat,G: $o > nat > nat,R3: $o > nat > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups8507830703676809646_o_nat
            @ ^ [X4: $o] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups8507830703676809646_o_nat
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8416_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B5: set_nat,G: int > nat > nat,R3: int > nat > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X4: int] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X4: int] : ( G @ X4 @ Y4 )
                @ ( collect_int
                  @ ^ [X4: int] :
                      ( ( member_int @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8417_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B5: set_nat,G: code_integer > nat > nat,R3: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups7237345082560585321er_nat
            @ ^ [X4: code_integer] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups7237345082560585321er_nat
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8418_sum_Oswap__restrict,axiom,
    ! [A4: set_o,B5: set_int,G: $o > int > int,R3: $o > int > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups8505340233167759370_o_int
            @ ^ [X4: $o] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y4: int] :
                ( groups8505340233167759370_o_int
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8419_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B5: set_int,G: nat > int > int,R3: nat > int > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups3539618377306564664at_int
            @ ^ [X4: nat] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y4: int] :
                ( groups3539618377306564664at_int
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8420_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B5: set_int,G: code_integer > int > int,R3: code_integer > int > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups7234854612051535045er_int
            @ ^ [X4: code_integer] :
                ( groups4538972089207619220nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y4: int] :
                ( groups7234854612051535045er_int
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8421_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B5: set_o,G: nat > $o > nat,R3: nat > $o > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_o @ B5 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] :
                ( groups8507830703676809646_o_nat @ ( G @ X4 )
                @ ( collect_o
                  @ ^ [Y4: $o] :
                      ( ( member_o @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups8507830703676809646_o_nat
            @ ^ [Y4: $o] :
                ( groups3542108847815614940at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8422_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B5: set_int,G: nat > int > nat,R3: nat > int > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] :
                ( groups4541462559716669496nt_nat @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups4541462559716669496nt_nat
            @ ^ [Y4: int] :
                ( groups3542108847815614940at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8423_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B5: set_Code_integer,G: nat > code_integer > nat,R3: nat > code_integer > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] :
                ( groups7237345082560585321er_nat @ ( G @ X4 )
                @ ( collect_Code_integer
                  @ ^ [Y4: code_integer] :
                      ( ( member_Code_integer @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups7237345082560585321er_nat
            @ ^ [Y4: code_integer] :
                ( groups3542108847815614940at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8424_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > nat > nat,R3: nat > nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] :
                ( groups3542108847815614940at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y4: nat] :
                ( groups3542108847815614940at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8425_prod_Oswap__restrict,axiom,
    ! [A4: set_o,B5: set_nat,G: $o > nat > nat,R3: $o > nat > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3504817904513533571_o_nat
            @ ^ [X4: $o] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups708209901874060359at_nat
            @ ^ [Y4: nat] :
                ( groups3504817904513533571_o_nat
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8426_prod_Oswap__restrict,axiom,
    ! [A4: set_int,B5: set_nat,G: int > nat > nat,R3: int > nat > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups1707563613775114915nt_nat
            @ ^ [X4: int] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups708209901874060359at_nat
            @ ^ [Y4: nat] :
                ( groups1707563613775114915nt_nat
                @ ^ [X4: int] : ( G @ X4 @ Y4 )
                @ ( collect_int
                  @ ^ [X4: int] :
                      ( ( member_int @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8427_prod_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B5: set_nat,G: code_integer > nat > nat,R3: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3190895334310489300er_nat
            @ ^ [X4: code_integer] :
                ( groups708209901874060359at_nat @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups708209901874060359at_nat
            @ ^ [Y4: nat] :
                ( groups3190895334310489300er_nat
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8428_prod_Oswap__restrict,axiom,
    ! [A4: set_o,B5: set_nat,G: $o > nat > int,R3: $o > nat > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3502327434004483295_o_int
            @ ^ [X4: $o] :
                ( groups705719431365010083at_int @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups705719431365010083at_int
            @ ^ [Y4: nat] :
                ( groups3502327434004483295_o_int
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8429_prod_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B5: set_nat,G: code_integer > nat > int,R3: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3188404863801439024er_int
            @ ^ [X4: code_integer] :
                ( groups705719431365010083at_int @ ( G @ X4 )
                @ ( collect_nat
                  @ ^ [Y4: nat] :
                      ( ( member_nat @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups705719431365010083at_int
            @ ^ [Y4: nat] :
                ( groups3188404863801439024er_int
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8430_prod_Oswap__restrict,axiom,
    ! [A4: set_o,B5: set_int,G: $o > int > int,R3: $o > int > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups3502327434004483295_o_int
            @ ^ [X4: $o] :
                ( groups1705073143266064639nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y4: int] :
                ( groups3502327434004483295_o_int
                @ ^ [X4: $o] : ( G @ X4 @ Y4 )
                @ ( collect_o
                  @ ^ [X4: $o] :
                      ( ( member_o @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8431_prod_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B5: set_int,G: code_integer > int > int,R3: code_integer > int > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups3188404863801439024er_int
            @ ^ [X4: code_integer] :
                ( groups1705073143266064639nt_int @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y4: int] :
                ( groups3188404863801439024er_int
                @ ^ [X4: code_integer] : ( G @ X4 @ Y4 )
                @ ( collect_Code_integer
                  @ ^ [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8432_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B5: set_o,G: nat > $o > nat,R3: nat > $o > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_o @ B5 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X4: nat] :
                ( groups3504817904513533571_o_nat @ ( G @ X4 )
                @ ( collect_o
                  @ ^ [Y4: $o] :
                      ( ( member_o @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3504817904513533571_o_nat
            @ ^ [Y4: $o] :
                ( groups708209901874060359at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8433_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B5: set_int,G: nat > int > nat,R3: nat > int > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X4: nat] :
                ( groups1707563613775114915nt_nat @ ( G @ X4 )
                @ ( collect_int
                  @ ^ [Y4: int] :
                      ( ( member_int @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups1707563613775114915nt_nat
            @ ^ [Y4: int] :
                ( groups708209901874060359at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8434_prod_Oswap__restrict,axiom,
    ! [A4: set_nat,B5: set_Code_integer,G: nat > code_integer > nat,R3: nat > code_integer > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X4: nat] :
                ( groups3190895334310489300er_nat @ ( G @ X4 )
                @ ( collect_Code_integer
                  @ ^ [Y4: code_integer] :
                      ( ( member_Code_integer @ Y4 @ B5 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ A4 )
          = ( groups3190895334310489300er_nat
            @ ^ [Y4: code_integer] :
                ( groups708209901874060359at_nat
                @ ^ [X4: nat] : ( G @ X4 @ Y4 )
                @ ( collect_nat
                  @ ^ [X4: nat] :
                      ( ( member_nat @ X4 @ A4 )
                      & ( R3 @ X4 @ Y4 ) ) ) )
            @ B5 ) ) ) ) ).

% prod.swap_restrict
thf(fact_8435_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_8436_or__greater__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K2 @ ( bit_se1409905431419307370or_int @ K2 @ L ) ) ) ).

% or_greater_eq
thf(fact_8437_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N5: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N5 )
          | ~ ( bit_se1146084159140164899it_int @ B @ N5 ) )
     => ( ( plus_plus_int @ A @ B )
        = ( bit_se1409905431419307370or_int @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_8438_disjunctive__add,axiom,
    ! [A: nat,B: nat] :
      ( ! [N5: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N5 )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N5 ) )
     => ( ( plus_plus_nat @ A @ B )
        = ( bit_se1412395901928357646or_nat @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_8439_infinite__growing,axiom,
    ! [X7: set_Code_integer] :
      ( ( X7 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X7 )
           => ? [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ X7 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa3 ) ) )
       => ~ ( finite6017078050557962740nteger @ X7 ) ) ) ).

% infinite_growing
thf(fact_8440_infinite__growing,axiom,
    ! [X7: set_o] :
      ( ( X7 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ X7 )
           => ? [Xa3: $o] :
                ( ( member_o @ Xa3 @ X7 )
                & ( ord_less_o @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_o @ X7 ) ) ) ).

% infinite_growing
thf(fact_8441_infinite__growing,axiom,
    ! [X7: set_rat] :
      ( ( X7 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X7 )
           => ? [Xa3: rat] :
                ( ( member_rat @ Xa3 @ X7 )
                & ( ord_less_rat @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_rat @ X7 ) ) ) ).

% infinite_growing
thf(fact_8442_infinite__growing,axiom,
    ! [X7: set_num] :
      ( ( X7 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X7 )
           => ? [Xa3: num] :
                ( ( member_num @ Xa3 @ X7 )
                & ( ord_less_num @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_num @ X7 ) ) ) ).

% infinite_growing
thf(fact_8443_infinite__growing,axiom,
    ! [X7: set_nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ? [Xa3: nat] :
                ( ( member_nat @ Xa3 @ X7 )
                & ( ord_less_nat @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_nat @ X7 ) ) ) ).

% infinite_growing
thf(fact_8444_infinite__growing,axiom,
    ! [X7: set_int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X7 )
           => ? [Xa3: int] :
                ( ( member_int @ Xa3 @ X7 )
                & ( ord_less_int @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_int @ X7 ) ) ) ).

% infinite_growing
thf(fact_8445_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S )
            & ~ ? [Xa3: code_integer] :
                  ( ( member_Code_integer @ Xa3 @ S )
                  & ( ord_le6747313008572928689nteger @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8446_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ S )
            & ~ ? [Xa3: $o] :
                  ( ( member_o @ Xa3 @ S )
                  & ( ord_less_o @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8447_ex__min__if__finite,axiom,
    ! [S: set_assn] :
      ( ( finite_finite_assn @ S )
     => ( ( S != bot_bot_set_assn )
       => ? [X3: assn] :
            ( ( member_assn @ X3 @ S )
            & ~ ? [Xa3: assn] :
                  ( ( member_assn @ Xa3 @ S )
                  & ( ord_less_assn @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8448_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( ( finite_finite_rat @ S )
     => ( ( S != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S )
            & ~ ? [Xa3: rat] :
                  ( ( member_rat @ Xa3 @ S )
                  & ( ord_less_rat @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8449_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( ( finite_finite_num @ S )
     => ( ( S != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S )
            & ~ ? [Xa3: num] :
                  ( ( member_num @ Xa3 @ S )
                  & ( ord_less_num @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8450_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S )
            & ~ ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ S )
                  & ( ord_less_nat @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8451_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S )
            & ~ ? [Xa3: int] :
                  ( ( member_int @ Xa3 @ S )
                  & ( ord_less_int @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8452_plus__and__or,axiom,
    ! [X: int,Y: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
      = ( plus_plus_int @ X @ Y ) ) ).

% plus_and_or
thf(fact_8453_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_8454_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_8455_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_8456_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_8457_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_8458_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_8459_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_8460_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_8461_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_8462_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_8463_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_8464_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_8465_push__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).

% push_bit_take_bit
thf(fact_8466_push__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N2 ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).

% push_bit_take_bit
thf(fact_8467_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > code_integer,Y: $o > code_integer] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I: $o] :
              ( ( member_o @ I @ I5 )
              & ( ( X @ I )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( Y @ I )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I ) @ ( Y @ I ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8468_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > code_integer,Y: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I ) @ ( Y @ I ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8469_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > code_integer,Y: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I5 )
              & ( ( X @ I )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( Y @ I )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I ) @ ( Y @ I ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8470_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > code_integer,Y: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I: code_integer] :
              ( ( member_Code_integer @ I @ I5 )
              & ( ( X @ I )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( Y @ I )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I ) @ ( Y @ I ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8471_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > rat,Y: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I: $o] :
              ( ( member_o @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( plus_plus_rat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8472_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( plus_plus_rat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8473_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( plus_plus_rat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8474_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > rat,Y: code_integer > rat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I: code_integer] :
              ( ( member_Code_integer @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_rat ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_rat ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( plus_plus_rat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8475_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > nat,Y: $o > nat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I: $o] :
              ( ( member_o @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( plus_plus_nat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8476_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > nat,Y: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( plus_plus_nat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8477_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > code_integer,Y: $o > code_integer] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I: $o] :
              ( ( member_o @ I @ I5 )
              & ( ( X @ I )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8478_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > code_integer,Y: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8479_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > code_integer,Y: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I5 )
              & ( ( X @ I )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8480_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > code_integer,Y: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I: code_integer] :
              ( ( member_Code_integer @ I @ I5 )
              & ( ( X @ I )
               != one_one_Code_integer ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_Code_integer ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8481_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > assn,Y: $o > assn] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I: $o] :
              ( ( member_o @ I @ I5 )
              & ( ( X @ I )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_assn ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( times_times_assn @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8482_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > assn,Y: nat > assn] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_assn ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( times_times_assn @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8483_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > assn,Y: int > assn] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I5 )
              & ( ( X @ I )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_assn ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( times_times_assn @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8484_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > assn,Y: code_integer > assn] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I: code_integer] :
              ( ( member_Code_integer @ I @ I5 )
              & ( ( X @ I )
               != one_one_assn ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_assn ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( times_times_assn @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8485_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X: $o > rat,Y: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I: $o] :
              ( ( member_o @ I @ I5 )
              & ( ( X @ I )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( times_times_rat @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8486_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( times_times_rat @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8487_sum_Ointer__filter,axiom,
    ! [A4: set_o,G: $o > code_integer,P2: $o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups4406642042086082107nteger @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups4406642042086082107nteger
          @ ^ [X4: $o] : ( if_Code_integer @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8488_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > code_integer,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups7501900531339628137nteger @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups7501900531339628137nteger
          @ ^ [X4: nat] : ( if_Code_integer @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8489_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: int > code_integer,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7873554091576472773nteger @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups7873554091576472773nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8490_sum_Ointer__filter,axiom,
    ! [A4: set_Code_integer,G: code_integer > code_integer,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups879477027807139574nteger @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups879477027807139574nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8491_sum_Ointer__filter,axiom,
    ! [A4: set_o,G: $o > rat,P2: $o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups7872700643590313910_o_rat @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups7872700643590313910_o_rat
          @ ^ [X4: $o] : ( if_rat @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8492_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > rat,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8493_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: int > rat,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8494_sum_Ointer__filter,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8495_sum_Ointer__filter,axiom,
    ! [A4: set_o,G: $o > nat,P2: $o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups8507830703676809646_o_nat @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups8507830703676809646_o_nat
          @ ^ [X4: $o] : ( if_nat @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8496_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: int > nat,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P2 @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8497_filter__preserves__multiset,axiom,
    ! [M6: list_nat > nat,P2: 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 @ ( P2 @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8498_filter__preserves__multiset,axiom,
    ! [M6: set_Pr958786334691620121nt_int > nat,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8499_filter__preserves__multiset,axiom,
    ! [M6: set_nat > nat,P2: 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 @ ( P2 @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8500_filter__preserves__multiset,axiom,
    ! [M6: nat > nat,P2: 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 @ ( P2 @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8501_filter__preserves__multiset,axiom,
    ! [M6: int > nat,P2: 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 @ ( P2 @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8502_filter__preserves__multiset,axiom,
    ! [M6: code_integer > nat,P2: 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 @ ( P2 @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8503_filter__preserves__multiset,axiom,
    ! [M6: product_prod_nat_nat > nat,P2: 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 @ ( P2 @ X4 ) @ ( M6 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8504_prod_Ointer__filter,axiom,
    ! [A4: set_o,G: $o > code_integer,P2: $o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups7694694392188491536nteger @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups7694694392188491536nteger
          @ ^ [X4: $o] : ( if_Code_integer @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8505_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > code_integer,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3455450783089532116nteger @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups3455450783089532116nteger
          @ ^ [X4: nat] : ( if_Code_integer @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8506_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: int > code_integer,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3827104343326376752nteger @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups3827104343326376752nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8507_prod_Ointer__filter,axiom,
    ! [A4: set_Code_integer,G: code_integer > code_integer,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups3674199335183972705nteger @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups3674199335183972705nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8508_prod_Ointer__filter,axiom,
    ! [A4: set_o,G: $o > assn,P2: $o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups5301882518646026715o_assn @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups5301882518646026715o_assn
          @ ^ [X4: $o] : ( if_assn @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_assn )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8509_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > assn,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups6906906614972039071t_assn @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups6906906614972039071t_assn
          @ ^ [X4: nat] : ( if_assn @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_assn )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8510_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: int > assn,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7882442080178216443t_assn @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups7882442080178216443t_assn
          @ ^ [X4: int] : ( if_assn @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_assn )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8511_prod_Ointer__filter,axiom,
    ! [A4: set_Code_integer,G: code_integer > assn,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1304777262505850412r_assn @ G
          @ ( collect_Code_integer
            @ ^ [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups1304777262505850412r_assn
          @ ^ [X4: code_integer] : ( if_assn @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_assn )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8512_prod_Ointer__filter,axiom,
    ! [A4: set_o,G: $o > rat,P2: $o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups2869687844427037835_o_rat @ G
          @ ( collect_o
            @ ^ [X4: $o] :
                ( ( member_o @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups2869687844427037835_o_rat
          @ ^ [X4: $o] : ( if_rat @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_rat )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8513_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > rat,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups73079841787564623at_rat @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups73079841787564623at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P2 @ X4 ) @ ( G @ X4 ) @ one_one_rat )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_8514_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_8515_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ A4 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups4406642042086082107nteger @ F @ A4 )
            = zero_z3403309356797280102nteger )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8516_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups7501900531339628137nteger @ F @ A4 )
            = zero_z3403309356797280102nteger )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8517_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups7873554091576472773nteger @ F @ A4 )
            = zero_z3403309356797280102nteger )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8518_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups879477027807139574nteger @ F @ A4 )
            = zero_z3403309356797280102nteger )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8519_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8520_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8521_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8522_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups6602215022474089585er_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8523_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,F: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( ( groups8507830703676809646_o_nat @ F @ A4 )
            = zero_zero_nat )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8524_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
            = zero_zero_nat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8525_sum__le__included,axiom,
    ! [S2: set_nat,T6: set_nat,G: nat > code_integer,I2: nat > nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T6 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ S2 ) @ ( groups7501900531339628137nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8526_sum__le__included,axiom,
    ! [S2: set_nat,T6: set_int,G: int > code_integer,I2: int > nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa3: int] :
                    ( ( member_int @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ S2 ) @ ( groups7873554091576472773nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8527_sum__le__included,axiom,
    ! [S2: set_nat,T6: set_Code_integer,G: code_integer > code_integer,I2: code_integer > nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa3: code_integer] :
                    ( ( member_Code_integer @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ S2 ) @ ( groups879477027807139574nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8528_sum__le__included,axiom,
    ! [S2: set_int,T6: set_nat,G: nat > code_integer,I2: nat > int,F: int > code_integer] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_nat @ T6 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ S2 ) @ ( groups7501900531339628137nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8529_sum__le__included,axiom,
    ! [S2: set_int,T6: set_int,G: int > code_integer,I2: int > int,F: int > code_integer] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa3: int] :
                    ( ( member_int @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ S2 ) @ ( groups7873554091576472773nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8530_sum__le__included,axiom,
    ! [S2: set_int,T6: set_Code_integer,G: code_integer > code_integer,I2: code_integer > int,F: int > code_integer] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa3: code_integer] :
                    ( ( member_Code_integer @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ S2 ) @ ( groups879477027807139574nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8531_sum__le__included,axiom,
    ! [S2: set_Code_integer,T6: set_nat,G: nat > code_integer,I2: nat > code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite_finite_nat @ T6 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ? [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ F @ S2 ) @ ( groups7501900531339628137nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8532_sum__le__included,axiom,
    ! [S2: set_Code_integer,T6: set_int,G: int > code_integer,I2: int > code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ? [Xa3: int] :
                    ( ( member_int @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ F @ S2 ) @ ( groups7873554091576472773nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8533_sum__le__included,axiom,
    ! [S2: set_Code_integer,T6: set_Code_integer,G: code_integer > code_integer,I2: code_integer > code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T6 )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S2 )
               => ? [Xa3: code_integer] :
                    ( ( member_Code_integer @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ F @ S2 ) @ ( groups879477027807139574nteger @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8534_sum__le__included,axiom,
    ! [S2: set_nat,T6: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T6 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T6 )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ T6 )
                    & ( ( I2 @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G @ T6 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8535_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P2: set_Code_integer > $o,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S8 )
             => ( ! [Y5: code_integer] :
                    ( ( member_Code_integer @ Y5 @ S8 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_Code_integer @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8536_finite__ranking__induct,axiom,
    ! [S: set_o,P2: set_o > $o,F: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X3: $o,S8: set_o] :
              ( ( finite_finite_o @ S8 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S8 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_o @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8537_finite__ranking__induct,axiom,
    ! [S: set_nat,P2: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X3: nat,S8: set_nat] :
              ( ( finite_finite_nat @ S8 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S8 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_nat @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8538_finite__ranking__induct,axiom,
    ! [S: set_int,P2: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X3: int,S8: set_int] :
              ( ( finite_finite_int @ S8 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S8 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_int @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8539_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P2: set_Code_integer > $o,F: code_integer > num] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S8 )
             => ( ! [Y5: code_integer] :
                    ( ( member_Code_integer @ Y5 @ S8 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_Code_integer @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8540_finite__ranking__induct,axiom,
    ! [S: set_o,P2: set_o > $o,F: $o > num] :
      ( ( finite_finite_o @ S )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X3: $o,S8: set_o] :
              ( ( finite_finite_o @ S8 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S8 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_o @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8541_finite__ranking__induct,axiom,
    ! [S: set_nat,P2: set_nat > $o,F: nat > num] :
      ( ( finite_finite_nat @ S )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X3: nat,S8: set_nat] :
              ( ( finite_finite_nat @ S8 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S8 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_nat @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8542_finite__ranking__induct,axiom,
    ! [S: set_int,P2: set_int > $o,F: int > num] :
      ( ( finite_finite_int @ S )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X3: int,S8: set_int] :
              ( ( finite_finite_int @ S8 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S8 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_int @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8543_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P2: set_Code_integer > $o,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S8 )
             => ( ! [Y5: code_integer] :
                    ( ( member_Code_integer @ Y5 @ S8 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_Code_integer @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8544_finite__ranking__induct,axiom,
    ! [S: set_o,P2: set_o > $o,F: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X3: $o,S8: set_o] :
              ( ( finite_finite_o @ S8 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S8 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P2 @ S8 )
                 => ( P2 @ ( insert_o @ X3 @ S8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8545_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: nat] :
              ( ( member_nat @ X6 @ A4 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8546_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A4 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8547_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: code_integer] :
              ( ( member_Code_integer @ X6 @ A4 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8548_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A4 )
              & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8549_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: code_integer] :
              ( ( member_Code_integer @ X6 @ A4 )
              & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8550_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: nat] :
              ( ( member_nat @ X6 @ A4 )
              & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8551_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: code_integer] :
              ( ( member_Code_integer @ X6 @ A4 )
              & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_int @ ( groups7234854612051535045er_int @ F @ A4 ) @ ( groups7234854612051535045er_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8552_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: nat] :
              ( ( member_nat @ X6 @ A4 )
              & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8553_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A4 )
              & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8554_sum__strict__mono__ex1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat,G: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X6 @ A4 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups342789780944988191at_rat @ F @ A4 ) @ ( groups342789780944988191at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8555_finite__linorder__max__induct,axiom,
    ! [A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [B3: code_integer,A9: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A9 )
             => ( ! [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ A9 )
                   => ( ord_le6747313008572928689nteger @ X6 @ B3 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_Code_integer @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8556_finite__linorder__max__induct,axiom,
    ! [A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [B3: $o,A9: set_o] :
              ( ( finite_finite_o @ A9 )
             => ( ! [X6: $o] :
                    ( ( member_o @ X6 @ A9 )
                   => ( ord_less_o @ X6 @ B3 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_o @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8557_finite__linorder__max__induct,axiom,
    ! [A4: set_rat,P2: set_rat > $o] :
      ( ( finite_finite_rat @ A4 )
     => ( ( P2 @ bot_bot_set_rat )
       => ( ! [B3: rat,A9: set_rat] :
              ( ( finite_finite_rat @ A9 )
             => ( ! [X6: rat] :
                    ( ( member_rat @ X6 @ A9 )
                   => ( ord_less_rat @ X6 @ B3 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_rat @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8558_finite__linorder__max__induct,axiom,
    ! [A4: set_num,P2: set_num > $o] :
      ( ( finite_finite_num @ A4 )
     => ( ( P2 @ bot_bot_set_num )
       => ( ! [B3: num,A9: set_num] :
              ( ( finite_finite_num @ A9 )
             => ( ! [X6: num] :
                    ( ( member_num @ X6 @ A9 )
                   => ( ord_less_num @ X6 @ B3 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_num @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8559_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [B3: nat,A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A9 )
                   => ( ord_less_nat @ X6 @ B3 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_nat @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8560_finite__linorder__max__induct,axiom,
    ! [A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [B3: int,A9: set_int] :
              ( ( finite_finite_int @ A9 )
             => ( ! [X6: int] :
                    ( ( member_int @ X6 @ A9 )
                   => ( ord_less_int @ X6 @ B3 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_int @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8561_finite__linorder__min__induct,axiom,
    ! [A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [B3: code_integer,A9: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A9 )
             => ( ! [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ A9 )
                   => ( ord_le6747313008572928689nteger @ B3 @ X6 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_Code_integer @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8562_finite__linorder__min__induct,axiom,
    ! [A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [B3: $o,A9: set_o] :
              ( ( finite_finite_o @ A9 )
             => ( ! [X6: $o] :
                    ( ( member_o @ X6 @ A9 )
                   => ( ord_less_o @ B3 @ X6 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_o @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8563_finite__linorder__min__induct,axiom,
    ! [A4: set_rat,P2: set_rat > $o] :
      ( ( finite_finite_rat @ A4 )
     => ( ( P2 @ bot_bot_set_rat )
       => ( ! [B3: rat,A9: set_rat] :
              ( ( finite_finite_rat @ A9 )
             => ( ! [X6: rat] :
                    ( ( member_rat @ X6 @ A9 )
                   => ( ord_less_rat @ B3 @ X6 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_rat @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8564_finite__linorder__min__induct,axiom,
    ! [A4: set_num,P2: set_num > $o] :
      ( ( finite_finite_num @ A4 )
     => ( ( P2 @ bot_bot_set_num )
       => ( ! [B3: num,A9: set_num] :
              ( ( finite_finite_num @ A9 )
             => ( ! [X6: num] :
                    ( ( member_num @ X6 @ A9 )
                   => ( ord_less_num @ B3 @ X6 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_num @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8565_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [B3: nat,A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A9 )
                   => ( ord_less_nat @ B3 @ X6 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_nat @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8566_finite__linorder__min__induct,axiom,
    ! [A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [B3: int,A9: set_int] :
              ( ( finite_finite_int @ A9 )
             => ( ! [X6: int] :
                    ( ( member_int @ X6 @ A9 )
                   => ( ord_less_int @ B3 @ X6 ) )
               => ( ( P2 @ A9 )
                 => ( P2 @ ( insert_int @ B3 @ A9 ) ) ) ) )
         => ( P2 @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8567_sum_Orelated,axiom,
    ! [R3: code_integer > code_integer > $o,S: set_nat,H: nat > code_integer,G: nat > code_integer] :
      ( ( R3 @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_p5714425477246183910nteger @ X13 @ Y12 ) @ ( plus_p5714425477246183910nteger @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups7501900531339628137nteger @ H @ S ) @ ( groups7501900531339628137nteger @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8568_sum_Orelated,axiom,
    ! [R3: code_integer > code_integer > $o,S: set_int,H: int > code_integer,G: int > code_integer] :
      ( ( R3 @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_p5714425477246183910nteger @ X13 @ Y12 ) @ ( plus_p5714425477246183910nteger @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups7873554091576472773nteger @ H @ S ) @ ( groups7873554091576472773nteger @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8569_sum_Orelated,axiom,
    ! [R3: code_integer > code_integer > $o,S: set_Code_integer,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( R3 @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_p5714425477246183910nteger @ X13 @ Y12 ) @ ( plus_p5714425477246183910nteger @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups879477027807139574nteger @ H @ S ) @ ( groups879477027807139574nteger @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8570_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S: set_nat,H: nat > rat,G: nat > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_plus_rat @ X13 @ Y12 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups2906978787729119204at_rat @ H @ S ) @ ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8571_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S: set_int,H: int > rat,G: int > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_plus_rat @ X13 @ Y12 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups3906332499630173760nt_rat @ H @ S ) @ ( groups3906332499630173760nt_rat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8572_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S: set_Code_integer,H: code_integer > rat,G: code_integer > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_plus_rat @ X13 @ Y12 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups6602215022474089585er_rat @ H @ S ) @ ( groups6602215022474089585er_rat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8573_sum_Orelated,axiom,
    ! [R3: nat > nat > $o,S: set_int,H: int > nat,G: int > nat] :
      ( ( R3 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X13: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_plus_nat @ X13 @ Y12 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups4541462559716669496nt_nat @ H @ S ) @ ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8574_sum_Orelated,axiom,
    ! [R3: nat > nat > $o,S: set_Code_integer,H: code_integer > nat,G: code_integer > nat] :
      ( ( R3 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X13: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_plus_nat @ X13 @ Y12 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups7237345082560585321er_nat @ H @ S ) @ ( groups7237345082560585321er_nat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8575_sum_Orelated,axiom,
    ! [R3: int > int > $o,S: set_nat,H: nat > int,G: nat > int] :
      ( ( R3 @ zero_zero_int @ zero_zero_int )
     => ( ! [X13: int,Y12: int,X22: int,Y22: int] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_plus_int @ X13 @ Y12 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups3539618377306564664at_int @ H @ S ) @ ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8576_sum_Orelated,axiom,
    ! [R3: int > int > $o,S: set_Code_integer,H: code_integer > int,G: code_integer > int] :
      ( ( R3 @ zero_zero_int @ zero_zero_int )
     => ( ! [X13: int,Y12: int,X22: int,Y22: int] :
            ( ( ( R3 @ X13 @ X22 )
              & ( R3 @ Y12 @ Y22 ) )
           => ( R3 @ ( plus_plus_int @ X13 @ Y12 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups7234854612051535045er_int @ H @ S ) @ ( groups7234854612051535045er_int @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8577_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8578_sum__strict__mono,axiom,
    ! [A4: set_o,F: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ ( groups7872700643590313910_o_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8579_sum__strict__mono,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8580_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8581_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8582_sum__strict__mono,axiom,
    ! [A4: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( groups8507830703676809646_o_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8583_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8584_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups7234854612051535045er_int @ F @ A4 ) @ ( groups7234854612051535045er_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8585_sum__strict__mono,axiom,
    ! [A4: set_o,F: $o > int,G: $o > int] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A4 ) @ ( groups8505340233167759370_o_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8586_sum__strict__mono,axiom,
    ! [A4: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8587_sum_Oinsert__if,axiom,
    ! [A4: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X @ A4 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A4 ) )
            = ( groups7872700643590313910_o_rat @ G @ A4 ) ) )
        & ( ~ ( member_o @ X @ A4 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A4 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8588_sum_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X @ A4 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A4 ) )
            = ( groups2906978787729119204at_rat @ G @ A4 ) ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A4 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8589_sum_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X @ A4 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A4 ) )
            = ( groups3906332499630173760nt_rat @ G @ A4 ) ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A4 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8590_sum_Oinsert__if,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( member_Code_integer @ X @ A4 )
         => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A4 ) )
            = ( groups6602215022474089585er_rat @ G @ A4 ) ) )
        & ( ~ ( member_Code_integer @ X @ A4 )
         => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A4 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8591_sum_Oinsert__if,axiom,
    ! [A4: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X @ A4 )
         => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A4 ) )
            = ( groups8507830703676809646_o_nat @ G @ A4 ) ) )
        & ( ~ ( member_o @ X @ A4 )
         => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A4 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8592_sum_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X @ A4 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A4 ) )
            = ( groups4541462559716669496nt_nat @ G @ A4 ) ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A4 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8593_sum_Oinsert__if,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( member_Code_integer @ X @ A4 )
         => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A4 ) )
            = ( groups7237345082560585321er_nat @ G @ A4 ) ) )
        & ( ~ ( member_Code_integer @ X @ A4 )
         => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A4 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8594_sum_Oinsert__if,axiom,
    ! [A4: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X @ A4 )
         => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X @ A4 ) )
            = ( groups8505340233167759370_o_int @ G @ A4 ) ) )
        & ( ~ ( member_o @ X @ A4 )
         => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X @ A4 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8595_sum_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X @ A4 )
         => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A4 ) )
            = ( groups3539618377306564664at_int @ G @ A4 ) ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A4 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8596_sum_Oinsert__if,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ( member_Code_integer @ X @ A4 )
         => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A4 ) )
            = ( groups7234854612051535045er_int @ G @ A4 ) ) )
        & ( ~ ( member_Code_integer @ X @ A4 )
         => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A4 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8597_bit__push__bit__iff__int,axiom,
    ! [M: nat,K2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K2 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_8598_prod__dvd__prod__subset,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( dvd_dvd_nat @ ( groups3190895334310489300er_nat @ F @ A4 ) @ ( groups3190895334310489300er_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8599_prod__dvd__prod__subset,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
       => ( dvd_dvd_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) @ ( groups4077766827762148844at_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8600_prod__dvd__prod__subset,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( dvd_dvd_int @ ( groups3188404863801439024er_int @ F @ A4 ) @ ( groups3188404863801439024er_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8601_prod__dvd__prod__subset,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
       => ( dvd_dvd_int @ ( groups4075276357253098568at_int @ F @ A4 ) @ ( groups4075276357253098568at_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8602_prod__dvd__prod__subset,axiom,
    ! [B5: set_int,A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8603_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8604_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8605_prod__dvd__prod__subset,axiom,
    ! [B5: set_int,A4: set_int,F: int > int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8606_prod__dvd__prod__subset2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups3504817904513533571_o_nat @ F @ A4 ) @ ( groups3504817904513533571_o_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8607_prod__dvd__prod__subset2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups3190895334310489300er_nat @ F @ A4 ) @ ( groups3190895334310489300er_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8608_prod__dvd__prod__subset2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > int,G: $o > int] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups3502327434004483295_o_int @ F @ A4 ) @ ( groups3502327434004483295_o_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8609_prod__dvd__prod__subset2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups3188404863801439024er_int @ F @ A4 ) @ ( groups3188404863801439024er_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8610_prod__dvd__prod__subset2,axiom,
    ! [B5: set_int,A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8611_prod__dvd__prod__subset2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8612_prod__dvd__prod__subset2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8613_prod__dvd__prod__subset2,axiom,
    ! [B5: set_int,A4: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8614_prod__dvd__prod__subset2,axiom,
    ! [B5: set_set_nat,A4: set_set_nat,F: set_nat > nat,G: set_nat > nat] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
       => ( ! [A3: set_nat] :
              ( ( member_set_nat @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups4248547760180025341at_nat @ F @ A4 ) @ ( groups4248547760180025341at_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8615_prod__dvd__prod__subset2,axiom,
    ! [B5: set_set_nat,A4: set_set_nat,F: set_nat > int,G: set_nat > int] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
       => ( ! [A3: set_nat] :
              ( ( member_set_nat @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups4246057289670975065at_int @ F @ A4 ) @ ( groups4246057289670975065at_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8616_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K3: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L2 ) ) ) ) ).

% concat_bit_eq
thf(fact_8617_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_8618_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: $o > code_integer,B5: code_integer,I2: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups4406642042086082107nteger @ F @ S2 )
            = B5 )
         => ( ( member_o @ I2 @ S2 )
           => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8619_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: nat > code_integer,B5: code_integer,I2: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7501900531339628137nteger @ F @ S2 )
            = B5 )
         => ( ( member_nat @ I2 @ S2 )
           => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8620_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > code_integer,B5: code_integer,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7873554091576472773nteger @ F @ S2 )
            = B5 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8621_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,F: code_integer > code_integer,B5: code_integer,I2: code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ S2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups879477027807139574nteger @ F @ S2 )
            = B5 )
         => ( ( member_Code_integer @ I2 @ S2 )
           => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8622_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: $o > rat,B5: 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 )
            = B5 )
         => ( ( member_o @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8623_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: nat > rat,B5: 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 )
            = B5 )
         => ( ( member_nat @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8624_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > rat,B5: 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 )
            = B5 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8625_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Code_integer,F: code_integer > rat,B5: 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 )
            = B5 )
         => ( ( member_Code_integer @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8626_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: $o > nat,B5: 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 )
            = B5 )
         => ( ( member_o @ I2 @ S2 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8627_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > nat,B5: 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 )
            = B5 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8628_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_8629_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_8630_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_8631_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_8632_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_8633_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_8634_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_8635_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_8636_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_8637_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_8638_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_8639_add__mset__in__multiset,axiom,
    ! [M6: set_Pr958786334691620121nt_int > nat,A: set_Pr958786334691620121nt_int] :
      ( ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8640_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_8641_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_8642_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_8643_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_8644_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_8645_sum_Ointer__restrict,axiom,
    ! [A4: set_o,G: $o > code_integer,B5: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups4406642042086082107nteger @ G @ ( inf_inf_set_o @ A4 @ B5 ) )
        = ( groups4406642042086082107nteger
          @ ^ [X4: $o] : ( if_Code_integer @ ( member_o @ X4 @ B5 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8646_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G: int > code_integer,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7873554091576472773nteger @ G @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups7873554091576472773nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( member_int @ X4 @ B5 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8647_sum_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,G: code_integer > code_integer,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups879477027807139574nteger @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
        = ( groups879477027807139574nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( member_Code_integer @ X4 @ B5 ) @ ( G @ X4 ) @ zero_z3403309356797280102nteger )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8648_sum_Ointer__restrict,axiom,
    ! [A4: set_o,G: $o > rat,B5: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups7872700643590313910_o_rat @ G @ ( inf_inf_set_o @ A4 @ B5 ) )
        = ( groups7872700643590313910_o_rat
          @ ^ [X4: $o] : ( if_rat @ ( member_o @ X4 @ B5 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8649_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G: int > rat,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( if_rat @ ( member_int @ X4 @ B5 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8650_sum_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
        = ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( member_Code_integer @ X4 @ B5 ) @ ( G @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8651_sum_Ointer__restrict,axiom,
    ! [A4: set_o,G: $o > nat,B5: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups8507830703676809646_o_nat @ G @ ( inf_inf_set_o @ A4 @ B5 ) )
        = ( groups8507830703676809646_o_nat
          @ ^ [X4: $o] : ( if_nat @ ( member_o @ X4 @ B5 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8652_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G: int > nat,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( if_nat @ ( member_int @ X4 @ B5 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8653_sum_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,G: code_integer > nat,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
        = ( groups7237345082560585321er_nat
          @ ^ [X4: code_integer] : ( if_nat @ ( member_Code_integer @ X4 @ B5 ) @ ( G @ X4 ) @ zero_zero_nat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8654_sum_Ointer__restrict,axiom,
    ! [A4: set_o,G: $o > int,B5: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups8505340233167759370_o_int @ G @ ( inf_inf_set_o @ A4 @ B5 ) )
        = ( groups8505340233167759370_o_int
          @ ^ [X4: $o] : ( if_int @ ( member_o @ X4 @ B5 ) @ ( G @ X4 ) @ zero_zero_int )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_8655_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups7501900531339628137nteger @ G
          @ ( minus_minus_set_nat @ A4
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) )
        = ( groups7501900531339628137nteger @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8656_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7873554091576472773nteger @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) )
        = ( groups7873554091576472773nteger @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8657_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups879477027807139574nteger @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_z3403309356797280102nteger ) ) ) )
        = ( groups879477027807139574nteger @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8658_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( minus_minus_set_nat @ A4
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8659_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8660_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8661_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8662_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups7237345082560585321er_nat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8663_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3539618377306564664at_int @ G
          @ ( minus_minus_set_nat @ A4
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = zero_zero_int ) ) ) )
        = ( groups3539618377306564664at_int @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8664_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7234854612051535045er_int @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = zero_zero_int ) ) ) )
        = ( groups7234854612051535045er_int @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8665_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_8666_diff__preserves__multiset,axiom,
    ! [M6: set_Pr958786334691620121nt_int > nat,N7: set_Pr958786334691620121nt_int > nat] :
      ( ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( M6 @ X4 ) ) ) )
     => ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8667_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_8668_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_8669_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_8670_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_8671_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_8672_prod_Ointer__restrict,axiom,
    ! [A4: set_o,G: $o > code_integer,B5: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups7694694392188491536nteger @ G @ ( inf_inf_set_o @ A4 @ B5 ) )
        = ( groups7694694392188491536nteger
          @ ^ [X4: $o] : ( if_Code_integer @ ( member_o @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8673_prod_Ointer__restrict,axiom,
    ! [A4: set_int,G: int > code_integer,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3827104343326376752nteger @ G @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups3827104343326376752nteger
          @ ^ [X4: int] : ( if_Code_integer @ ( member_int @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8674_prod_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,G: code_integer > code_integer,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups3674199335183972705nteger @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
        = ( groups3674199335183972705nteger
          @ ^ [X4: code_integer] : ( if_Code_integer @ ( member_Code_integer @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_Code_integer )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8675_prod_Ointer__restrict,axiom,
    ! [A4: set_o,G: $o > assn,B5: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups5301882518646026715o_assn @ G @ ( inf_inf_set_o @ A4 @ B5 ) )
        = ( groups5301882518646026715o_assn
          @ ^ [X4: $o] : ( if_assn @ ( member_o @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_assn )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8676_prod_Ointer__restrict,axiom,
    ! [A4: set_int,G: int > assn,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups7882442080178216443t_assn
          @ ^ [X4: int] : ( if_assn @ ( member_int @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_assn )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8677_prod_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,G: code_integer > assn,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
        = ( groups1304777262505850412r_assn
          @ ^ [X4: code_integer] : ( if_assn @ ( member_Code_integer @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_assn )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8678_prod_Ointer__restrict,axiom,
    ! [A4: set_o,G: $o > rat,B5: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups2869687844427037835_o_rat @ G @ ( inf_inf_set_o @ A4 @ B5 ) )
        = ( groups2869687844427037835_o_rat
          @ ^ [X4: $o] : ( if_rat @ ( member_o @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_rat )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8679_prod_Ointer__restrict,axiom,
    ! [A4: set_int,G: int > rat,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups1072433553688619179nt_rat
          @ ^ [X4: int] : ( if_rat @ ( member_int @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_rat )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8680_prod_Ointer__restrict,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
        = ( groups2555765274223993564er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( member_Code_integer @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_rat )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8681_prod_Ointer__restrict,axiom,
    ! [A4: set_o,G: $o > nat,B5: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups3504817904513533571_o_nat @ G @ ( inf_inf_set_o @ A4 @ B5 ) )
        = ( groups3504817904513533571_o_nat
          @ ^ [X4: $o] : ( if_nat @ ( member_o @ X4 @ B5 ) @ ( G @ X4 ) @ one_one_nat )
          @ A4 ) ) ) ).

% prod.inter_restrict
thf(fact_8682_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3455450783089532116nteger @ G
          @ ( minus_minus_set_nat @ A4
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3455450783089532116nteger @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8683_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3827104343326376752nteger @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3827104343326376752nteger @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8684_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups3674199335183972705nteger @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3674199335183972705nteger @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8685_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups6906906614972039071t_assn @ G
          @ ( minus_minus_set_nat @ A4
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = one_one_assn ) ) ) )
        = ( groups6906906614972039071t_assn @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8686_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > assn] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7882442080178216443t_assn @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_assn ) ) ) )
        = ( groups7882442080178216443t_assn @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8687_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1304777262505850412r_assn @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = one_one_assn ) ) ) )
        = ( groups1304777262505850412r_assn @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8688_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups73079841787564623at_rat @ G
          @ ( minus_minus_set_nat @ A4
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = one_one_rat ) ) ) )
        = ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8689_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1072433553688619179nt_rat @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_rat ) ) ) )
        = ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8690_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups2555765274223993564er_rat @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X4: code_integer] :
                  ( ( G @ X4 )
                  = one_one_rat ) ) ) )
        = ( groups2555765274223993564er_rat @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8691_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1707563613775114915nt_nat @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8692_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_8693_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_8694_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_8695_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_8696_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_8697_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_8698_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_8699_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_8700_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_8701_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_8702_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_8703_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_8704_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_8705_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_8706_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_8707_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_8708_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_8709_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_8710_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_8711_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_8712_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_8713_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_8714_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_8715_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_8716_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_8717_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_8718_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_8719_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_8720_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_8721_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_8722_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_8723_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_8724_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_8725_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_8726_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_8727_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_8728_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_8729_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_8730_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_8731_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_8732_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_8733_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_8734_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_8735_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_8736_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups4406642042086082107nteger @ G @ T7 )
              = ( groups4406642042086082107nteger @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8737_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7501900531339628137nteger @ G @ T7 )
              = ( groups7501900531339628137nteger @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8738_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups879477027807139574nteger @ G @ T7 )
              = ( groups879477027807139574nteger @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8739_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7872700643590313910_o_rat @ G @ T7 )
              = ( groups7872700643590313910_o_rat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8740_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2906978787729119204at_rat @ G @ T7 )
              = ( groups2906978787729119204at_rat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8741_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups6602215022474089585er_rat @ G @ T7 )
              = ( groups6602215022474089585er_rat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8742_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G @ T7 )
              = ( groups8507830703676809646_o_nat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8743_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7237345082560585321er_nat @ G @ T7 )
              = ( groups7237345082560585321er_nat @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8744_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8505340233167759370_o_int @ G @ T7 )
              = ( groups8505340233167759370_o_int @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8745_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > int,H: nat > int] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3539618377306564664at_int @ G @ T7 )
              = ( groups3539618377306564664at_int @ H @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8746_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,H: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( H @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups4406642042086082107nteger @ G @ S )
              = ( groups4406642042086082107nteger @ H @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8747_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,H: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( H @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7501900531339628137nteger @ G @ S )
              = ( groups7501900531339628137nteger @ H @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8748_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( H @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups879477027807139574nteger @ G @ S )
              = ( groups879477027807139574nteger @ H @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8749_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,H: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8750_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,H: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( H @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2906978787729119204at_rat @ G @ S )
              = ( groups2906978787729119204at_rat @ H @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8751_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8752_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8753_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8754_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,H: $o > int,G: $o > int] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8755_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,H: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( H @ X3 )
                = zero_zero_int ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3539618377306564664at_int @ G @ S )
              = ( groups3539618377306564664at_int @ H @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8756_sum_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7501900531339628137nteger @ G @ T7 )
            = ( groups7501900531339628137nteger @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8757_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups879477027807139574nteger @ G @ T7 )
            = ( groups879477027807139574nteger @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8758_sum_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T7 )
            = ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8759_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ T7 )
            = ( groups6602215022474089585er_rat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8760_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ T7 )
            = ( groups7237345082560585321er_nat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8761_sum_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T7 )
            = ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8762_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ T7 )
            = ( groups7234854612051535045er_int @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8763_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7873554091576472773nteger @ G @ T7 )
            = ( groups7873554091576472773nteger @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8764_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,G: int > rat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T7 )
            = ( groups3906332499630173760nt_rat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8765_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,G: int > nat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ T7 )
            = ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8766_sum_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7501900531339628137nteger @ G @ S )
            = ( groups7501900531339628137nteger @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8767_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups879477027807139574nteger @ G @ S )
            = ( groups879477027807139574nteger @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8768_sum_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S )
            = ( groups2906978787729119204at_rat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8769_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ S )
            = ( groups6602215022474089585er_rat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8770_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ S )
            = ( groups7237345082560585321er_nat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8771_sum_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S )
            = ( groups3539618377306564664at_int @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8772_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ S )
            = ( groups7234854612051535045er_int @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8773_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7873554091576472773nteger @ G @ S )
            = ( groups7873554091576472773nteger @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8774_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,G: int > rat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S )
            = ( groups3906332499630173760nt_rat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8775_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,G: int > nat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ S )
            = ( groups4541462559716669496nt_nat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8776_sum_Osame__carrierI,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups4406642042086082107nteger @ G @ C4 )
                  = ( groups4406642042086082107nteger @ H @ C4 ) )
               => ( ( groups4406642042086082107nteger @ G @ A4 )
                  = ( groups4406642042086082107nteger @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8777_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups7501900531339628137nteger @ G @ C4 )
                  = ( groups7501900531339628137nteger @ H @ C4 ) )
               => ( ( groups7501900531339628137nteger @ G @ A4 )
                  = ( groups7501900531339628137nteger @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8778_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups879477027807139574nteger @ G @ C4 )
                  = ( groups879477027807139574nteger @ H @ C4 ) )
               => ( ( groups879477027807139574nteger @ G @ A4 )
                  = ( groups879477027807139574nteger @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8779_sum_Osame__carrierI,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups7872700643590313910_o_rat @ G @ C4 )
                  = ( groups7872700643590313910_o_rat @ H @ C4 ) )
               => ( ( groups7872700643590313910_o_rat @ G @ A4 )
                  = ( groups7872700643590313910_o_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8780_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups2906978787729119204at_rat @ G @ C4 )
                  = ( groups2906978787729119204at_rat @ H @ C4 ) )
               => ( ( groups2906978787729119204at_rat @ G @ A4 )
                  = ( groups2906978787729119204at_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8781_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups6602215022474089585er_rat @ G @ C4 )
                  = ( groups6602215022474089585er_rat @ H @ C4 ) )
               => ( ( groups6602215022474089585er_rat @ G @ A4 )
                  = ( groups6602215022474089585er_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8782_sum_Osame__carrierI,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G @ C4 )
                  = ( groups8507830703676809646_o_nat @ H @ C4 ) )
               => ( ( groups8507830703676809646_o_nat @ G @ A4 )
                  = ( groups8507830703676809646_o_nat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8783_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups7237345082560585321er_nat @ G @ C4 )
                  = ( groups7237345082560585321er_nat @ H @ C4 ) )
               => ( ( groups7237345082560585321er_nat @ G @ A4 )
                  = ( groups7237345082560585321er_nat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8784_sum_Osame__carrierI,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_int ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_int ) )
             => ( ( ( groups8505340233167759370_o_int @ G @ C4 )
                  = ( groups8505340233167759370_o_int @ H @ C4 ) )
               => ( ( groups8505340233167759370_o_int @ G @ A4 )
                  = ( groups8505340233167759370_o_int @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8785_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > int,H: nat > int] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_int ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_int ) )
             => ( ( ( groups3539618377306564664at_int @ G @ C4 )
                  = ( groups3539618377306564664at_int @ H @ C4 ) )
               => ( ( groups3539618377306564664at_int @ G @ A4 )
                  = ( groups3539618377306564664at_int @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8786_sum_Osame__carrier,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups4406642042086082107nteger @ G @ A4 )
                  = ( groups4406642042086082107nteger @ H @ B5 ) )
                = ( ( groups4406642042086082107nteger @ G @ C4 )
                  = ( groups4406642042086082107nteger @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8787_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups7501900531339628137nteger @ G @ A4 )
                  = ( groups7501900531339628137nteger @ H @ B5 ) )
                = ( ( groups7501900531339628137nteger @ G @ C4 )
                  = ( groups7501900531339628137nteger @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8788_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_z3403309356797280102nteger ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_z3403309356797280102nteger ) )
             => ( ( ( groups879477027807139574nteger @ G @ A4 )
                  = ( groups879477027807139574nteger @ H @ B5 ) )
                = ( ( groups879477027807139574nteger @ G @ C4 )
                  = ( groups879477027807139574nteger @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8789_sum_Osame__carrier,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups7872700643590313910_o_rat @ G @ A4 )
                  = ( groups7872700643590313910_o_rat @ H @ B5 ) )
                = ( ( groups7872700643590313910_o_rat @ G @ C4 )
                  = ( groups7872700643590313910_o_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8790_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups2906978787729119204at_rat @ G @ A4 )
                  = ( groups2906978787729119204at_rat @ H @ B5 ) )
                = ( ( groups2906978787729119204at_rat @ G @ C4 )
                  = ( groups2906978787729119204at_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8791_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups6602215022474089585er_rat @ G @ A4 )
                  = ( groups6602215022474089585er_rat @ H @ B5 ) )
                = ( ( groups6602215022474089585er_rat @ G @ C4 )
                  = ( groups6602215022474089585er_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8792_sum_Osame__carrier,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G @ A4 )
                  = ( groups8507830703676809646_o_nat @ H @ B5 ) )
                = ( ( groups8507830703676809646_o_nat @ G @ C4 )
                  = ( groups8507830703676809646_o_nat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8793_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups7237345082560585321er_nat @ G @ A4 )
                  = ( groups7237345082560585321er_nat @ H @ B5 ) )
                = ( ( groups7237345082560585321er_nat @ G @ C4 )
                  = ( groups7237345082560585321er_nat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8794_sum_Osame__carrier,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > int,H: $o > int] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_int ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_int ) )
             => ( ( ( groups8505340233167759370_o_int @ G @ A4 )
                  = ( groups8505340233167759370_o_int @ H @ B5 ) )
                = ( ( groups8505340233167759370_o_int @ G @ C4 )
                  = ( groups8505340233167759370_o_int @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8795_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > int,H: nat > int] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_int ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_int ) )
             => ( ( ( groups3539618377306564664at_int @ G @ A4 )
                  = ( groups3539618377306564664at_int @ H @ B5 ) )
                = ( ( groups3539618377306564664at_int @ G @ C4 )
                  = ( groups3539618377306564664at_int @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8796_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8797_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups6602215022474089585er_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups6602215022474089585er_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8798_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups7237345082560585321er_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups7237345082560585321er_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8799_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3539618377306564664at_int @ G @ A4 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8800_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups7234854612051535045er_int @ G @ A4 )
          = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups7234854612051535045er_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8801_sum_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups3906332499630173760nt_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8802_sum_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups4541462559716669496nt_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8803_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3542108847815614940at_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3542108847815614940at_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8804_sum_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups4538972089207619220nt_int @ G @ A4 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8805_sum_Osubset__diff,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
     => ( ( finite6177210948735845034at_nat @ A4 )
       => ( ( groups342789780944988191at_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) @ ( groups342789780944988191at_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8806_sum__diff,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B5 @ A4 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_8807_sum__diff,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
       => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_8808_sum__diff,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B5 @ A4 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_8809_sum__diff,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
       => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A4 @ B5 ) )
          = ( minus_minus_int @ ( groups7234854612051535045er_int @ F @ A4 ) @ ( groups7234854612051535045er_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_8810_sum__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
       => ( ( groups342789780944988191at_rat @ F @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( groups342789780944988191at_rat @ F @ A4 ) @ ( groups342789780944988191at_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_8811_sum__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
       => ( ( groups975429370522433651at_int @ F @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
          = ( minus_minus_int @ ( groups975429370522433651at_int @ F @ A4 ) @ ( groups975429370522433651at_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_8812_sum__diff,axiom,
    ! [A4: set_int,B5: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_8813_sum__diff,axiom,
    ! [A4: set_int,B5: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_8814_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > assn] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups6906906614972039071t_assn @ G @ A4 )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups6906906614972039071t_assn @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8815_prod_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,G: code_integer > assn] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups1304777262505850412r_assn @ G @ A4 )
          = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups1304777262505850412r_assn @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8816_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups73079841787564623at_rat @ G @ A4 )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8817_prod_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups2555765274223993564er_rat @ G @ A4 )
          = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups2555765274223993564er_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8818_prod_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups3190895334310489300er_nat @ G @ A4 )
          = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups3190895334310489300er_nat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8819_prod_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups3188404863801439024er_int @ G @ A4 )
          = ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups3188404863801439024er_int @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8820_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > assn] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups7882442080178216443t_assn @ G @ A4 )
          = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups7882442080178216443t_assn @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8821_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups1072433553688619179nt_rat @ G @ A4 )
          = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8822_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups1707563613775114915nt_nat @ G @ A4 )
          = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8823_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups708209901874060359at_nat @ G @ A4 )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8824_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7694694392188491536nteger @ G @ T7 )
              = ( groups7694694392188491536nteger @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8825_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3455450783089532116nteger @ G @ T7 )
              = ( groups3455450783089532116nteger @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8826_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3674199335183972705nteger @ G @ T7 )
              = ( groups3674199335183972705nteger @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8827_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,G: $o > assn,H: $o > assn] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5301882518646026715o_assn @ G @ T7 )
              = ( groups5301882518646026715o_assn @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8828_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > assn,H: nat > assn] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups6906906614972039071t_assn @ G @ T7 )
              = ( groups6906906614972039071t_assn @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8829_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > assn,H: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups1304777262505850412r_assn @ G @ T7 )
              = ( groups1304777262505850412r_assn @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8830_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G @ T7 )
              = ( groups2869687844427037835_o_rat @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8831_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups73079841787564623at_rat @ G @ T7 )
              = ( groups73079841787564623at_rat @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8832_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups2555765274223993564er_rat @ G @ T7 )
              = ( groups2555765274223993564er_rat @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8833_prod_Omono__neutral__cong__right,axiom,
    ! [T7: set_o,S: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G @ T7 )
              = ( groups3504817904513533571_o_nat @ H @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8834_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,H: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( H @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7694694392188491536nteger @ G @ S )
              = ( groups7694694392188491536nteger @ H @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8835_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,H: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( H @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3455450783089532116nteger @ G @ S )
              = ( groups3455450783089532116nteger @ H @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8836_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,H: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( H @ X3 )
                = one_one_Code_integer ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3674199335183972705nteger @ G @ S )
              = ( groups3674199335183972705nteger @ H @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8837_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,H: $o > assn,G: $o > assn] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ S ) )
             => ( ( H @ X3 )
                = one_one_assn ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5301882518646026715o_assn @ G @ S )
              = ( groups5301882518646026715o_assn @ H @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8838_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,H: nat > assn,G: nat > assn] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( H @ X3 )
                = one_one_assn ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups6906906614972039071t_assn @ G @ S )
              = ( groups6906906614972039071t_assn @ H @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8839_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,H: code_integer > assn,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8840_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,H: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8841_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_nat,S: set_nat,H: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( H @ X3 )
                = one_one_rat ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups73079841787564623at_rat @ G @ S )
              = ( groups73079841787564623at_rat @ H @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8842_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8843_prod_Omono__neutral__cong__left,axiom,
    ! [T7: set_o,S: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T7 )
     => ( ( ord_less_eq_set_o @ S @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( minus_minus_set_o @ T7 @ 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 @ T7 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8844_prod_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G @ T7 )
            = ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8845_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3674199335183972705nteger @ G @ T7 )
            = ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8846_prod_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups6906906614972039071t_assn @ G @ T7 )
            = ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8847_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups1304777262505850412r_assn @ G @ T7 )
            = ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8848_prod_Omono__neutral__right,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ T7 )
            = ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8849_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ T7 )
            = ( groups2555765274223993564er_rat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8850_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups3190895334310489300er_nat @ G @ T7 )
            = ( groups3190895334310489300er_nat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8851_prod_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups3188404863801439024er_int @ G @ T7 )
            = ( groups3188404863801439024er_int @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8852_prod_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G @ T7 )
            = ( groups3827104343326376752nteger @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8853_prod_Omono__neutral__right,axiom,
    ! [T7: set_int,S: set_int,G: int > assn] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups7882442080178216443t_assn @ G @ T7 )
            = ( groups7882442080178216443t_assn @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8854_prod_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G @ S )
            = ( groups3455450783089532116nteger @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8855_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3674199335183972705nteger @ G @ S )
            = ( groups3674199335183972705nteger @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8856_prod_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups6906906614972039071t_assn @ G @ S )
            = ( groups6906906614972039071t_assn @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8857_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups1304777262505850412r_assn @ G @ S )
            = ( groups1304777262505850412r_assn @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8858_prod_Omono__neutral__left,axiom,
    ! [T7: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T7 )
     => ( ( ord_less_eq_set_nat @ S @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ S )
            = ( groups73079841787564623at_rat @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8859_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ S )
            = ( groups2555765274223993564er_rat @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8860_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups3190895334310489300er_nat @ G @ S )
            = ( groups3190895334310489300er_nat @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8861_prod_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups3188404863801439024er_int @ G @ S )
            = ( groups3188404863801439024er_int @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8862_prod_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G @ S )
            = ( groups3827104343326376752nteger @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8863_prod_Omono__neutral__left,axiom,
    ! [T7: set_int,S: set_int,G: int > assn] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups7882442080178216443t_assn @ G @ S )
            = ( groups7882442080178216443t_assn @ G @ T7 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8864_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_Code_integer ) )
             => ( ( ( groups7694694392188491536nteger @ G @ C4 )
                  = ( groups7694694392188491536nteger @ H @ C4 ) )
               => ( ( groups7694694392188491536nteger @ G @ A4 )
                  = ( groups7694694392188491536nteger @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8865_prod_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3455450783089532116nteger @ G @ C4 )
                  = ( groups3455450783089532116nteger @ H @ C4 ) )
               => ( ( groups3455450783089532116nteger @ G @ A4 )
                  = ( groups3455450783089532116nteger @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8866_prod_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3674199335183972705nteger @ G @ C4 )
                  = ( groups3674199335183972705nteger @ H @ C4 ) )
               => ( ( groups3674199335183972705nteger @ G @ A4 )
                  = ( groups3674199335183972705nteger @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8867_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > assn,H: $o > assn] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_assn ) )
             => ( ( ( groups5301882518646026715o_assn @ G @ C4 )
                  = ( groups5301882518646026715o_assn @ H @ C4 ) )
               => ( ( groups5301882518646026715o_assn @ G @ A4 )
                  = ( groups5301882518646026715o_assn @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8868_prod_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > assn,H: nat > assn] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_assn ) )
             => ( ( ( groups6906906614972039071t_assn @ G @ C4 )
                  = ( groups6906906614972039071t_assn @ H @ C4 ) )
               => ( ( groups6906906614972039071t_assn @ G @ A4 )
                  = ( groups6906906614972039071t_assn @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8869_prod_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > assn,H: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_assn ) )
             => ( ( ( groups1304777262505850412r_assn @ G @ C4 )
                  = ( groups1304777262505850412r_assn @ H @ C4 ) )
               => ( ( groups1304777262505850412r_assn @ G @ A4 )
                  = ( groups1304777262505850412r_assn @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8870_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G @ C4 )
                  = ( groups2869687844427037835_o_rat @ H @ C4 ) )
               => ( ( groups2869687844427037835_o_rat @ G @ A4 )
                  = ( groups2869687844427037835_o_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8871_prod_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups73079841787564623at_rat @ G @ C4 )
                  = ( groups73079841787564623at_rat @ H @ C4 ) )
               => ( ( groups73079841787564623at_rat @ G @ A4 )
                  = ( groups73079841787564623at_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8872_prod_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups2555765274223993564er_rat @ G @ C4 )
                  = ( groups2555765274223993564er_rat @ H @ C4 ) )
               => ( ( groups2555765274223993564er_rat @ G @ A4 )
                  = ( groups2555765274223993564er_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8873_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G @ C4 )
                  = ( groups3504817904513533571_o_nat @ H @ C4 ) )
               => ( ( groups3504817904513533571_o_nat @ G @ A4 )
                  = ( groups3504817904513533571_o_nat @ H @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8874_prod_Osame__carrier,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > code_integer,H: $o > code_integer] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_Code_integer ) )
             => ( ( ( groups7694694392188491536nteger @ G @ A4 )
                  = ( groups7694694392188491536nteger @ H @ B5 ) )
                = ( ( groups7694694392188491536nteger @ G @ C4 )
                  = ( groups7694694392188491536nteger @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8875_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > code_integer,H: nat > code_integer] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3455450783089532116nteger @ G @ A4 )
                  = ( groups3455450783089532116nteger @ H @ B5 ) )
                = ( ( groups3455450783089532116nteger @ G @ C4 )
                  = ( groups3455450783089532116nteger @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8876_prod_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > code_integer,H: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3674199335183972705nteger @ G @ A4 )
                  = ( groups3674199335183972705nteger @ H @ B5 ) )
                = ( ( groups3674199335183972705nteger @ G @ C4 )
                  = ( groups3674199335183972705nteger @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8877_prod_Osame__carrier,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > assn,H: $o > assn] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_assn ) )
             => ( ( ( groups5301882518646026715o_assn @ G @ A4 )
                  = ( groups5301882518646026715o_assn @ H @ B5 ) )
                = ( ( groups5301882518646026715o_assn @ G @ C4 )
                  = ( groups5301882518646026715o_assn @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8878_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > assn,H: nat > assn] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_assn ) )
             => ( ( ( groups6906906614972039071t_assn @ G @ A4 )
                  = ( groups6906906614972039071t_assn @ H @ B5 ) )
                = ( ( groups6906906614972039071t_assn @ G @ C4 )
                  = ( groups6906906614972039071t_assn @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8879_prod_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > assn,H: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_assn ) )
             => ( ( ( groups1304777262505850412r_assn @ G @ A4 )
                  = ( groups1304777262505850412r_assn @ H @ B5 ) )
                = ( ( groups1304777262505850412r_assn @ G @ C4 )
                  = ( groups1304777262505850412r_assn @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8880_prod_Osame__carrier,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G @ A4 )
                  = ( groups2869687844427037835_o_rat @ H @ B5 ) )
                = ( ( groups2869687844427037835_o_rat @ G @ C4 )
                  = ( groups2869687844427037835_o_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8881_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups73079841787564623at_rat @ G @ A4 )
                  = ( groups73079841787564623at_rat @ H @ B5 ) )
                = ( ( groups73079841787564623at_rat @ G @ C4 )
                  = ( groups73079841787564623at_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8882_prod_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: code_integer] :
                  ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups2555765274223993564er_rat @ G @ A4 )
                  = ( groups2555765274223993564er_rat @ H @ B5 ) )
                = ( ( groups2555765274223993564er_rat @ G @ C4 )
                  = ( groups2555765274223993564er_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8883_prod_Osame__carrier,axiom,
    ! [C4: set_o,A4: set_o,B5: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A4 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G @ A4 )
                  = ( groups3504817904513533571_o_nat @ H @ B5 ) )
                = ( ( groups3504817904513533571_o_nat @ G @ C4 )
                  = ( groups3504817904513533571_o_nat @ H @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8884_sum_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8885_sum_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) ) @ ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) )
          = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A4 ) @ ( groups6602215022474089585er_rat @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8886_sum_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8887_sum_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) ) @ ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) )
          = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A4 ) @ ( groups7237345082560585321er_nat @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8888_sum_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) ) @ ( groups7234854612051535045er_int @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) )
          = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A4 ) @ ( groups7234854612051535045er_int @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8889_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A4 ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8890_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A4 ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8891_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8892_sum_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8893_sum_Ounion__inter,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) ) @ ( groups342789780944988191at_rat @ G @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) )
          = ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ A4 ) @ ( groups342789780944988191at_rat @ G @ B5 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8894_sum_OInt__Diff,axiom,
    ! [A4: set_int,G: int > rat,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G @ A4 )
        = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8895_sum_OInt__Diff,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G @ A4 )
        = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8896_sum_OInt__Diff,axiom,
    ! [A4: set_int,G: int > nat,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G @ A4 )
        = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8897_sum_OInt__Diff,axiom,
    ! [A4: set_Code_integer,G: code_integer > nat,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat @ G @ A4 )
        = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8898_sum_OInt__Diff,axiom,
    ! [A4: set_Code_integer,G: code_integer > int,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7234854612051535045er_int @ G @ A4 )
        = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8899_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: nat > rat,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G @ A4 )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8900_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: nat > int,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3539618377306564664at_int @ G @ A4 )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8901_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: nat > nat,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3542108847815614940at_nat @ G @ A4 )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8902_sum_OInt__Diff,axiom,
    ! [A4: set_int,G: int > int,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4538972089207619220nt_int @ G @ A4 )
        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8903_sum_OInt__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat,B5: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( groups342789780944988191at_rat @ G @ A4 )
        = ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8904_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G: int > assn] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A4 ) @ ( groups7882442080178216443t_assn @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8905_prod_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) )
          = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A4 ) @ ( groups1304777262505850412r_assn @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8906_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A4 ) @ ( groups6906906614972039071t_assn @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8907_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8908_prod_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) )
          = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A4 ) @ ( groups2555765274223993564er_rat @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8909_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A4 ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8910_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8911_prod_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) )
          = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ A4 ) @ ( groups3190895334310489300er_nat @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8912_prod_Ounion__inter,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) )
          = ( times_times_int @ ( groups3188404863801439024er_int @ G @ A4 ) @ ( groups3188404863801439024er_int @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8913_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A4 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_8914_sum__diff__nat,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A4 )
       => ( ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8915_sum__diff__nat,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8916_sum__diff__nat,axiom,
    ! [B5: set_int,A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8917_sum__diff__nat,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8918_push__bit__mask__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N2 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).

% push_bit_mask_eq
thf(fact_8919_sum_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8920_sum_OIf__cases,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ H @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) @ ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8921_sum_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8922_sum_OIf__cases,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat
          @ ^ [X4: code_integer] : ( if_nat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ H @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) @ ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A4 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8923_sum_OIf__cases,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,H: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7234854612051535045er_int
          @ ^ [X4: code_integer] : ( if_int @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_int @ ( groups7234854612051535045er_int @ H @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) @ ( groups7234854612051535045er_int @ G @ ( inf_in1364745209274528805nteger @ A4 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8924_sum_OIf__cases,axiom,
    ! [A4: set_nat,P2: nat > $o,H: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8925_sum_OIf__cases,axiom,
    ! [A4: set_nat,P2: nat > $o,H: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3539618377306564664at_int
          @ ^ [X4: nat] : ( if_int @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8926_sum_OIf__cases,axiom,
    ! [A4: set_nat,P2: nat > $o,H: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( if_nat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8927_sum_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X4: int] : ( if_int @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8928_sum_OIf__cases,axiom,
    ! [A4: set_list_nat,P2: list_nat > $o,H: list_nat > rat,G: list_nat > rat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( groups3760926236672600436at_rat
          @ ^ [X4: list_nat] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_rat @ ( groups3760926236672600436at_rat @ H @ ( inf_inf_set_list_nat @ A4 @ ( collect_list_nat @ P2 ) ) ) @ ( groups3760926236672600436at_rat @ G @ ( inf_inf_set_list_nat @ A4 @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P2 ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8929_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > assn,G: int > assn] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7882442080178216443t_assn
          @ ^ [X4: int] : ( if_assn @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_assn @ ( groups7882442080178216443t_assn @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8930_prod_OIf__cases,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,H: code_integer > assn,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1304777262505850412r_assn
          @ ^ [X4: code_integer] : ( if_assn @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_assn @ ( groups1304777262505850412r_assn @ H @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A4 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8931_prod_OIf__cases,axiom,
    ! [A4: set_nat,P2: nat > $o,H: nat > assn,G: nat > assn] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups6906906614972039071t_assn
          @ ^ [X4: nat] : ( if_assn @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_assn @ ( groups6906906614972039071t_assn @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8932_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1072433553688619179nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_rat @ ( groups1072433553688619179nt_rat @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8933_prod_OIf__cases,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,H: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups2555765274223993564er_rat
          @ ^ [X4: code_integer] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_rat @ ( groups2555765274223993564er_rat @ H @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8934_prod_OIf__cases,axiom,
    ! [A4: set_nat,P2: nat > $o,H: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups73079841787564623at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8935_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1707563613775114915nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_nat @ ( groups1707563613775114915nt_nat @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups1707563613775114915nt_nat @ G @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8936_prod_OIf__cases,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,H: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups3190895334310489300er_nat
          @ ^ [X4: code_integer] : ( if_nat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_nat @ ( groups3190895334310489300er_nat @ H @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A4 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8937_prod_OIf__cases,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o,H: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups3188404863801439024er_int
          @ ^ [X4: code_integer] : ( if_int @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_int @ ( groups3188404863801439024er_int @ H @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A4 @ ( uminus804700908173204444nteger @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8938_prod_OIf__cases,axiom,
    ! [A4: set_nat,P2: nat > $o,H: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups708209901874060359at_nat
          @ ^ [X4: nat] : ( if_nat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_8939_prod__mono__strict,axiom,
    ! [A4: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ A4 )
           => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
              & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bo3990330152332043303nteger )
         => ( ord_le6747313008572928689nteger @ ( groups3674199335183972705nteger @ F @ A4 ) @ ( groups3674199335183972705nteger @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8940_prod__mono__strict,axiom,
    ! [A4: set_o,F: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ A4 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ A4 )
           => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
              & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_o )
         => ( ord_le6747313008572928689nteger @ ( groups7694694392188491536nteger @ F @ A4 ) @ ( groups7694694392188491536nteger @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8941_prod__mono__strict,axiom,
    ! [A4: set_nat,F: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ A4 )
           => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
              & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_le6747313008572928689nteger @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8942_prod__mono__strict,axiom,
    ! [A4: set_int,F: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A4 )
           => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
              & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_le6747313008572928689nteger @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8943_prod__mono__strict,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bo3990330152332043303nteger )
         => ( ord_less_rat @ ( groups2555765274223993564er_rat @ F @ A4 ) @ ( groups2555765274223993564er_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8944_prod__mono__strict,axiom,
    ! [A4: set_o,F: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_o )
         => ( ord_less_rat @ ( groups2869687844427037835_o_rat @ F @ A4 ) @ ( groups2869687844427037835_o_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8945_prod__mono__strict,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8946_prod__mono__strict,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8947_prod__mono__strict,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [I3: code_integer] :
            ( ( member_Code_integer @ I3 @ A4 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bo3990330152332043303nteger )
         => ( ord_less_nat @ ( groups3190895334310489300er_nat @ F @ A4 ) @ ( groups3190895334310489300er_nat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8948_prod__mono__strict,axiom,
    ! [A4: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ A4 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_o )
         => ( ord_less_nat @ ( groups3504817904513533571_o_nat @ F @ A4 ) @ ( groups3504817904513533571_o_nat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8949_sum__mono2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups4406642042086082107nteger @ F @ A4 ) @ ( groups4406642042086082107nteger @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8950_sum__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A4 ) @ ( groups7501900531339628137nteger @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8951_sum__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ! [B3: code_integer] :
              ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ F @ A4 ) @ ( groups879477027807139574nteger @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8952_sum__mono2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ ( groups7872700643590313910_o_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8953_sum__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8954_sum__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ! [B3: code_integer] :
              ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8955_sum__mono2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > nat] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B5 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( groups8507830703676809646_o_nat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8956_sum__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ! [B3: code_integer] :
              ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8957_sum__mono2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > int] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B5 @ A4 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
         => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A4 ) @ ( groups8505340233167759370_o_int @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8958_sum__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_8959_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7873554091576472773nteger @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_p5714425477246183910nteger @ ( groups7873554091576472773nteger @ G @ A4 ) @ ( groups7873554091576472773nteger @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8960_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups879477027807139574nteger @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( plus_p5714425477246183910nteger @ ( groups879477027807139574nteger @ G @ A4 ) @ ( groups879477027807139574nteger @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8961_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8962_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A4 ) @ ( groups6602215022474089585er_rat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8963_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8964_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A4 ) @ ( groups7237345082560585321er_nat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8965_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A4 ) @ ( groups7234854612051535045er_int @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8966_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7501900531339628137nteger @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ A4 ) @ ( groups7501900531339628137nteger @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8967_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A4 ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8968_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A4 ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8969_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A4 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8970_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: code_integer > nat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A4 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8971_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: code_integer > int,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A4 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8972_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > rat,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A4 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8973_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > nat,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A4 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8974_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > int,X: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X @ A4 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8975_sum_Oinsert__remove,axiom,
    ! [A4: set_nat,G: nat > rat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A4 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8976_sum_Oinsert__remove,axiom,
    ! [A4: set_nat,G: nat > int,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A4 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8977_sum_Oinsert__remove,axiom,
    ! [A4: set_int,G: int > rat,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A4 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8978_sum_Oinsert__remove,axiom,
    ! [A4: set_int,G: int > nat,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A4 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8979_sum_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( groups6602215022474089585er_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8980_sum_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( groups7237345082560585321er_nat @ G @ A4 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8981_sum_Oremove,axiom,
    ! [A4: set_Code_integer,X: code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ X @ A4 )
       => ( ( groups7234854612051535045er_int @ G @ A4 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8982_sum_Oremove,axiom,
    ! [A4: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ( groups7872700643590313910_o_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8983_sum_Oremove,axiom,
    ! [A4: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ( groups8507830703676809646_o_nat @ G @ A4 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8984_sum_Oremove,axiom,
    ! [A4: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X @ A4 )
       => ( ( groups8505340233167759370_o_int @ G @ A4 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8985_sum_Oremove,axiom,
    ! [A4: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8986_sum_Oremove,axiom,
    ! [A4: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( groups3539618377306564664at_int @ G @ A4 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8987_sum_Oremove,axiom,
    ! [A4: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( groups3906332499630173760nt_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8988_sum_Oremove,axiom,
    ! [A4: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( groups4541462559716669496nt_nat @ G @ A4 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8989_sum__Un,axiom,
    ! [A4: set_int,B5: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8990_sum__Un,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups6602215022474089585er_rat @ F @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) @ ( groups6602215022474089585er_rat @ F @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8991_sum__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8992_sum__Un,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups7234854612051535045er_int @ F @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups7234854612051535045er_int @ F @ A4 ) @ ( groups7234854612051535045er_int @ F @ B5 ) ) @ ( groups7234854612051535045er_int @ F @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8993_sum__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8994_sum__Un,axiom,
    ! [A4: set_int,B5: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups4538972089207619220nt_int @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) @ ( groups4538972089207619220nt_int @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8995_sum__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ( groups342789780944988191at_rat @ F @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ F @ A4 ) @ ( groups342789780944988191at_rat @ F @ B5 ) ) @ ( groups342789780944988191at_rat @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8996_sum__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ( groups975429370522433651at_int @ F @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups975429370522433651at_int @ F @ A4 ) @ ( groups975429370522433651at_int @ F @ B5 ) ) @ ( groups975429370522433651at_int @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8997_sum__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,F: produc4166570645942440679at_nat > rat] :
      ( ( finite1918287321285529104at_nat @ A4 )
     => ( ( finite1918287321285529104at_nat @ B5 )
       => ( ( groups6424767709179651461at_rat @ F @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups6424767709179651461at_rat @ F @ A4 ) @ ( groups6424767709179651461at_rat @ F @ B5 ) ) @ ( groups6424767709179651461at_rat @ F @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8998_sum__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > rat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( finite4343798906461161616at_nat @ B5 )
       => ( ( groups3225780264831618053at_rat @ F @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups3225780264831618053at_rat @ F @ A4 ) @ ( groups3225780264831618053at_rat @ F @ B5 ) ) @ ( groups3225780264831618053at_rat @ F @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8999_sum_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( ( inf_in1364745209274528805nteger @ A4 @ B5 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A4 ) @ ( groups6602215022474089585er_rat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9000_sum_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( ( inf_in1364745209274528805nteger @ A4 @ B5 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A4 ) @ ( groups7237345082560585321er_nat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9001_sum_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( ( inf_in1364745209274528805nteger @ A4 @ B5 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A4 ) @ ( groups7234854612051535045er_int @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9002_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B5: set_o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B5 )
       => ( ( ( inf_inf_set_o @ A4 @ B5 )
            = bot_bot_set_o )
         => ( ( groups7872700643590313910_o_rat @ G @ ( sup_sup_set_o @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups7872700643590313910_o_rat @ G @ A4 ) @ ( groups7872700643590313910_o_rat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9003_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B5: set_o,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B5 )
       => ( ( ( inf_inf_set_o @ A4 @ B5 )
            = bot_bot_set_o )
         => ( ( groups8507830703676809646_o_nat @ G @ ( sup_sup_set_o @ A4 @ B5 ) )
            = ( plus_plus_nat @ ( groups8507830703676809646_o_nat @ G @ A4 ) @ ( groups8507830703676809646_o_nat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9004_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B5: set_o,G: $o > int] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B5 )
       => ( ( ( inf_inf_set_o @ A4 @ B5 )
            = bot_bot_set_o )
         => ( ( groups8505340233167759370_o_int @ G @ ( sup_sup_set_o @ A4 @ B5 ) )
            = ( plus_plus_int @ ( groups8505340233167759370_o_int @ G @ A4 ) @ ( groups8505340233167759370_o_int @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9005_sum_Ounion__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ( inf_inf_set_nat @ A4 @ B5 )
            = bot_bot_set_nat )
         => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A4 ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9006_sum_Ounion__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ( inf_inf_set_nat @ A4 @ B5 )
            = bot_bot_set_nat )
         => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A4 ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9007_sum_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ( inf_inf_set_int @ A4 @ B5 )
            = bot_bot_set_int )
         => ( ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9008_sum_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ( inf_inf_set_int @ A4 @ B5 )
            = bot_bot_set_int )
         => ( ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_9009_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G @ A4 ) @ ( groups3827104343326376752nteger @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9010_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3674199335183972705nteger @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ A4 ) @ ( groups3674199335183972705nteger @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9011_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ A4 ) @ ( groups3455450783089532116nteger @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9012_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G: int > assn] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A4 ) @ ( groups7882442080178216443t_assn @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9013_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A4 ) @ ( groups1304777262505850412r_assn @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9014_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_assn ) )
         => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A4 ) @ ( groups6906906614972039071t_assn @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9015_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9016_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A4 ) @ ( groups2555765274223993564er_rat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9017_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A4 ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9018_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups1707563613775114915nt_nat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_9019_prod_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( ( inf_in1364745209274528805nteger @ A4 @ B5 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A4 ) @ ( groups1304777262505850412r_assn @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9020_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B5: set_o,G: $o > assn] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B5 )
       => ( ( ( inf_inf_set_o @ A4 @ B5 )
            = bot_bot_set_o )
         => ( ( groups5301882518646026715o_assn @ G @ ( sup_sup_set_o @ A4 @ B5 ) )
            = ( times_times_assn @ ( groups5301882518646026715o_assn @ G @ A4 ) @ ( groups5301882518646026715o_assn @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9021_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ( inf_inf_set_nat @ A4 @ B5 )
            = bot_bot_set_nat )
         => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A4 ) @ ( groups6906906614972039071t_assn @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9022_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G: int > assn] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ( inf_inf_set_int @ A4 @ B5 )
            = bot_bot_set_int )
         => ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A4 ) @ ( groups7882442080178216443t_assn @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9023_prod_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( ( inf_in1364745209274528805nteger @ A4 @ B5 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A4 ) @ ( groups2555765274223993564er_rat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9024_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B5: set_o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B5 )
       => ( ( ( inf_inf_set_o @ A4 @ B5 )
            = bot_bot_set_o )
         => ( ( groups2869687844427037835_o_rat @ G @ ( sup_sup_set_o @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups2869687844427037835_o_rat @ G @ A4 ) @ ( groups2869687844427037835_o_rat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9025_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ( inf_inf_set_nat @ A4 @ B5 )
            = bot_bot_set_nat )
         => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A4 ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9026_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ( inf_inf_set_int @ A4 @ B5 )
            = bot_bot_set_int )
         => ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9027_prod_Ounion__disjoint,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( ( inf_in1364745209274528805nteger @ A4 @ B5 )
            = bot_bo3990330152332043303nteger )
         => ( ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ A4 ) @ ( groups3190895334310489300er_nat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9028_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B5: set_o,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B5 )
       => ( ( ( inf_inf_set_o @ A4 @ B5 )
            = bot_bot_set_o )
         => ( ( groups3504817904513533571_o_nat @ G @ ( sup_sup_set_o @ A4 @ B5 ) )
            = ( times_times_nat @ ( groups3504817904513533571_o_nat @ G @ A4 ) @ ( groups3504817904513533571_o_nat @ G @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9029_sum_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9030_sum_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups6602215022474089585er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9031_sum_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9032_sum_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups7237345082560585321er_nat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9033_sum_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups7234854612051535045er_int @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9034_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9035_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9036_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3542108847815614940at_nat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9037_sum_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9038_sum_Ounion__diff2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ( groups342789780944988191at_rat @ G @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) @ ( groups342789780944988191at_rat @ G @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_9039_sum__Un2,axiom,
    ! [A4: set_int,B5: set_int,F: int > rat] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A4 @ B5 ) )
     => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9040_sum__Un2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
     => ( ( groups6602215022474089585er_rat @ F @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups6602215022474089585er_rat @ F @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9041_sum__Un2,axiom,
    ! [A4: set_int,B5: set_int,F: int > nat] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A4 @ B5 ) )
     => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9042_sum__Un2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
     => ( ( groups7237345082560585321er_nat @ F @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups7237345082560585321er_nat @ F @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9043_sum__Un2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
     => ( ( groups7234854612051535045er_int @ F @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups7234854612051535045er_int @ F @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9044_sum__Un2,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A4 @ B5 ) )
     => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9045_sum__Un2,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A4 @ B5 ) )
     => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9046_sum__Un2,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A4 @ B5 ) )
     => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9047_sum__Un2,axiom,
    ! [A4: set_int,B5: set_int,F: int > int] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A4 @ B5 ) )
     => ( ( groups4538972089207619220nt_int @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups4538972089207619220nt_int @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9048_sum__Un2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
     => ( ( groups342789780944988191at_rat @ F @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ F @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) @ ( groups342789780944988191at_rat @ F @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) @ ( groups342789780944988191at_rat @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ) ).

% sum_Un2
thf(fact_9049_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G: int > assn] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( times_times_assn @ ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups7882442080178216443t_assn @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9050_prod_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( times_times_assn @ ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9051_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > assn] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( times_times_assn @ ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9052_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups1072433553688619179nt_rat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9053_prod_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9054_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9055_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups1707563613775114915nt_nat @ G @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( times_times_nat @ ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups1707563613775114915nt_nat @ G @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9056_prod_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( times_times_nat @ ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9057_prod_Ounion__diff2,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( times_times_int @ ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9058_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups708209901874060359at_nat @ G @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( times_times_nat @ ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_9059_sum__Un__nat,axiom,
    ! [A4: set_int,B5: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_9060_sum__Un__nat,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( groups7237345082560585321er_nat @ F @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ F @ B5 ) ) @ ( groups7237345082560585321er_nat @ F @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_9061_sum__Un__nat,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ( groups977919841031483927at_nat @ F @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ F @ B5 ) ) @ ( groups977919841031483927at_nat @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_9062_sum__Un__nat,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,F: produc4166570645942440679at_nat > nat] :
      ( ( finite1918287321285529104at_nat @ A4 )
     => ( ( finite1918287321285529104at_nat @ B5 )
       => ( ( groups7059897769266147197at_nat @ F @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups7059897769266147197at_nat @ F @ A4 ) @ ( groups7059897769266147197at_nat @ F @ B5 ) ) @ ( groups7059897769266147197at_nat @ F @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_9063_sum__Un__nat,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( finite4343798906461161616at_nat @ B5 )
       => ( ( groups3860910324918113789at_nat @ F @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3860910324918113789at_nat @ F @ A4 ) @ ( groups3860910324918113789at_nat @ F @ B5 ) ) @ ( groups3860910324918113789at_nat @ F @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_9064_sum__Un__nat,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_9065_sum_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > rat,C2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups6602215022474089585er_rat @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups6602215022474089585er_rat @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9066_sum_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > nat,C2: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K3: code_integer] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups7237345082560585321er_nat @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K3: code_integer] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups7237345082560585321er_nat @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9067_sum_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > int,C2: code_integer > int] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups7234854612051535045er_int
              @ ^ [K3: code_integer] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_int @ ( B @ A ) @ ( groups7234854612051535045er_int @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups7234854612051535045er_int
              @ ^ [K3: code_integer] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups7234854612051535045er_int @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9068_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: $o > rat,C2: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups7872700643590313910_o_rat @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups7872700643590313910_o_rat @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9069_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: $o > nat,C2: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups8507830703676809646_o_nat @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups8507830703676809646_o_nat @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9070_sum_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: $o > int,C2: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups8505340233167759370_o_int
              @ ^ [K3: $o] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_int @ ( B @ A ) @ ( groups8505340233167759370_o_int @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups8505340233167759370_o_int
              @ ^ [K3: $o] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups8505340233167759370_o_int @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9071_sum_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: nat > rat,C2: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups2906978787729119204at_rat @ C2 @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups2906978787729119204at_rat @ C2 @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9072_sum_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: nat > int,C2: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups3539618377306564664at_int
              @ ^ [K3: nat] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_int @ ( B @ A ) @ ( groups3539618377306564664at_int @ C2 @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups3539618377306564664at_int
              @ ^ [K3: nat] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups3539618377306564664at_int @ C2 @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9073_sum_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: int > rat,C2: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C2 @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups3906332499630173760nt_rat @ C2 @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9074_sum_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: int > nat,C2: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups4541462559716669496nt_nat @ C2 @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups4541462559716669496nt_nat @ C2 @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_9075_sum__div__partition,axiom,
    ! [A4: set_int,F: int > nat,B: nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( divide_divide_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ B )
        = ( plus_plus_nat
          @ ( groups4541462559716669496nt_nat
            @ ^ [A5: int] : ( divide_divide_nat @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_int @ A4
              @ ( collect_int
                @ ^ [A5: int] : ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups4541462559716669496nt_nat @ F
              @ ( inf_inf_set_int @ A4
                @ ( collect_int
                  @ ^ [A5: int] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9076_sum__div__partition,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat,B: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( divide_divide_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ B )
        = ( plus_plus_nat
          @ ( groups7237345082560585321er_nat
            @ ^ [A5: code_integer] : ( divide_divide_nat @ ( F @ A5 ) @ B )
            @ ( inf_in1364745209274528805nteger @ A4
              @ ( collect_Code_integer
                @ ^ [A5: code_integer] : ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups7237345082560585321er_nat @ F
              @ ( inf_in1364745209274528805nteger @ A4
                @ ( collect_Code_integer
                  @ ^ [A5: code_integer] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9077_sum__div__partition,axiom,
    ! [A4: set_Code_integer,F: code_integer > int,B: int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( divide_divide_int @ ( groups7234854612051535045er_int @ F @ A4 ) @ B )
        = ( plus_plus_int
          @ ( groups7234854612051535045er_int
            @ ^ [A5: code_integer] : ( divide_divide_int @ ( F @ A5 ) @ B )
            @ ( inf_in1364745209274528805nteger @ A4
              @ ( collect_Code_integer
                @ ^ [A5: code_integer] : ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups7234854612051535045er_int @ F
              @ ( inf_in1364745209274528805nteger @ A4
                @ ( collect_Code_integer
                  @ ^ [A5: code_integer] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9078_sum__div__partition,axiom,
    ! [A4: set_nat,F: nat > int,B: int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( divide_divide_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ B )
        = ( plus_plus_int
          @ ( groups3539618377306564664at_int
            @ ^ [A5: nat] : ( divide_divide_int @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_nat @ A4
              @ ( collect_nat
                @ ^ [A5: nat] : ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups3539618377306564664at_int @ F
              @ ( inf_inf_set_nat @ A4
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9079_sum__div__partition,axiom,
    ! [A4: set_int,F: int > code_natural,B: code_natural] :
      ( ( finite_finite_int @ A4 )
     => ( ( divide5121882707175180666atural @ ( groups6697149243333190288atural @ F @ A4 ) @ B )
        = ( plus_p4538020629002901425atural
          @ ( groups6697149243333190288atural
            @ ^ [A5: int] : ( divide5121882707175180666atural @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_int @ A4
              @ ( collect_int
                @ ^ [A5: int] : ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide5121882707175180666atural
            @ ( groups6697149243333190288atural @ F
              @ ( inf_inf_set_int @ A4
                @ ( collect_int
                  @ ^ [A5: int] :
                      ~ ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9080_sum__div__partition,axiom,
    ! [A4: set_Code_integer,F: code_integer > code_natural,B: code_natural] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( divide5121882707175180666atural @ ( groups8926444216418632897atural @ F @ A4 ) @ B )
        = ( plus_p4538020629002901425atural
          @ ( groups8926444216418632897atural
            @ ^ [A5: code_integer] : ( divide5121882707175180666atural @ ( F @ A5 ) @ B )
            @ ( inf_in1364745209274528805nteger @ A4
              @ ( collect_Code_integer
                @ ^ [A5: code_integer] : ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide5121882707175180666atural
            @ ( groups8926444216418632897atural @ F
              @ ( inf_in1364745209274528805nteger @ A4
                @ ( collect_Code_integer
                  @ ^ [A5: code_integer] :
                      ~ ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9081_sum__div__partition,axiom,
    ! [A4: set_nat,F: nat > code_natural,B: code_natural] :
      ( ( finite_finite_nat @ A4 )
     => ( ( divide5121882707175180666atural @ ( groups6325495683096345652atural @ F @ A4 ) @ B )
        = ( plus_p4538020629002901425atural
          @ ( groups6325495683096345652atural
            @ ^ [A5: nat] : ( divide5121882707175180666atural @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_nat @ A4
              @ ( collect_nat
                @ ^ [A5: nat] : ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide5121882707175180666atural
            @ ( groups6325495683096345652atural @ F
              @ ( inf_inf_set_nat @ A4
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ~ ( dvd_dvd_Code_natural @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9082_sum__div__partition,axiom,
    ! [A4: set_nat,F: nat > nat,B: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( divide_divide_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ B )
        = ( plus_plus_nat
          @ ( groups3542108847815614940at_nat
            @ ^ [A5: nat] : ( divide_divide_nat @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_nat @ A4
              @ ( collect_nat
                @ ^ [A5: nat] : ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups3542108847815614940at_nat @ F
              @ ( inf_inf_set_nat @ A4
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9083_sum__div__partition,axiom,
    ! [A4: set_int,F: int > int,B: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( divide_divide_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ B )
        = ( plus_plus_int
          @ ( groups4538972089207619220nt_int
            @ ^ [A5: int] : ( divide_divide_int @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_int @ A4
              @ ( collect_int
                @ ^ [A5: int] : ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups4538972089207619220nt_int @ F
              @ ( inf_inf_set_int @ A4
                @ ( collect_int
                  @ ^ [A5: int] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9084_sum__div__partition,axiom,
    ! [A4: set_list_nat,F: list_nat > nat,B: nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( divide_divide_nat @ ( groups4396056296759096172at_nat @ F @ A4 ) @ B )
        = ( plus_plus_nat
          @ ( groups4396056296759096172at_nat
            @ ^ [A5: list_nat] : ( divide_divide_nat @ ( F @ A5 ) @ B )
            @ ( inf_inf_set_list_nat @ A4
              @ ( collect_list_nat
                @ ^ [A5: list_nat] : ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups4396056296759096172at_nat @ F
              @ ( inf_inf_set_list_nat @ A4
                @ ( collect_list_nat
                  @ ^ [A5: list_nat] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A5 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_9085_prod_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > assn,C2: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K3: code_integer] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_assn @ ( B @ A ) @ ( groups1304777262505850412r_assn @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K3: code_integer] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups1304777262505850412r_assn @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9086_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: $o > assn,C2: $o > assn] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K3: $o] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_assn @ ( B @ A ) @ ( groups5301882518646026715o_assn @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K3: $o] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups5301882518646026715o_assn @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9087_prod_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: nat > assn,C2: nat > assn] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K3: nat] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_assn @ ( B @ A ) @ ( groups6906906614972039071t_assn @ C2 @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K3: nat] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups6906906614972039071t_assn @ C2 @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9088_prod_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: int > assn,C2: int > assn] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K3: int] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_assn @ ( B @ A ) @ ( groups7882442080178216443t_assn @ C2 @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K3: int] : ( if_assn @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups7882442080178216443t_assn @ C2 @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9089_prod_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > rat,C2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups2555765274223993564er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_rat @ ( B @ A ) @ ( groups2555765274223993564er_rat @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups2555765274223993564er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups2555765274223993564er_rat @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9090_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: $o > rat,C2: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_rat @ ( B @ A ) @ ( groups2869687844427037835_o_rat @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups2869687844427037835_o_rat @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9091_prod_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: nat > rat,C2: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_rat @ ( B @ A ) @ ( groups73079841787564623at_rat @ C2 @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups73079841787564623at_rat @ C2 @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9092_prod_Odelta__remove,axiom,
    ! [S: set_int,A: int,B: int > rat,C2: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_rat @ ( B @ A ) @ ( groups1072433553688619179nt_rat @ C2 @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups1072433553688619179nt_rat @ C2 @ ( minus_minus_set_int @ S @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9093_prod_Odelta__remove,axiom,
    ! [S: set_Code_integer,A: code_integer,B: code_integer > nat,C2: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( ( member_Code_integer @ A @ S )
         => ( ( groups3190895334310489300er_nat
              @ ^ [K3: code_integer] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_nat @ ( B @ A ) @ ( groups3190895334310489300er_nat @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S )
         => ( ( groups3190895334310489300er_nat
              @ ^ [K3: code_integer] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups3190895334310489300er_nat @ C2 @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9094_prod_Odelta__remove,axiom,
    ! [S: set_o,A: $o,B: $o > nat,C2: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A @ S )
         => ( ( groups3504817904513533571_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( times_times_nat @ ( B @ A ) @ ( groups3504817904513533571_o_nat @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S )
         => ( ( groups3504817904513533571_o_nat
              @ ^ [K3: $o] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S )
            = ( groups3504817904513533571_o_nat @ C2 @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9095_effect__of__listI,axiom,
    ! [A: array_nat,H5: heap_e7401611519738050253t_unit,Xs: list_nat,H: heap_e7401611519738050253t_unit] :
      ( ( ( produc1210945592254923927t_unit @ A @ H5 )
        = ( array_alloc_nat @ Xs @ H ) )
     => ( heap_T1188250621483087889ay_nat @ ( array_of_list_nat @ Xs ) @ H @ H5 @ A @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_nat @ Xs ) ) ) ) ).

% effect_of_listI
thf(fact_9096_effect__of__listI,axiom,
    ! [A: array_o,H5: heap_e7401611519738050253t_unit,Xs: list_o,H: heap_e7401611519738050253t_unit] :
      ( ( ( produc3742984184817956739t_unit @ A @ H5 )
        = ( array_alloc_o @ Xs @ H ) )
     => ( heap_T1873992626244518189rray_o @ ( array_of_list_o @ Xs ) @ H @ H5 @ A @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_o @ Xs ) ) ) ) ).

% effect_of_listI
thf(fact_9097_effect__of__listI,axiom,
    ! [A: array_int,H5: heap_e7401611519738050253t_unit,Xs: list_int,H: heap_e7401611519738050253t_unit] :
      ( ( ( produc2888458824036673339t_unit @ A @ H5 )
        = ( array_alloc_int @ Xs @ H ) )
     => ( heap_T6233771638828666989ay_int @ ( array_of_list_int @ Xs ) @ H @ H5 @ A @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_int @ Xs ) ) ) ) ).

% effect_of_listI
thf(fact_9098_sum__strict__mono2,axiom,
    ! [B5: set_o,A4: set_o,B: $o,F: $o > code_integer] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B5 @ A4 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ B5 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups4406642042086082107nteger @ F @ A4 ) @ ( groups4406642042086082107nteger @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9099_sum__strict__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,B: nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A4 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ B5 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups7501900531339628137nteger @ F @ A4 ) @ ( groups7501900531339628137nteger @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9100_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,B: code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B5 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups879477027807139574nteger @ F @ A4 ) @ ( groups879477027807139574nteger @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9101_sum__strict__mono2,axiom,
    ! [B5: set_o,A4: set_o,B: $o,F: $o > rat] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ ( groups7872700643590313910_o_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9102_sum__strict__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,B: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9103_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,B: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9104_sum__strict__mono2,axiom,
    ! [B5: set_o,A4: set_o,B: $o,F: $o > nat] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B5 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( groups8507830703676809646_o_nat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9105_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,B: code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9106_sum__strict__mono2,axiom,
    ! [B5: set_o,A4: set_o,B: $o,F: $o > int] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B5 @ A4 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X3: $o] :
                  ( ( member_o @ X3 @ B5 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
             => ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A4 ) @ ( groups8505340233167759370_o_int @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9107_sum__strict__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,B: nat,F: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A4 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ B5 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_9108_member__le__sum,axiom,
    ! [I2: code_integer,A4: set_Code_integer,F: code_integer > code_integer] :
      ( ( member_Code_integer @ I2 @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups879477027807139574nteger @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9109_member__le__sum,axiom,
    ! [I2: $o,A4: set_o,F: $o > code_integer] :
      ( ( member_o @ I2 @ A4 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( minus_minus_set_o @ A4 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite_finite_o @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups4406642042086082107nteger @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9110_member__le__sum,axiom,
    ! [I2: nat,A4: set_nat,F: nat > code_integer] :
      ( ( member_nat @ I2 @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite_finite_nat @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups7501900531339628137nteger @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9111_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F: int > code_integer] :
      ( ( member_int @ I2 @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups7873554091576472773nteger @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9112_member__le__sum,axiom,
    ! [I2: code_integer,A4: set_Code_integer,F: code_integer > rat] :
      ( ( member_Code_integer @ I2 @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups6602215022474089585er_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9113_member__le__sum,axiom,
    ! [I2: $o,A4: set_o,F: $o > rat] :
      ( ( member_o @ I2 @ A4 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( minus_minus_set_o @ A4 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_o @ A4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups7872700643590313910_o_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9114_member__le__sum,axiom,
    ! [I2: nat,A4: set_nat,F: nat > rat] :
      ( ( member_nat @ I2 @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_nat @ A4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9115_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F: int > rat] :
      ( ( member_int @ I2 @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9116_member__le__sum,axiom,
    ! [I2: code_integer,A4: set_Code_integer,F: code_integer > nat] :
      ( ( member_Code_integer @ I2 @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( groups7237345082560585321er_nat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9117_member__le__sum,axiom,
    ! [I2: $o,A4: set_o,F: $o > nat] :
      ( ( member_o @ I2 @ A4 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( minus_minus_set_o @ A4 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( finite_finite_o @ A4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( groups8507830703676809646_o_nat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_9118_prod__mono2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B3 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A4 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A4 ) @ ( groups7694694392188491536nteger @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9119_prod__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B3 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A4 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9120_prod__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ! [B3: code_integer] :
              ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B3 ) ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ A4 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3674199335183972705nteger @ F @ A4 ) @ ( groups3674199335183972705nteger @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9121_prod__mono2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A4 ) @ ( groups2869687844427037835_o_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9122_prod__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9123_prod__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ! [B3: code_integer] :
              ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups2555765274223993564er_rat @ F @ A4 ) @ ( groups2555765274223993564er_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9124_prod__mono2,axiom,
    ! [B5: set_o,A4: set_o,F: $o > int] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A4 @ B5 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B5 @ A4 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A4 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A4 ) @ ( groups3502327434004483295_o_int @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9125_prod__mono2,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ! [B3: code_integer] :
              ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B5 @ A4 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ A4 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups3188404863801439024er_int @ F @ A4 ) @ ( groups3188404863801439024er_int @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9126_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9127_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9128_prod__Un,axiom,
    ! [A4: set_int,B5: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups1072433553688619179nt_rat @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) @ ( groups1072433553688619179nt_rat @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_9129_prod__Un,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups2555765274223993564er_rat @ F @ ( sup_su848401254843788991nteger @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups2555765274223993564er_rat @ F @ A4 ) @ ( groups2555765274223993564er_rat @ F @ B5 ) ) @ ( groups2555765274223993564er_rat @ F @ ( inf_in1364745209274528805nteger @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_9130_prod__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups3442636767675653108at_rat @ F @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups3442636767675653108at_rat @ F @ A4 ) @ ( groups3442636767675653108at_rat @ F @ B5 ) ) @ ( groups3442636767675653108at_rat @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_9131_prod__Un,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat,F: produc4166570645942440679at_nat > rat] :
      ( ( finite1918287321285529104at_nat @ A4 )
     => ( ( finite1918287321285529104at_nat @ B5 )
       => ( ! [X3: produc4166570645942440679at_nat] :
              ( ( member6689249552917799696at_nat @ X3 @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups5938585286922990810at_rat @ F @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups5938585286922990810at_rat @ F @ A4 ) @ ( groups5938585286922990810at_rat @ F @ B5 ) ) @ ( groups5938585286922990810at_rat @ F @ ( inf_in1697001100524423349at_nat @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_9132_prod__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > rat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( finite4343798906461161616at_nat @ B5 )
       => ( ! [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups8874911130973611098at_rat @ F @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups8874911130973611098at_rat @ F @ A4 ) @ ( groups8874911130973611098at_rat @ F @ B5 ) ) @ ( groups8874911130973611098at_rat @ F @ ( inf_in7913087082777306421at_nat @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_9133_prod__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( F @ X3 )
               != zero_zero_rat ) )
         => ( ( groups73079841787564623at_rat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) @ ( groups73079841787564623at_rat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_9134_or__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se9127793120404214118atural @ A @ one_one_Code_natural )
      = ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_9135_or__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_9136_or__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_9137_or__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_9138_one__or__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ A )
      = ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_9139_one__or__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_9140_one__or__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_9141_one__or__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_9142_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_9143_or__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9144_take__bit__sum,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N: nat,A5: code_integer] :
          ( groups7501900531339628137nteger
          @ ^ [K3: nat] : ( bit_se7788150548672797655nteger @ K3 @ ( zero_n356916108424825756nteger @ ( bit_se9216721137139052372nteger @ A5 @ K3 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% take_bit_sum
thf(fact_9145_take__bit__sum,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,A5: int] :
          ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( bit_se545348938243370406it_int @ K3 @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ A5 @ K3 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% take_bit_sum
thf(fact_9146_take__bit__sum,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,A5: nat] :
          ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( bit_se547839408752420682it_nat @ K3 @ ( zero_n2687167440665602831ol_nat @ ( bit_se1148574629649215175it_nat @ A5 @ K3 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% take_bit_sum
thf(fact_9147_or__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9148_or__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9149_of__list__def,axiom,
    ( array_of_list_nat
    = ( ^ [Xs3: list_nat] :
          ( heap_T4325564685996367510ay_nat
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( produc8657510640128716596it_nat
              @ ^ [R5: array_nat,H7: heap_e7401611519738050253t_unit] : ( produc4907525368653469954it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_nat @ Xs3 ) ) ) )
              @ ( array_alloc_nat @ Xs3 @ H6 ) ) ) ) ) ).

% of_list_def
thf(fact_9150_of__list__def,axiom,
    ( array_of_list_o
    = ( ^ [Xs3: list_o] :
          ( heap_T8367841184088661864rray_o
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( produc5512802776183222702it_nat
              @ ^ [R5: array_o,H7: heap_e7401611519738050253t_unit] : ( produc2890842502232952598it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_o @ Xs3 ) ) ) )
              @ ( array_alloc_o @ Xs3 @ H6 ) ) ) ) ) ).

% of_list_def
thf(fact_9151_of__list__def,axiom,
    ( array_of_list_int
    = ( ^ [Xs3: list_int] :
          ( heap_T147713666487170802ay_int
          @ ^ [H6: heap_e7401611519738050253t_unit] :
              ( produc3016605123915955252it_nat
              @ ^ [R5: array_int,H7: heap_e7401611519738050253t_unit] : ( produc5351973125643376990it_nat @ R5 @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ one_one_nat @ ( size_size_list_int @ Xs3 ) ) ) )
              @ ( array_alloc_int @ Xs3 @ H6 ) ) ) ) ) ).

% of_list_def
thf(fact_9152_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_9153_finite__Collect__less__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K2 ) ) ) ).

% finite_Collect_less_nat
thf(fact_9154_finite__Collect__subsets,axiom,
    ! [A4: set_Pr958786334691620121nt_int] :
      ( ( finite2998713641127702882nt_int @ A4 )
     => ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [B6: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_9155_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_9156_finite__Collect__subsets,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [B6: set_Code_integer] : ( ord_le7084787975880047091nteger @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_9157_finite__Collect__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_9158_finite__Collect__subsets,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_9159_finite__induct__select,axiom,
    ! [S: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( P2 @ bot_bo228742789529271731at_nat )
       => ( ! [T8: set_Pr4329608150637261639at_nat] :
              ( ( ord_le2604355607129572851at_nat @ T8 @ S )
             => ( ( P2 @ T8 )
               => ? [X6: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X6 @ ( minus_3314409938677909166at_nat @ S @ T8 ) )
                    & ( P2 @ ( insert9069300056098147895at_nat @ X6 @ T8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_induct_select
thf(fact_9160_finite__induct__select,axiom,
    ! [S: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [T8: set_Code_integer] :
              ( ( ord_le1307284697595431911nteger @ T8 @ S )
             => ( ( P2 @ T8 )
               => ? [X6: code_integer] :
                    ( ( member_Code_integer @ X6 @ ( minus_2355218937544613996nteger @ S @ T8 ) )
                    & ( P2 @ ( insert_Code_integer @ X6 @ T8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_induct_select
thf(fact_9161_finite__induct__select,axiom,
    ! [S: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ S )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [T8: set_o] :
              ( ( ord_less_set_o @ T8 @ S )
             => ( ( P2 @ T8 )
               => ? [X6: $o] :
                    ( ( member_o @ X6 @ ( minus_minus_set_o @ S @ T8 ) )
                    & ( P2 @ ( insert_o @ X6 @ T8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_induct_select
thf(fact_9162_finite__induct__select,axiom,
    ! [S: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [T8: set_nat] :
              ( ( ord_less_set_nat @ T8 @ S )
             => ( ( P2 @ T8 )
               => ? [X6: nat] :
                    ( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T8 ) )
                    & ( P2 @ ( insert_nat @ X6 @ T8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_induct_select
thf(fact_9163_finite__induct__select,axiom,
    ! [S: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ S )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [T8: set_int] :
              ( ( ord_less_set_int @ T8 @ S )
             => ( ( P2 @ T8 )
               => ? [X6: int] :
                    ( ( member_int @ X6 @ ( minus_minus_set_int @ S @ T8 ) )
                    & ( P2 @ ( insert_int @ X6 @ T8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_induct_select
thf(fact_9164_finite__induct__select,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( P2 @ bot_bo2099793752762293965at_nat )
       => ( ! [T8: set_Pr1261947904930325089at_nat] :
              ( ( ord_le7866589430770878221at_nat @ T8 @ S )
             => ( ( P2 @ T8 )
               => ? [X6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X6 @ ( minus_1356011639430497352at_nat @ S @ T8 ) )
                    & ( P2 @ ( insert8211810215607154385at_nat @ X6 @ T8 ) ) ) ) )
         => ( P2 @ S ) ) ) ) ).

% finite_induct_select
thf(fact_9165_finite__Un,axiom,
    ! [F5: set_int,G4: set_int] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ F5 @ G4 ) )
      = ( ( finite_finite_int @ F5 )
        & ( finite_finite_int @ G4 ) ) ) ).

% finite_Un
thf(fact_9166_finite__Un,axiom,
    ! [F5: set_Code_integer,G4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ F5 @ G4 ) )
      = ( ( finite6017078050557962740nteger @ F5 )
        & ( finite6017078050557962740nteger @ G4 ) ) ) ).

% finite_Un
thf(fact_9167_finite__Un,axiom,
    ! [F5: set_Pr1261947904930325089at_nat,G4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F5 @ G4 ) )
      = ( ( finite6177210948735845034at_nat @ F5 )
        & ( finite6177210948735845034at_nat @ G4 ) ) ) ).

% finite_Un
thf(fact_9168_finite__Un,axiom,
    ! [F5: set_Pr8551490117392284871at_nat,G4: set_Pr8551490117392284871at_nat] :
      ( ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ F5 @ G4 ) )
      = ( ( finite1918287321285529104at_nat @ F5 )
        & ( finite1918287321285529104at_nat @ G4 ) ) ) ).

% finite_Un
thf(fact_9169_finite__Un,axiom,
    ! [F5: set_Pr4329608150637261639at_nat,G4: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F5 @ G4 ) )
      = ( ( finite4343798906461161616at_nat @ F5 )
        & ( finite4343798906461161616at_nat @ G4 ) ) ) ).

% finite_Un
thf(fact_9170_finite__Un,axiom,
    ! [F5: set_nat,G4: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F5 @ G4 ) )
      = ( ( finite_finite_nat @ F5 )
        & ( finite_finite_nat @ G4 ) ) ) ).

% finite_Un
thf(fact_9171_finite__Collect__conjI,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
        | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q2 ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_9172_finite__Collect__conjI,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ( ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ P2 ) )
        | ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ Q2 ) ) )
     => ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_9173_finite__Collect__conjI,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q2 ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_9174_finite__Collect__conjI,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
        | ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_9175_finite__Collect__conjI,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P2 ) )
        | ( finite_finite_int @ ( collect_int @ Q2 ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_9176_finite__Collect__conjI,axiom,
    ! [P2: code_integer > $o,Q2: code_integer > $o] :
      ( ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P2 ) )
        | ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q2 ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_9177_finite__Collect__conjI,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q2 ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_9178_finite__Collect__disjI,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_9179_finite__Collect__disjI,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o,Q2: set_Pr958786334691620121nt_int > $o] :
      ( ( finite8744585540193469122nt_int
        @ ( collec5210948495886036740nt_int
          @ ^ [X4: set_Pr958786334691620121nt_int] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ P2 ) )
        & ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_9180_finite__Collect__disjI,axiom,
    ! [P2: set_nat > $o,Q2: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_9181_finite__Collect__disjI,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
        & ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_9182_finite__Collect__disjI,axiom,
    ! [P2: int > $o,Q2: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P2 ) )
        & ( finite_finite_int @ ( collect_int @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_9183_finite__Collect__disjI,axiom,
    ! [P2: code_integer > $o,Q2: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P2 ) )
        & ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_9184_finite__Collect__disjI,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( P2 @ X4 )
              | ( Q2 @ X4 ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_9185_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I: int] :
            ( ( ord_less_eq_int @ A @ I )
            & ( ord_less_eq_int @ I @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_9186_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I: int] :
            ( ( ord_less_int @ A @ I )
            & ( ord_less_int @ I @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_9187_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_9188_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_9189_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I: int] :
            ( ( ord_less_eq_int @ A @ I )
            & ( ord_less_int @ I @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_9190_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I: int] :
            ( ( ord_less_int @ A @ I )
            & ( ord_less_eq_int @ I @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_9191_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M6 )
     => ? [N5: nat] :
        ! [X6: list_nat] :
          ( ( member_list_nat @ X6 @ M6 )
         => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N5 ) ) ) ).

% finite_maxlen
thf(fact_9192_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( ( finite_finite_list_o @ M6 )
     => ? [N5: nat] :
        ! [X6: list_o] :
          ( ( member_list_o @ X6 @ M6 )
         => ( ord_less_nat @ ( size_size_list_o @ X6 ) @ N5 ) ) ) ).

% finite_maxlen
thf(fact_9193_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( ( finite3922522038869484883st_int @ M6 )
     => ? [N5: nat] :
        ! [X6: list_int] :
          ( ( member_list_int @ X6 @ M6 )
         => ( ord_less_nat @ ( size_size_list_int @ X6 ) @ N5 ) ) ) ).

% finite_maxlen
thf(fact_9194_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_9195_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_9196_not__finite__existsD,axiom,
    ! [P2: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
     => ? [X_1: list_nat] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_9197_not__finite__existsD,axiom,
    ! [P2: set_Pr958786334691620121nt_int > $o] :
      ( ~ ( finite8744585540193469122nt_int @ ( collec5210948495886036740nt_int @ P2 ) )
     => ? [X_1: set_Pr958786334691620121nt_int] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_9198_not__finite__existsD,axiom,
    ! [P2: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
     => ? [X_1: set_nat] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_9199_not__finite__existsD,axiom,
    ! [P2: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ? [X_1: nat] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_9200_not__finite__existsD,axiom,
    ! [P2: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P2 ) )
     => ? [X_1: int] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_9201_not__finite__existsD,axiom,
    ! [P2: code_integer > $o] :
      ( ~ ( finite6017078050557962740nteger @ ( collect_Code_integer @ P2 ) )
     => ? [X_1: code_integer] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_9202_not__finite__existsD,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ? [X_1: product_prod_nat_nat] : ( P2 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_9203_pigeonhole__infinite__rel,axiom,
    ! [A4: set_o,B5: set_nat,R3: $o > nat > $o] :
      ( ~ ( finite_finite_o @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A4 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B5 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A5: $o] :
                        ( ( member_o @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9204_pigeonhole__infinite__rel,axiom,
    ! [A4: set_o,B5: set_int,R3: $o > int > $o] :
      ( ~ ( finite_finite_o @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A4 )
             => ? [Xa3: int] :
                  ( ( member_int @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B5 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A5: $o] :
                        ( ( member_o @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9205_pigeonhole__infinite__rel,axiom,
    ! [A4: set_o,B5: set_Code_integer,R3: $o > code_integer > $o] :
      ( ~ ( finite_finite_o @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A4 )
             => ? [Xa3: code_integer] :
                  ( ( member_Code_integer @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B5 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A5: $o] :
                        ( ( member_o @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9206_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B5: set_nat,R3: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B5 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9207_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B5: set_int,R3: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa3: int] :
                  ( ( member_int @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B5 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9208_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B5: set_Code_integer,R3: nat > code_integer > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa3: code_integer] :
                  ( ( member_Code_integer @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B5 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9209_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B5: set_nat,R3: int > nat > $o] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A4 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B5 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A5: int] :
                        ( ( member_int @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9210_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B5: set_int,R3: int > int > $o] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A4 )
             => ? [Xa3: int] :
                  ( ( member_int @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B5 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A5: int] :
                        ( ( member_int @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9211_pigeonhole__infinite__rel,axiom,
    ! [A4: set_int,B5: set_Code_integer,R3: int > code_integer > $o] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A4 )
             => ? [Xa3: code_integer] :
                  ( ( member_Code_integer @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B5 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A5: int] :
                        ( ( member_int @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9212_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Code_integer,B5: set_nat,R3: code_integer > nat > $o] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B5 )
                  & ( R3 @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B5 )
              & ~ ( finite6017078050557962740nteger
                  @ ( collect_Code_integer
                    @ ^ [A5: code_integer] :
                        ( ( member_Code_integer @ A5 @ A4 )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_9213_finite__has__minimal2,axiom,
    ! [A4: set_o,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ A @ A4 )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A4 )
            & ( ord_less_eq_o @ X3 @ A )
            & ! [Xa3: $o] :
                ( ( member_o @ Xa3 @ A4 )
               => ( ( ord_less_eq_o @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9214_finite__has__minimal2,axiom,
    ! [A4: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( finite8744585540193469122nt_int @ A4 )
     => ( ( member2340774599025711042nt_int @ A @ A4 )
       => ? [X3: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X3 @ A4 )
            & ( ord_le2843351958646193337nt_int @ X3 @ A )
            & ! [Xa3: set_Pr958786334691620121nt_int] :
                ( ( member2340774599025711042nt_int @ Xa3 @ A4 )
               => ( ( ord_le2843351958646193337nt_int @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9215_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ( ord_less_eq_set_nat @ X3 @ A )
            & ! [Xa3: set_nat] :
                ( ( member_set_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9216_finite__has__minimal2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ A @ A4 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ( ord_le3102999989581377725nteger @ X3 @ A )
            & ! [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ A4 )
               => ( ( ord_le3102999989581377725nteger @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9217_finite__has__minimal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( member_set_int @ A @ A4 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A4 )
            & ( ord_less_eq_set_int @ X3 @ A )
            & ! [Xa3: set_int] :
                ( ( member_set_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_int @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9218_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ( ord_less_eq_rat @ X3 @ A )
            & ! [Xa3: rat] :
                ( ( member_rat @ Xa3 @ A4 )
               => ( ( ord_less_eq_rat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9219_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ( ord_less_eq_num @ X3 @ A )
            & ! [Xa3: num] :
                ( ( member_num @ Xa3 @ A4 )
               => ( ( ord_less_eq_num @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9220_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9221_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ( ord_less_eq_int @ X3 @ A )
            & ! [Xa3: int] :
                ( ( member_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_int @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_9222_finite__has__maximal2,axiom,
    ! [A4: set_o,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ A @ A4 )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A4 )
            & ( ord_less_eq_o @ A @ X3 )
            & ! [Xa3: $o] :
                ( ( member_o @ Xa3 @ A4 )
               => ( ( ord_less_eq_o @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9223_finite__has__maximal2,axiom,
    ! [A4: set_se6260736226359567993nt_int,A: set_Pr958786334691620121nt_int] :
      ( ( finite8744585540193469122nt_int @ A4 )
     => ( ( member2340774599025711042nt_int @ A @ A4 )
       => ? [X3: set_Pr958786334691620121nt_int] :
            ( ( member2340774599025711042nt_int @ X3 @ A4 )
            & ( ord_le2843351958646193337nt_int @ A @ X3 )
            & ! [Xa3: set_Pr958786334691620121nt_int] :
                ( ( member2340774599025711042nt_int @ Xa3 @ A4 )
               => ( ( ord_le2843351958646193337nt_int @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9224_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ( ord_less_eq_set_nat @ A @ X3 )
            & ! [Xa3: set_nat] :
                ( ( member_set_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9225_finite__has__maximal2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ A @ A4 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ( ord_le3102999989581377725nteger @ A @ X3 )
            & ! [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ A4 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9226_finite__has__maximal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( member_set_int @ A @ A4 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A4 )
            & ( ord_less_eq_set_int @ A @ X3 )
            & ! [Xa3: set_int] :
                ( ( member_set_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9227_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ( ord_less_eq_rat @ A @ X3 )
            & ! [Xa3: rat] :
                ( ( member_rat @ Xa3 @ A4 )
               => ( ( ord_less_eq_rat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9228_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ( ord_less_eq_num @ A @ X3 )
            & ! [Xa3: num] :
                ( ( member_num @ Xa3 @ A4 )
               => ( ( ord_less_eq_num @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9229_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9230_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ( ord_less_eq_int @ A @ X3 )
            & ! [Xa3: int] :
                ( ( member_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_int @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_9231_rev__finite__subset,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_9232_rev__finite__subset,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( finite6017078050557962740nteger @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_9233_rev__finite__subset,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
       => ( finite6177210948735845034at_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_9234_rev__finite__subset,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( finite_finite_int @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_9235_infinite__super,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T7 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T7 ) ) ) ).

% infinite_super
thf(fact_9236_infinite__super,axiom,
    ! [S: set_Code_integer,T7: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ S @ T7 )
     => ( ~ ( finite6017078050557962740nteger @ S )
       => ~ ( finite6017078050557962740nteger @ T7 ) ) ) ).

% infinite_super
thf(fact_9237_infinite__super,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ~ ( finite6177210948735845034at_nat @ S )
       => ~ ( finite6177210948735845034at_nat @ T7 ) ) ) ).

% infinite_super
thf(fact_9238_infinite__super,axiom,
    ! [S: set_int,T7: set_int] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ~ ( finite_finite_int @ S )
       => ~ ( finite_finite_int @ T7 ) ) ) ).

% infinite_super
thf(fact_9239_finite__subset,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( finite_finite_nat @ B5 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_9240_finite__subset,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( finite6017078050557962740nteger @ A4 ) ) ) ).

% finite_subset
thf(fact_9241_finite__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( finite6177210948735845034at_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_9242_finite__subset,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( finite_finite_int @ B5 )
       => ( finite_finite_int @ A4 ) ) ) ).

% finite_subset
thf(fact_9243_finite__UnI,axiom,
    ! [F5: set_int,G4: set_int] :
      ( ( finite_finite_int @ F5 )
     => ( ( finite_finite_int @ G4 )
       => ( finite_finite_int @ ( sup_sup_set_int @ F5 @ G4 ) ) ) ) ).

% finite_UnI
thf(fact_9244_finite__UnI,axiom,
    ! [F5: set_Code_integer,G4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( finite6017078050557962740nteger @ G4 )
       => ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ F5 @ G4 ) ) ) ) ).

% finite_UnI
thf(fact_9245_finite__UnI,axiom,
    ! [F5: set_Pr1261947904930325089at_nat,G4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ F5 )
     => ( ( finite6177210948735845034at_nat @ G4 )
       => ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F5 @ G4 ) ) ) ) ).

% finite_UnI
thf(fact_9246_finite__UnI,axiom,
    ! [F5: set_Pr8551490117392284871at_nat,G4: set_Pr8551490117392284871at_nat] :
      ( ( finite1918287321285529104at_nat @ F5 )
     => ( ( finite1918287321285529104at_nat @ G4 )
       => ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ F5 @ G4 ) ) ) ) ).

% finite_UnI
thf(fact_9247_finite__UnI,axiom,
    ! [F5: set_Pr4329608150637261639at_nat,G4: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ F5 )
     => ( ( finite4343798906461161616at_nat @ G4 )
       => ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F5 @ G4 ) ) ) ) ).

% finite_UnI
thf(fact_9248_finite__UnI,axiom,
    ! [F5: set_nat,G4: set_nat] :
      ( ( finite_finite_nat @ F5 )
     => ( ( finite_finite_nat @ G4 )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F5 @ G4 ) ) ) ) ).

% finite_UnI
thf(fact_9249_Un__infinite,axiom,
    ! [S: set_int,T7: set_int] :
      ( ~ ( finite_finite_int @ S )
     => ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T7 ) ) ) ).

% Un_infinite
thf(fact_9250_Un__infinite,axiom,
    ! [S: set_Code_integer,T7: set_Code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S )
     => ~ ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ S @ T7 ) ) ) ).

% Un_infinite
thf(fact_9251_Un__infinite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T7 ) ) ) ).

% Un_infinite
thf(fact_9252_Un__infinite,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ~ ( finite1918287321285529104at_nat @ S )
     => ~ ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ S @ T7 ) ) ) ).

% Un_infinite
thf(fact_9253_Un__infinite,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S )
     => ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T7 ) ) ) ).

% Un_infinite
thf(fact_9254_Un__infinite,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T7 ) ) ) ).

% Un_infinite
thf(fact_9255_infinite__Un,axiom,
    ! [S: set_int,T7: set_int] :
      ( ( ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T7 ) ) )
      = ( ~ ( finite_finite_int @ S )
        | ~ ( finite_finite_int @ T7 ) ) ) ).

% infinite_Un
thf(fact_9256_infinite__Un,axiom,
    ! [S: set_Code_integer,T7: set_Code_integer] :
      ( ( ~ ( finite6017078050557962740nteger @ ( sup_su848401254843788991nteger @ S @ T7 ) ) )
      = ( ~ ( finite6017078050557962740nteger @ S )
        | ~ ( finite6017078050557962740nteger @ T7 ) ) ) ).

% infinite_Un
thf(fact_9257_infinite__Un,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T7 ) ) )
      = ( ~ ( finite6177210948735845034at_nat @ S )
        | ~ ( finite6177210948735845034at_nat @ T7 ) ) ) ).

% infinite_Un
thf(fact_9258_infinite__Un,axiom,
    ! [S: set_Pr8551490117392284871at_nat,T7: set_Pr8551490117392284871at_nat] :
      ( ( ~ ( finite1918287321285529104at_nat @ ( sup_su3035147773818789531at_nat @ S @ T7 ) ) )
      = ( ~ ( finite1918287321285529104at_nat @ S )
        | ~ ( finite1918287321285529104at_nat @ T7 ) ) ) ).

% infinite_Un
thf(fact_9259_infinite__Un,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T7: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T7 ) ) )
      = ( ~ ( finite4343798906461161616at_nat @ S )
        | ~ ( finite4343798906461161616at_nat @ T7 ) ) ) ).

% infinite_Un
thf(fact_9260_infinite__Un,axiom,
    ! [S: set_nat,T7: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T7 ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T7 ) ) ) ).

% infinite_Un
thf(fact_9261_finite__psubset__induct,axiom,
    ! [A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [A9: set_nat] :
            ( ( finite_finite_nat @ A9 )
           => ( ! [B10: set_nat] :
                  ( ( ord_less_set_nat @ B10 @ A9 )
                 => ( P2 @ B10 ) )
             => ( P2 @ A9 ) ) )
       => ( P2 @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_9262_finite__psubset__induct,axiom,
    ! [A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ! [A9: set_int] :
            ( ( finite_finite_int @ A9 )
           => ( ! [B10: set_int] :
                  ( ( ord_less_set_int @ B10 @ A9 )
                 => ( P2 @ B10 ) )
             => ( P2 @ A9 ) ) )
       => ( P2 @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_9263_finite__psubset__induct,axiom,
    ! [A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [A9: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ A9 )
           => ( ! [B10: set_Code_integer] :
                  ( ( ord_le1307284697595431911nteger @ B10 @ A9 )
                 => ( P2 @ B10 ) )
             => ( P2 @ A9 ) ) )
       => ( P2 @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_9264_finite__psubset__induct,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ! [A9: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ A9 )
           => ( ! [B10: set_Pr1261947904930325089at_nat] :
                  ( ( ord_le7866589430770878221at_nat @ B10 @ A9 )
                 => ( P2 @ B10 ) )
             => ( P2 @ A9 ) ) )
       => ( P2 @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_9265_Suc__0__or__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_9266_or__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_9267_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9268_finite__has__maximal,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ! [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ A4 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_9269_finite__has__maximal,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A4 )
            & ! [Xa3: $o] :
                ( ( member_o @ Xa3 @ A4 )
               => ( ( ord_less_eq_o @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_9270_finite__has__maximal,axiom,
    ! [A4: set_set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A4 )
            & ! [Xa3: set_int] :
                ( ( member_set_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_9271_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ! [Xa3: rat] :
                ( ( member_rat @ Xa3 @ A4 )
               => ( ( ord_less_eq_rat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_9272_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ! [Xa3: num] :
                ( ( member_num @ Xa3 @ A4 )
               => ( ( ord_less_eq_num @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_9273_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_9274_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ! [Xa3: int] :
                ( ( member_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_int @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_9275_finite__has__minimal,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ! [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ A4 )
               => ( ( ord_le3102999989581377725nteger @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_9276_finite__has__minimal,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A4 )
            & ! [Xa3: $o] :
                ( ( member_o @ Xa3 @ A4 )
               => ( ( ord_less_eq_o @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_9277_finite__has__minimal,axiom,
    ! [A4: set_set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A4 )
            & ! [Xa3: set_int] :
                ( ( member_set_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_int @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_9278_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ! [Xa3: rat] :
                ( ( member_rat @ Xa3 @ A4 )
               => ( ( ord_less_eq_rat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_9279_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ! [Xa3: num] :
                ( ( member_num @ Xa3 @ A4 )
               => ( ( ord_less_eq_num @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_9280_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_9281_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ! [Xa3: int] :
                ( ( member_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_int @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_9282_finite__subset__induct_H,axiom,
    ! [F5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F5 )
     => ( ( ord_le1268244103169919719at_nat @ F5 @ A4 )
       => ( ( P2 @ bot_bo228742789529271731at_nat )
         => ( ! [A3: produc3843707927480180839at_nat,F6: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F6 )
               => ( ( member8757157785044589968at_nat @ A3 @ A4 )
                 => ( ( ord_le1268244103169919719at_nat @ F6 @ A4 )
                   => ( ~ ( member8757157785044589968at_nat @ A3 @ F6 )
                     => ( ( P2 @ F6 )
                       => ( P2 @ ( insert9069300056098147895at_nat @ A3 @ F6 ) ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_9283_finite__subset__induct_H,axiom,
    ! [F5: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int,P2: set_se6260736226359567993nt_int > $o] :
      ( ( finite8744585540193469122nt_int @ F5 )
     => ( ( ord_le483042692224249369nt_int @ F5 @ A4 )
       => ( ( P2 @ bot_bo1488462491386950373nt_int )
         => ( ! [A3: set_Pr958786334691620121nt_int,F6: set_se6260736226359567993nt_int] :
                ( ( finite8744585540193469122nt_int @ F6 )
               => ( ( member2340774599025711042nt_int @ A3 @ A4 )
                 => ( ( ord_le483042692224249369nt_int @ F6 @ A4 )
                   => ( ~ ( member2340774599025711042nt_int @ A3 @ F6 )
                     => ( ( P2 @ F6 )
                       => ( P2 @ ( insert8897473484851387113nt_int @ A3 @ F6 ) ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_9284_finite__subset__induct_H,axiom,
    ! [F5: set_set_nat,A4: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F5 )
     => ( ( ord_le6893508408891458716et_nat @ F5 @ A4 )
       => ( ( P2 @ bot_bot_set_set_nat )
         => ( ! [A3: set_nat,F6: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F6 )
               => ( ( member_set_nat @ A3 @ A4 )
                 => ( ( ord_le6893508408891458716et_nat @ F6 @ A4 )
                   => ( ~ ( member_set_nat @ A3 @ F6 )
                     => ( ( P2 @ F6 )
                       => ( P2 @ ( insert_set_nat @ A3 @ F6 ) ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_9285_finite__subset__induct_H,axiom,
    ! [F5: set_Code_integer,A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger @ F5 @ A4 )
       => ( ( P2 @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( member_Code_integer @ A3 @ A4 )
                 => ( ( ord_le7084787975880047091nteger @ F6 @ A4 )
                   => ( ~ ( member_Code_integer @ A3 @ F6 )
                     => ( ( P2 @ F6 )
                       => ( P2 @ ( insert_Code_integer @ A3 @ F6 ) ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_9286_finite__subset__induct_H,axiom,
    ! [F5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F5 )
     => ( ( ord_le3146513528884898305at_nat @ F5 @ A4 )
       => ( ( P2 @ bot_bo2099793752762293965at_nat )
         => ( ! [A3: product_prod_nat_nat,F6: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F6 )
               => ( ( member8440522571783428010at_nat @ A3 @ A4 )
                 => ( ( ord_le3146513528884898305at_nat @ F6 @ A4 )
                   => ( ~ ( member8440522571783428010at_nat @ A3 @ F6 )
                     => ( ( P2 @ F6 )
                       => ( P2 @ ( insert8211810215607154385at_nat @ A3 @ F6 ) ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_9287_finite__subset__induct_H,axiom,
    ! [F5: set_o,A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o @ F5 @ A4 )
       => ( ( P2 @ bot_bot_set_o )
         => ( ! [A3: $o,F6: set_o] :
                ( ( finite_finite_o @ F6 )
               => ( ( member_o @ A3 @ A4 )
                 => ( ( ord_less_eq_set_o @ F6 @ A4 )
                   => ( ~ ( member_o @ A3 @ F6 )
                     => ( ( P2 @ F6 )
                       => ( P2 @ ( insert_o @ A3 @ F6 ) ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_9288_finite__subset__induct_H,axiom,
    ! [F5: set_nat,A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat @ F5 @ A4 )
       => ( ( P2 @ bot_bot_set_nat )
         => ( ! [A3: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A3 @ A4 )
                 => ( ( ord_less_eq_set_nat @ F6 @ A4 )
                   => ( ~ ( member_nat @ A3 @ F6 )
                     => ( ( P2 @ F6 )
                       => ( P2 @ ( insert_nat @ A3 @ F6 ) ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_9289_finite__subset__induct_H,axiom,
    ! [F5: set_int,A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F5 )
     => ( ( ord_less_eq_set_int @ F5 @ A4 )
       => ( ( P2 @ bot_bot_set_int )
         => ( ! [A3: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( member_int @ A3 @ A4 )
                 => ( ( ord_less_eq_set_int @ F6 @ A4 )
                   => ( ~ ( member_int @ A3 @ F6 )
                     => ( ( P2 @ F6 )
                       => ( P2 @ ( insert_int @ A3 @ F6 ) ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_9290_finite__subset__induct,axiom,
    ! [F5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F5 )
     => ( ( ord_le1268244103169919719at_nat @ F5 @ A4 )
       => ( ( P2 @ bot_bo228742789529271731at_nat )
         => ( ! [A3: produc3843707927480180839at_nat,F6: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F6 )
               => ( ( member8757157785044589968at_nat @ A3 @ A4 )
                 => ( ~ ( member8757157785044589968at_nat @ A3 @ F6 )
                   => ( ( P2 @ F6 )
                     => ( P2 @ ( insert9069300056098147895at_nat @ A3 @ F6 ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_9291_finite__subset__induct,axiom,
    ! [F5: set_se6260736226359567993nt_int,A4: set_se6260736226359567993nt_int,P2: set_se6260736226359567993nt_int > $o] :
      ( ( finite8744585540193469122nt_int @ F5 )
     => ( ( ord_le483042692224249369nt_int @ F5 @ A4 )
       => ( ( P2 @ bot_bo1488462491386950373nt_int )
         => ( ! [A3: set_Pr958786334691620121nt_int,F6: set_se6260736226359567993nt_int] :
                ( ( finite8744585540193469122nt_int @ F6 )
               => ( ( member2340774599025711042nt_int @ A3 @ A4 )
                 => ( ~ ( member2340774599025711042nt_int @ A3 @ F6 )
                   => ( ( P2 @ F6 )
                     => ( P2 @ ( insert8897473484851387113nt_int @ A3 @ F6 ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_9292_finite__subset__induct,axiom,
    ! [F5: set_set_nat,A4: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F5 )
     => ( ( ord_le6893508408891458716et_nat @ F5 @ A4 )
       => ( ( P2 @ bot_bot_set_set_nat )
         => ( ! [A3: set_nat,F6: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F6 )
               => ( ( member_set_nat @ A3 @ A4 )
                 => ( ~ ( member_set_nat @ A3 @ F6 )
                   => ( ( P2 @ F6 )
                     => ( P2 @ ( insert_set_nat @ A3 @ F6 ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_9293_finite__subset__induct,axiom,
    ! [F5: set_Code_integer,A4: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger @ F5 @ A4 )
       => ( ( P2 @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( member_Code_integer @ A3 @ A4 )
                 => ( ~ ( member_Code_integer @ A3 @ F6 )
                   => ( ( P2 @ F6 )
                     => ( P2 @ ( insert_Code_integer @ A3 @ F6 ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_9294_finite__subset__induct,axiom,
    ! [F5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F5 )
     => ( ( ord_le3146513528884898305at_nat @ F5 @ A4 )
       => ( ( P2 @ bot_bo2099793752762293965at_nat )
         => ( ! [A3: product_prod_nat_nat,F6: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F6 )
               => ( ( member8440522571783428010at_nat @ A3 @ A4 )
                 => ( ~ ( member8440522571783428010at_nat @ A3 @ F6 )
                   => ( ( P2 @ F6 )
                     => ( P2 @ ( insert8211810215607154385at_nat @ A3 @ F6 ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_9295_finite__subset__induct,axiom,
    ! [F5: set_o,A4: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o @ F5 @ A4 )
       => ( ( P2 @ bot_bot_set_o )
         => ( ! [A3: $o,F6: set_o] :
                ( ( finite_finite_o @ F6 )
               => ( ( member_o @ A3 @ A4 )
                 => ( ~ ( member_o @ A3 @ F6 )
                   => ( ( P2 @ F6 )
                     => ( P2 @ ( insert_o @ A3 @ F6 ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_9296_finite__subset__induct,axiom,
    ! [F5: set_nat,A4: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat @ F5 @ A4 )
       => ( ( P2 @ bot_bot_set_nat )
         => ( ! [A3: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A3 @ A4 )
                 => ( ~ ( member_nat @ A3 @ F6 )
                   => ( ( P2 @ F6 )
                     => ( P2 @ ( insert_nat @ A3 @ F6 ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_9297_finite__subset__induct,axiom,
    ! [F5: set_int,A4: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F5 )
     => ( ( ord_less_eq_set_int @ F5 @ A4 )
       => ( ( P2 @ bot_bot_set_int )
         => ( ! [A3: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( member_int @ A3 @ A4 )
                 => ( ~ ( member_int @ A3 @ F6 )
                   => ( ( P2 @ F6 )
                     => ( P2 @ ( insert_int @ A3 @ F6 ) ) ) ) ) )
           => ( P2 @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_9298_finite__remove__induct,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ B5 )
     => ( ( P2 @ bot_bo228742789529271731at_nat )
       => ( ! [A9: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A9 )
             => ( ( A9 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A9 @ B5 )
                 => ( ! [X6: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X6 @ A9 )
                       => ( P2 @ ( minus_3314409938677909166at_nat @ A9 @ ( insert9069300056098147895at_nat @ X6 @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_9299_finite__remove__induct,axiom,
    ! [B5: set_se6260736226359567993nt_int,P2: set_se6260736226359567993nt_int > $o] :
      ( ( finite8744585540193469122nt_int @ B5 )
     => ( ( P2 @ bot_bo1488462491386950373nt_int )
       => ( ! [A9: set_se6260736226359567993nt_int] :
              ( ( finite8744585540193469122nt_int @ A9 )
             => ( ( A9 != bot_bo1488462491386950373nt_int )
               => ( ( ord_le483042692224249369nt_int @ A9 @ B5 )
                 => ( ! [X6: set_Pr958786334691620121nt_int] :
                        ( ( member2340774599025711042nt_int @ X6 @ A9 )
                       => ( P2 @ ( minus_2612819937483484256nt_int @ A9 @ ( insert8897473484851387113nt_int @ X6 @ bot_bo1488462491386950373nt_int ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_9300_finite__remove__induct,axiom,
    ! [B5: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [A9: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A9 )
             => ( ( A9 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A9 @ B5 )
                 => ( ! [X6: set_nat] :
                        ( ( member_set_nat @ X6 @ A9 )
                       => ( P2 @ ( minus_2163939370556025621et_nat @ A9 @ ( insert_set_nat @ X6 @ bot_bot_set_set_nat ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_9301_finite__remove__induct,axiom,
    ! [B5: set_Code_integer,P2: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( P2 @ bot_bo3990330152332043303nteger )
       => ( ! [A9: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A9 )
             => ( ( A9 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A9 @ B5 )
                 => ( ! [X6: code_integer] :
                        ( ( member_Code_integer @ X6 @ A9 )
                       => ( P2 @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_9302_finite__remove__induct,axiom,
    ! [B5: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ B5 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [A9: set_o] :
              ( ( finite_finite_o @ A9 )
             => ( ( A9 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A9 @ B5 )
                 => ( ! [X6: $o] :
                        ( ( member_o @ X6 @ A9 )
                       => ( P2 @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_9303_finite__remove__induct,axiom,
    ! [B5: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ( A9 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A9 @ B5 )
                 => ( ! [X6: nat] :
                        ( ( member_nat @ X6 @ A9 )
                       => ( P2 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_9304_finite__remove__induct,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( P2 @ bot_bo2099793752762293965at_nat )
       => ( ! [A9: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A9 )
             => ( ( A9 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A9 @ B5 )
                 => ( ! [X6: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X6 @ A9 )
                       => ( P2 @ ( minus_1356011639430497352at_nat @ A9 @ ( insert8211810215607154385at_nat @ X6 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_9305_finite__remove__induct,axiom,
    ! [B5: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ B5 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [A9: set_int] :
              ( ( finite_finite_int @ A9 )
             => ( ( A9 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A9 @ B5 )
                 => ( ! [X6: int] :
                        ( ( member_int @ X6 @ A9 )
                       => ( P2 @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_9306_remove__induct,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > $o,B5: set_Pr4329608150637261639at_nat] :
      ( ( P2 @ bot_bo228742789529271731at_nat )
     => ( ( ~ ( finite4343798906461161616at_nat @ B5 )
         => ( P2 @ B5 ) )
       => ( ! [A9: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A9 )
             => ( ( A9 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A9 @ B5 )
                 => ( ! [X6: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X6 @ A9 )
                       => ( P2 @ ( minus_3314409938677909166at_nat @ A9 @ ( insert9069300056098147895at_nat @ X6 @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% remove_induct
thf(fact_9307_remove__induct,axiom,
    ! [P2: set_se6260736226359567993nt_int > $o,B5: set_se6260736226359567993nt_int] :
      ( ( P2 @ bot_bo1488462491386950373nt_int )
     => ( ( ~ ( finite8744585540193469122nt_int @ B5 )
         => ( P2 @ B5 ) )
       => ( ! [A9: set_se6260736226359567993nt_int] :
              ( ( finite8744585540193469122nt_int @ A9 )
             => ( ( A9 != bot_bo1488462491386950373nt_int )
               => ( ( ord_le483042692224249369nt_int @ A9 @ B5 )
                 => ( ! [X6: set_Pr958786334691620121nt_int] :
                        ( ( member2340774599025711042nt_int @ X6 @ A9 )
                       => ( P2 @ ( minus_2612819937483484256nt_int @ A9 @ ( insert8897473484851387113nt_int @ X6 @ bot_bo1488462491386950373nt_int ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% remove_induct
thf(fact_9308_remove__induct,axiom,
    ! [P2: set_set_nat > $o,B5: set_set_nat] :
      ( ( P2 @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B5 )
         => ( P2 @ B5 ) )
       => ( ! [A9: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A9 )
             => ( ( A9 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A9 @ B5 )
                 => ( ! [X6: set_nat] :
                        ( ( member_set_nat @ X6 @ A9 )
                       => ( P2 @ ( minus_2163939370556025621et_nat @ A9 @ ( insert_set_nat @ X6 @ bot_bot_set_set_nat ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% remove_induct
thf(fact_9309_remove__induct,axiom,
    ! [P2: set_Code_integer > $o,B5: set_Code_integer] :
      ( ( P2 @ bot_bo3990330152332043303nteger )
     => ( ( ~ ( finite6017078050557962740nteger @ B5 )
         => ( P2 @ B5 ) )
       => ( ! [A9: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A9 )
             => ( ( A9 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A9 @ B5 )
                 => ( ! [X6: code_integer] :
                        ( ( member_Code_integer @ X6 @ A9 )
                       => ( P2 @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% remove_induct
thf(fact_9310_remove__induct,axiom,
    ! [P2: set_o > $o,B5: set_o] :
      ( ( P2 @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B5 )
         => ( P2 @ B5 ) )
       => ( ! [A9: set_o] :
              ( ( finite_finite_o @ A9 )
             => ( ( A9 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A9 @ B5 )
                 => ( ! [X6: $o] :
                        ( ( member_o @ X6 @ A9 )
                       => ( P2 @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% remove_induct
thf(fact_9311_remove__induct,axiom,
    ! [P2: set_nat > $o,B5: set_nat] :
      ( ( P2 @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B5 )
         => ( P2 @ B5 ) )
       => ( ! [A9: set_nat] :
              ( ( finite_finite_nat @ A9 )
             => ( ( A9 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A9 @ B5 )
                 => ( ! [X6: nat] :
                        ( ( member_nat @ X6 @ A9 )
                       => ( P2 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% remove_induct
thf(fact_9312_remove__induct,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > $o,B5: set_Pr1261947904930325089at_nat] :
      ( ( P2 @ bot_bo2099793752762293965at_nat )
     => ( ( ~ ( finite6177210948735845034at_nat @ B5 )
         => ( P2 @ B5 ) )
       => ( ! [A9: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A9 )
             => ( ( A9 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A9 @ B5 )
                 => ( ! [X6: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X6 @ A9 )
                       => ( P2 @ ( minus_1356011639430497352at_nat @ A9 @ ( insert8211810215607154385at_nat @ X6 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% remove_induct
thf(fact_9313_remove__induct,axiom,
    ! [P2: set_int > $o,B5: set_int] :
      ( ( P2 @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B5 )
         => ( P2 @ B5 ) )
       => ( ! [A9: set_int] :
              ( ( finite_finite_int @ A9 )
             => ( ( A9 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A9 @ B5 )
                 => ( ! [X6: int] :
                        ( ( member_int @ X6 @ A9 )
                       => ( P2 @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
                   => ( P2 @ A9 ) ) ) ) )
         => ( P2 @ B5 ) ) ) ) ).

% remove_induct
thf(fact_9314_set__encode__insert,axiom,
    ! [A4: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ N2 @ A4 )
       => ( ( nat_set_encode @ ( insert_nat @ N2 @ A4 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).

% set_encode_insert
thf(fact_9315_finite__nat__bounded,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ? [K: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K ) ) ) ).

% finite_nat_bounded
thf(fact_9316_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S4: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_9317_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S4: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_9318_in__finite__psubset,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B5 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A4 @ B5 )
        & ( finite_finite_nat @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_9319_in__finite__psubset,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A4 @ B5 ) @ finite_psubset_int )
      = ( ( ord_less_set_int @ A4 @ B5 )
        & ( finite_finite_int @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_9320_in__finite__psubset,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer] :
      ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A4 @ B5 ) @ finite2416775604798480986nteger )
      = ( ( ord_le1307284697595431911nteger @ A4 @ B5 )
        & ( finite6017078050557962740nteger @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_9321_in__finite__psubset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B5 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
        & ( finite6177210948735845034at_nat @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_9322_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_9323_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_9324_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_9325_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_9326_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_9327_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_9328_unbounded__k__infinite,axiom,
    ! [K2: nat,S: set_nat] :
      ( ! [M5: nat] :
          ( ( ord_less_nat @ K2 @ M5 )
         => ? [N10: nat] :
              ( ( ord_less_nat @ M5 @ N10 )
              & ( member_nat @ N10 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_9329_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_9330_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_9331_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9332_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9333_sum__diff1_H__aux,axiom,
    ! [F5: set_Code_integer,I5: set_Code_integer,F: code_integer > code_integer,I2: code_integer] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( F @ I )
                 != zero_z3403309356797280102nteger ) ) )
          @ F5 )
       => ( ( ( member_Code_integer @ I2 @ I5 )
           => ( ( groups910942671188738463nteger @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
              = ( minus_8373710615458151222nteger @ ( groups910942671188738463nteger @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_Code_integer @ I2 @ I5 )
           => ( ( groups910942671188738463nteger @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
              = ( groups910942671188738463nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9334_sum__diff1_H__aux,axiom,
    ! [F5: set_o,I5: set_o,F: $o > code_integer,I2: $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( F @ I )
                 != zero_z3403309356797280102nteger ) ) )
          @ F5 )
       => ( ( ( member_o @ I2 @ I5 )
           => ( ( groups1402912129352969042nteger @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
              = ( minus_8373710615458151222nteger @ ( groups1402912129352969042nteger @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_o @ I2 @ I5 )
           => ( ( groups1402912129352969042nteger @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
              = ( groups1402912129352969042nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9335_sum__diff1_H__aux,axiom,
    ! [F5: set_nat,I5: set_nat,F: nat > code_integer,I2: nat] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( F @ I )
                 != zero_z3403309356797280102nteger ) ) )
          @ F5 )
       => ( ( ( member_nat @ I2 @ I5 )
           => ( ( groups555127423416065298nteger @ F @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
              = ( minus_8373710615458151222nteger @ ( groups555127423416065298nteger @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_nat @ I2 @ I5 )
           => ( ( groups555127423416065298nteger @ F @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
              = ( groups555127423416065298nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9336_sum__diff1_H__aux,axiom,
    ! [F5: set_Code_integer,I5: set_Code_integer,F: code_integer > rat,I2: code_integer] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( F @ I )
                 != zero_zero_rat ) ) )
          @ F5 )
       => ( ( ( member_Code_integer @ I2 @ I5 )
           => ( ( groups8878813951405302554er_rat @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
              = ( minus_minus_rat @ ( groups8878813951405302554er_rat @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_Code_integer @ I2 @ I5 )
           => ( ( groups8878813951405302554er_rat @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
              = ( groups8878813951405302554er_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9337_sum__diff1_H__aux,axiom,
    ! [F5: set_o,I5: set_o,F: $o > rat,I2: $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( F @ I )
                 != zero_zero_rat ) ) )
          @ F5 )
       => ( ( ( member_o @ I2 @ I5 )
           => ( ( groups3921277224699582669_o_rat @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
              = ( minus_minus_rat @ ( groups3921277224699582669_o_rat @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_o @ I2 @ I5 )
           => ( ( groups3921277224699582669_o_rat @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
              = ( groups3921277224699582669_o_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9338_sum__diff1_H__aux,axiom,
    ! [F5: set_nat,I5: set_nat,F: nat > rat,I2: nat] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( F @ I )
                 != zero_zero_rat ) ) )
          @ F5 )
       => ( ( ( member_nat @ I2 @ I5 )
           => ( ( groups1351286907653491341at_rat @ F @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
              = ( minus_minus_rat @ ( groups1351286907653491341at_rat @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_nat @ I2 @ I5 )
           => ( ( groups1351286907653491341at_rat @ F @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
              = ( groups1351286907653491341at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9339_sum__diff1_H__aux,axiom,
    ! [F5: set_Code_integer,I5: set_Code_integer,F: code_integer > int,I2: code_integer] :
      ( ( finite6017078050557962740nteger @ F5 )
     => ( ( ord_le7084787975880047091nteger
          @ ( collect_Code_integer
            @ ^ [I: code_integer] :
                ( ( member_Code_integer @ I @ I5 )
                & ( ( F @ I )
                 != zero_zero_int ) ) )
          @ F5 )
       => ( ( ( member_Code_integer @ I2 @ I5 )
           => ( ( groups288081504127972206er_int @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
              = ( minus_minus_int @ ( groups288081504127972206er_int @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_Code_integer @ I2 @ I5 )
           => ( ( groups288081504127972206er_int @ F @ ( minus_2355218937544613996nteger @ I5 @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
              = ( groups288081504127972206er_int @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9340_sum__diff1_H__aux,axiom,
    ! [F5: set_o,I5: set_o,F: $o > int,I2: $o] :
      ( ( finite_finite_o @ F5 )
     => ( ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [I: $o] :
                ( ( member_o @ I @ I5 )
                & ( ( F @ I )
                 != zero_zero_int ) ) )
          @ F5 )
       => ( ( ( member_o @ I2 @ I5 )
           => ( ( groups4553916814277028129_o_int @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
              = ( minus_minus_int @ ( groups4553916814277028129_o_int @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_o @ I2 @ I5 )
           => ( ( groups4553916814277028129_o_int @ F @ ( minus_minus_set_o @ I5 @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
              = ( groups4553916814277028129_o_int @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9341_sum__diff1_H__aux,axiom,
    ! [F5: set_nat,I5: set_nat,F: nat > int,I2: nat] :
      ( ( finite_finite_nat @ F5 )
     => ( ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( F @ I )
                 != zero_zero_int ) ) )
          @ F5 )
       => ( ( ( member_nat @ I2 @ I5 )
           => ( ( groups1983926497230936801at_int @ F @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
              = ( minus_minus_int @ ( groups1983926497230936801at_int @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_nat @ I2 @ I5 )
           => ( ( groups1983926497230936801at_int @ F @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
              = ( groups1983926497230936801at_int @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9342_sum__diff1_H__aux,axiom,
    ! [F5: set_int,I5: set_int,F: int > code_integer,I2: int] :
      ( ( finite_finite_int @ F5 )
     => ( ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( F @ I )
                 != zero_z3403309356797280102nteger ) ) )
          @ F5 )
       => ( ( ( member_int @ I2 @ I5 )
           => ( ( groups926780983652909934nteger @ F @ ( minus_minus_set_int @ I5 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
              = ( minus_8373710615458151222nteger @ ( groups926780983652909934nteger @ F @ I5 ) @ ( F @ I2 ) ) ) )
          & ( ~ ( member_int @ I2 @ I5 )
           => ( ( groups926780983652909934nteger @ F @ ( minus_minus_set_int @ I5 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
              = ( groups926780983652909934nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_9343_even__sum__iff,axiom,
    ! [A4: set_o,F: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups8507830703676809646_o_nat @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_o
            @ ( collect_o
              @ ^ [A5: $o] :
                  ( ( member_o @ A5 @ A4 )
                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9344_even__sum__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_int
            @ ( collect_int
              @ ^ [A5: int] :
                  ( ( member_int @ A5 @ A4 )
                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9345_even__sum__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups7237345082560585321er_nat @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite4902975817058060853nteger
            @ ( collect_Code_integer
              @ ^ [A5: code_integer] :
                  ( ( member_Code_integer @ A5 @ A4 )
                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9346_even__sum__iff,axiom,
    ! [A4: set_o,F: $o > int] :
      ( ( finite_finite_o @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups8505340233167759370_o_int @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_o
            @ ( collect_o
              @ ^ [A5: $o] :
                  ( ( member_o @ A5 @ A4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9347_even__sum__iff,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [A5: nat] :
                  ( ( member_nat @ A5 @ A4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9348_even__sum__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups7234854612051535045er_int @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite4902975817058060853nteger
            @ ( collect_Code_integer
              @ ^ [A5: code_integer] :
                  ( ( member_Code_integer @ A5 @ A4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9349_even__sum__iff,axiom,
    ! [A4: set_o,F: $o > code_natural] :
      ( ( finite_finite_o @ A4 )
     => ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups3230237193842799622atural @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_o
            @ ( collect_o
              @ ^ [A5: $o] :
                  ( ( member_o @ A5 @ A4 )
                  & ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9350_even__sum__iff,axiom,
    ! [A4: set_nat,F: nat > code_natural] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6325495683096345652atural @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [A5: nat] :
                  ( ( member_nat @ A5 @ A4 )
                  & ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9351_even__sum__iff,axiom,
    ! [A4: set_int,F: int > code_natural] :
      ( ( finite_finite_int @ A4 )
     => ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6697149243333190288atural @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite_card_int
            @ ( collect_int
              @ ^ [A5: int] :
                  ( ( member_int @ A5 @ A4 )
                  & ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9352_even__sum__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > code_natural] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups8926444216418632897atural @ F @ A4 ) )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
          @ ( finite4902975817058060853nteger
            @ ( collect_Code_integer
              @ ^ [A5: code_integer] :
                  ( ( member_Code_integer @ A5 @ A4 )
                  & ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( F @ A5 ) ) ) ) ) ) ) ) ).

% even_sum_iff
thf(fact_9353_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 )
                                  @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
                                  @ ( 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_9354_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_9355_max_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_9356_max_Obounded__iff,axiom,
    ! [B: num,C2: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_9357_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_9358_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_9359_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_9360_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_9361_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_9362_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_9363_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_9364_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_9365_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_9366_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_9367_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z3 )
      = ( ( ord_less_rat @ X @ Z3 )
        & ( ord_less_rat @ Y @ Z3 ) ) ) ).

% max_less_iff_conj
thf(fact_9368_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z3 )
      = ( ( ord_less_num @ X @ Z3 )
        & ( ord_less_num @ Y @ Z3 ) ) ) ).

% max_less_iff_conj
thf(fact_9369_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_nat @ X @ Z3 )
        & ( ord_less_nat @ Y @ Z3 ) ) ) ).

% max_less_iff_conj
thf(fact_9370_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z3 )
      = ( ( ord_less_int @ X @ Z3 )
        & ( ord_less_int @ Y @ Z3 ) ) ) ).

% max_less_iff_conj
thf(fact_9371_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_9372_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_9373_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_9374_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_9375_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_9376_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_9377_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_9378_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_9379_max__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_ma7524802468073614006at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% max_bot2
thf(fact_9380_max__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ X @ bot_bot_set_o )
      = X ) ).

% max_bot2
thf(fact_9381_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_9382_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_9383_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_9384_max__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_ma7524802468073614006at_nat @ bot_bo2099793752762293965at_nat @ X )
      = X ) ).

% max_bot
thf(fact_9385_max__bot,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X )
      = X ) ).

% max_bot
thf(fact_9386_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_9387_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_9388_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_9389_map__option__eq__Some,axiom,
    ! [F: num > num,Xo: option_num,Y: num] :
      ( ( ( map_option_num_num @ F @ Xo )
        = ( some_num @ Y ) )
      = ( ? [Z: num] :
            ( ( Xo
              = ( some_num @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9390_map__option__eq__Some,axiom,
    ! [F: num > produc7388388658123137530it_nat,Xo: option_num,Y: produc7388388658123137530it_nat] :
      ( ( ( map_op2490657901765117089it_nat @ F @ Xo )
        = ( some_P2818173045054083285it_nat @ Y ) )
      = ( ? [Z: num] :
            ( ( Xo
              = ( some_num @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9391_map__option__eq__Some,axiom,
    ! [F: num > produc3260487557148687353it_nat,Xo: option_num,Y: produc3260487557148687353it_nat] :
      ( ( ( map_op7586128837645442720it_nat @ F @ Xo )
        = ( some_P7913643980934408916it_nat @ Y ) )
      = ( ? [Z: num] :
            ( ( Xo
              = ( some_num @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9392_map__option__eq__Some,axiom,
    ! [F: num > produc8664842809031399944it_nat,Xo: option_num,Y: produc8664842809031399944it_nat] :
      ( ( ( map_op8118133509425879471it_nat @ F @ Xo )
        = ( some_P1914260805536162275it_nat @ Y ) )
      = ( ? [Z: num] :
            ( ( Xo
              = ( some_num @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9393_map__option__eq__Some,axiom,
    ! [F: produc7388388658123137530it_nat > num,Xo: option4065278094766928714it_nat,Y: num] :
      ( ( ( map_op6233829773304596897at_num @ F @ Xo )
        = ( some_num @ Y ) )
      = ( ? [Z: produc7388388658123137530it_nat] :
            ( ( Xo
              = ( some_P2818173045054083285it_nat @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9394_map__option__eq__Some,axiom,
    ! [F: produc3260487557148687353it_nat > num,Xo: option3562590408128118217it_nat,Y: num] :
      ( ( ( map_op5165227558541518752at_num @ F @ Xo )
        = ( some_num @ Y ) )
      = ( ? [Z: produc3260487557148687353it_nat] :
            ( ( Xo
              = ( some_P7913643980934408916it_nat @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9395_map__option__eq__Some,axiom,
    ! [F: produc8664842809031399944it_nat > num,Xo: option8956607266484857688it_nat,Y: num] :
      ( ( ( map_op2731889312448867759at_num @ F @ Xo )
        = ( some_num @ Y ) )
      = ( ? [Z: produc8664842809031399944it_nat] :
            ( ( Xo
              = ( some_P1914260805536162275it_nat @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9396_map__option__eq__Some,axiom,
    ! [F: produc7388388658123137530it_nat > produc7388388658123137530it_nat,Xo: option4065278094766928714it_nat,Y: produc7388388658123137530it_nat] :
      ( ( ( map_op8581638957091552385it_nat @ F @ Xo )
        = ( some_P2818173045054083285it_nat @ Y ) )
      = ( ? [Z: produc7388388658123137530it_nat] :
            ( ( Xo
              = ( some_P2818173045054083285it_nat @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9397_map__option__eq__Some,axiom,
    ! [F: produc3260487557148687353it_nat > produc7388388658123137530it_nat,Xo: option3562590408128118217it_nat,Y: produc7388388658123137530it_nat] :
      ( ( ( map_op7599720869725723008it_nat @ F @ Xo )
        = ( some_P2818173045054083285it_nat @ Y ) )
      = ( ? [Z: produc3260487557148687353it_nat] :
            ( ( Xo
              = ( some_P7913643980934408916it_nat @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9398_map__option__eq__Some,axiom,
    ! [F: produc8664842809031399944it_nat > produc7388388658123137530it_nat,Xo: option8956607266484857688it_nat,Y: produc7388388658123137530it_nat] :
      ( ( ( map_op604423251913076367it_nat @ F @ Xo )
        = ( some_P2818173045054083285it_nat @ Y ) )
      = ( ? [Z: produc8664842809031399944it_nat] :
            ( ( Xo
              = ( some_P1914260805536162275it_nat @ Z ) )
            & ( ( F @ Z )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_9399_None__eq__map__option__iff,axiom,
    ! [F: num > num,X: option_num] :
      ( ( none_num
        = ( map_option_num_num @ F @ X ) )
      = ( X = none_num ) ) ).

% None_eq_map_option_iff
thf(fact_9400_map__option__is__None,axiom,
    ! [F: num > num,Opt: option_num] :
      ( ( ( map_option_num_num @ F @ Opt )
        = none_num )
      = ( Opt = none_num ) ) ).

% map_option_is_None
thf(fact_9401_option_Omap__disc__iff,axiom,
    ! [F: num > num,A: option_num] :
      ( ( ( map_option_num_num @ F @ A )
        = none_num )
      = ( A = none_num ) ) ).

% option.map_disc_iff
thf(fact_9402_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_9403_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_9404_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_9405_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_9406_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_9407_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_9408_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_9409_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_eq_nat @ I @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_9410_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
       => ( ( ord_max_Code_natural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
          = ( numera5444537566228673987atural @ V ) ) )
      & ( ~ ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
       => ( ( ord_max_Code_natural @ ( numera5444537566228673987atural @ U ) @ ( numera5444537566228673987atural @ V ) )
          = ( numera5444537566228673987atural @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_9411_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_9412_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_9413_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_9414_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_9415_case__map__option,axiom,
    ! [G: produc8923325533196201883nteger > produc8923325533196201883nteger,H: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer,X: option_Code_integer] :
      ( ( case_o7134296353695833103nteger @ G @ H @ ( map_op3669829223712506439nteger @ F @ X ) )
      = ( case_o7134296353695833103nteger @ G @ ( comp_C1593894019821074884nteger @ H @ F ) @ X ) ) ).

% case_map_option
thf(fact_9416_case__map__option,axiom,
    ! [G: produc8923325533196201883nteger > produc8923325533196201883nteger,H: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,X: option_Code_integer] :
      ( ( case_o1441893360019914891nteger @ G @ H @ ( map_op370346339404370627nteger @ F @ X ) )
      = ( case_o7134296353695833103nteger @ G @ ( comp_C8797469213163452608nteger @ H @ F ) @ X ) ) ).

% case_map_option
thf(fact_9417_case__map__option,axiom,
    ! [G: nat,H: int > nat,F: int > int,X: option_int] :
      ( ( case_option_nat_int @ G @ H @ ( map_option_int_int @ F @ X ) )
      = ( case_option_nat_int @ G @ ( comp_int_nat_int @ H @ F ) @ X ) ) ).

% case_map_option
thf(fact_9418_case__map__option,axiom,
    ! [G: int,H: num > int,F: num > num,X: option_num] :
      ( ( case_option_int_num @ G @ H @ ( map_option_num_num @ F @ X ) )
      = ( case_option_int_num @ G @ ( comp_num_int_num @ H @ F ) @ X ) ) ).

% case_map_option
thf(fact_9419_case__map__option,axiom,
    ! [G: option_num,H: num > option_num,F: num > num,X: option_num] :
      ( ( case_o6005452278849405969um_num @ G @ H @ ( map_option_num_num @ F @ X ) )
      = ( case_o6005452278849405969um_num @ G @ ( comp_n6731957995704128387um_num @ H @ F ) @ X ) ) ).

% case_map_option
thf(fact_9420_case__map__option,axiom,
    ! [G: num,H: num > num,F: num > num,X: option_num] :
      ( ( case_option_num_num @ G @ H @ ( map_option_num_num @ F @ X ) )
      = ( case_option_num_num @ G @ ( comp_num_num_num @ H @ F ) @ X ) ) ).

% case_map_option
thf(fact_9421_prod__constant,axiom,
    ! [Y: nat,A4: set_int] :
      ( ( groups1707563613775114915nt_nat
        @ ^ [X4: int] : Y
        @ A4 )
      = ( power_power_nat @ Y @ ( finite_card_int @ A4 ) ) ) ).

% prod_constant
thf(fact_9422_prod__constant,axiom,
    ! [Y: nat,A4: set_list_nat] :
      ( ( groups2907647131375434839at_nat
        @ ^ [X4: list_nat] : Y
        @ A4 )
      = ( power_power_nat @ Y @ ( finite_card_list_nat @ A4 ) ) ) ).

% prod_constant
thf(fact_9423_prod__constant,axiom,
    ! [Y: nat,A4: set_set_nat] :
      ( ( groups4248547760180025341at_nat
        @ ^ [X4: set_nat] : Y
        @ A4 )
      = ( power_power_nat @ Y @ ( finite_card_set_nat @ A4 ) ) ) ).

% prod_constant
thf(fact_9424_prod__constant,axiom,
    ! [Y: int,A4: set_list_nat] :
      ( ( groups2905156660866384563at_int
        @ ^ [X4: list_nat] : Y
        @ A4 )
      = ( power_power_int @ Y @ ( finite_card_list_nat @ A4 ) ) ) ).

% prod_constant
thf(fact_9425_prod__constant,axiom,
    ! [Y: int,A4: set_set_nat] :
      ( ( groups4246057289670975065at_int
        @ ^ [X4: set_nat] : Y
        @ A4 )
      = ( power_power_int @ Y @ ( finite_card_set_nat @ A4 ) ) ) ).

% prod_constant
thf(fact_9426_prod__constant,axiom,
    ! [Y: nat,A4: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( power_power_nat @ Y @ ( finite_card_nat @ A4 ) ) ) ).

% prod_constant
thf(fact_9427_prod__constant,axiom,
    ! [Y: int,A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( power_power_int @ Y @ ( finite_card_nat @ A4 ) ) ) ).

% prod_constant
thf(fact_9428_prod__constant,axiom,
    ! [Y: int,A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X4: int] : Y
        @ A4 )
      = ( power_power_int @ Y @ ( finite_card_int @ A4 ) ) ) ).

% prod_constant
thf(fact_9429_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_9430_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_9431_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_9432_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_9433_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_9434_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_9435_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_9436_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_9437_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_9438_sum__constant,axiom,
    ! [Y: rat,A4: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9439_sum__constant,axiom,
    ! [Y: rat,A4: set_int] :
      ( ( groups3906332499630173760nt_rat
        @ ^ [X4: int] : Y
        @ A4 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9440_sum__constant,axiom,
    ! [Y: int,A4: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9441_sum__constant,axiom,
    ! [Y: nat,A4: set_int] :
      ( ( groups4541462559716669496nt_nat
        @ ^ [X4: int] : Y
        @ A4 )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9442_sum__constant,axiom,
    ! [Y: code_integer,A4: set_nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9443_sum__constant,axiom,
    ! [Y: code_integer,A4: set_int] :
      ( ( groups7873554091576472773nteger
        @ ^ [X4: int] : Y
        @ A4 )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9444_sum__constant,axiom,
    ! [Y: nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9445_sum__constant,axiom,
    ! [Y: int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : Y
        @ A4 )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_int @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9446_sum__constant,axiom,
    ! [Y: rat,A4: set_list_nat] :
      ( ( groups3760926236672600436at_rat
        @ ^ [X4: list_nat] : Y
        @ A4 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_list_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9447_sum__constant,axiom,
    ! [Y: rat,A4: set_set_nat] :
      ( ( groups7659867448343625626at_rat
        @ ^ [X4: set_nat] : Y
        @ A4 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_set_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_9448_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_9449_sum_Oinsert_H,axiom,
    ! [I5: set_o,P3: $o > code_integer,I2: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( ( member_o @ I2 @ I5 )
         => ( ( groups1402912129352969042nteger @ P3 @ ( insert_o @ I2 @ I5 ) )
            = ( groups1402912129352969042nteger @ P3 @ I5 ) ) )
        & ( ~ ( member_o @ I2 @ I5 )
         => ( ( groups1402912129352969042nteger @ P3 @ ( insert_o @ I2 @ I5 ) )
            = ( plus_p5714425477246183910nteger @ ( P3 @ I2 ) @ ( groups1402912129352969042nteger @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9450_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P3: nat > code_integer,I2: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( ( member_nat @ I2 @ I5 )
         => ( ( groups555127423416065298nteger @ P3 @ ( insert_nat @ I2 @ I5 ) )
            = ( groups555127423416065298nteger @ P3 @ I5 ) ) )
        & ( ~ ( member_nat @ I2 @ I5 )
         => ( ( groups555127423416065298nteger @ P3 @ ( insert_nat @ I2 @ I5 ) )
            = ( plus_p5714425477246183910nteger @ ( P3 @ I2 ) @ ( groups555127423416065298nteger @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9451_sum_Oinsert_H,axiom,
    ! [I5: set_int,P3: int > code_integer,I2: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( ( member_int @ I2 @ I5 )
         => ( ( groups926780983652909934nteger @ P3 @ ( insert_int @ I2 @ I5 ) )
            = ( groups926780983652909934nteger @ P3 @ I5 ) ) )
        & ( ~ ( member_int @ I2 @ I5 )
         => ( ( groups926780983652909934nteger @ P3 @ ( insert_int @ I2 @ I5 ) )
            = ( plus_p5714425477246183910nteger @ ( P3 @ I2 ) @ ( groups926780983652909934nteger @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9452_sum_Oinsert_H,axiom,
    ! [I5: set_Code_integer,P3: code_integer > code_integer,I2: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( ( member_Code_integer @ I2 @ I5 )
         => ( ( groups910942671188738463nteger @ P3 @ ( insert_Code_integer @ I2 @ I5 ) )
            = ( groups910942671188738463nteger @ P3 @ I5 ) ) )
        & ( ~ ( member_Code_integer @ I2 @ I5 )
         => ( ( groups910942671188738463nteger @ P3 @ ( insert_Code_integer @ I2 @ I5 ) )
            = ( plus_p5714425477246183910nteger @ ( P3 @ I2 ) @ ( groups910942671188738463nteger @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9453_sum_Oinsert_H,axiom,
    ! [I5: set_o,P3: $o > rat,I2: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_zero_rat ) ) ) )
     => ( ( ( member_o @ I2 @ I5 )
         => ( ( groups3921277224699582669_o_rat @ P3 @ ( insert_o @ I2 @ I5 ) )
            = ( groups3921277224699582669_o_rat @ P3 @ I5 ) ) )
        & ( ~ ( member_o @ I2 @ I5 )
         => ( ( groups3921277224699582669_o_rat @ P3 @ ( insert_o @ I2 @ I5 ) )
            = ( plus_plus_rat @ ( P3 @ I2 ) @ ( groups3921277224699582669_o_rat @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9454_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P3: nat > rat,I2: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_zero_rat ) ) ) )
     => ( ( ( member_nat @ I2 @ I5 )
         => ( ( groups1351286907653491341at_rat @ P3 @ ( insert_nat @ I2 @ I5 ) )
            = ( groups1351286907653491341at_rat @ P3 @ I5 ) ) )
        & ( ~ ( member_nat @ I2 @ I5 )
         => ( ( groups1351286907653491341at_rat @ P3 @ ( insert_nat @ I2 @ I5 ) )
            = ( plus_plus_rat @ ( P3 @ I2 ) @ ( groups1351286907653491341at_rat @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9455_sum_Oinsert_H,axiom,
    ! [I5: set_int,P3: int > rat,I2: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_zero_rat ) ) ) )
     => ( ( ( member_int @ I2 @ I5 )
         => ( ( groups2350640619554545897nt_rat @ P3 @ ( insert_int @ I2 @ I5 ) )
            = ( groups2350640619554545897nt_rat @ P3 @ I5 ) ) )
        & ( ~ ( member_int @ I2 @ I5 )
         => ( ( groups2350640619554545897nt_rat @ P3 @ ( insert_int @ I2 @ I5 ) )
            = ( plus_plus_rat @ ( P3 @ I2 ) @ ( groups2350640619554545897nt_rat @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9456_sum_Oinsert_H,axiom,
    ! [I5: set_Code_integer,P3: code_integer > rat,I2: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_zero_rat ) ) ) )
     => ( ( ( member_Code_integer @ I2 @ I5 )
         => ( ( groups8878813951405302554er_rat @ P3 @ ( insert_Code_integer @ I2 @ I5 ) )
            = ( groups8878813951405302554er_rat @ P3 @ I5 ) ) )
        & ( ~ ( member_Code_integer @ I2 @ I5 )
         => ( ( groups8878813951405302554er_rat @ P3 @ ( insert_Code_integer @ I2 @ I5 ) )
            = ( plus_plus_rat @ ( P3 @ I2 ) @ ( groups8878813951405302554er_rat @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9457_sum_Oinsert_H,axiom,
    ! [I5: set_o,P3: $o > nat,I2: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_o @ I2 @ I5 )
         => ( ( groups4556407284786078405_o_nat @ P3 @ ( insert_o @ I2 @ I5 ) )
            = ( groups4556407284786078405_o_nat @ P3 @ I5 ) ) )
        & ( ~ ( member_o @ I2 @ I5 )
         => ( ( groups4556407284786078405_o_nat @ P3 @ ( insert_o @ I2 @ I5 ) )
            = ( plus_plus_nat @ ( P3 @ I2 ) @ ( groups4556407284786078405_o_nat @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9458_sum_Oinsert_H,axiom,
    ! [I5: set_nat,P3: nat > nat,I2: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ I5 )
              & ( ( P3 @ X4 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_nat @ I2 @ I5 )
         => ( ( groups1986416967739987077at_nat @ P3 @ ( insert_nat @ I2 @ I5 ) )
            = ( groups1986416967739987077at_nat @ P3 @ I5 ) ) )
        & ( ~ ( member_nat @ I2 @ I5 )
         => ( ( groups1986416967739987077at_nat @ P3 @ ( insert_nat @ I2 @ I5 ) )
            = ( plus_plus_nat @ ( P3 @ I2 ) @ ( groups1986416967739987077at_nat @ P3 @ I5 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_9459_card__doubleton__eq__2__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_9460_card__doubleton__eq__2__iff,axiom,
    ! [A: list_nat,B: list_nat] :
      ( ( ( finite_card_list_nat @ ( insert_list_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_9461_card__doubleton__eq__2__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( finite_card_set_nat @ ( insert_set_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_9462_card__doubleton__eq__2__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_9463_card__doubleton__eq__2__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( finite_card_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_9464_card__doubleton__eq__2__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( finite_card_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_9465_card__doubleton__eq__2__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( finite_card_int @ ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_9466_sum__of__bool__eq,axiom,
    ! [A4: set_int,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X4: int] : ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri1316708129612266289at_nat @ ( finite_card_int @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9467_sum__of__bool__eq,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups7237345082560585321er_nat
            @ ^ [X4: code_integer] : ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri1316708129612266289at_nat @ ( finite4902975817058060853nteger @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9468_sum__of__bool__eq,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups7234854612051535045er_int
            @ ^ [X4: code_integer] : ( zero_n2684676970156552555ol_int @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri1314217659103216013at_int @ ( finite4902975817058060853nteger @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9469_sum__of__bool__eq,axiom,
    ! [A4: set_nat,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3539618377306564664at_int
            @ ^ [X4: nat] : ( zero_n2684676970156552555ol_int @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri1314217659103216013at_int @ ( finite_card_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9470_sum__of__bool__eq,axiom,
    ! [A4: set_int,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups7873554091576472773nteger
            @ ^ [X4: int] : ( zero_n356916108424825756nteger @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri4939895301339042750nteger @ ( finite_card_int @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9471_sum__of__bool__eq,axiom,
    ! [A4: set_Code_integer,P2: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ A4 )
       => ( ( groups879477027807139574nteger
            @ ^ [X4: code_integer] : ( zero_n356916108424825756nteger @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri4939895301339042750nteger @ ( finite4902975817058060853nteger @ ( inf_in1364745209274528805nteger @ A4 @ ( collect_Code_integer @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9472_sum__of__bool__eq,axiom,
    ! [A4: set_nat,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups7501900531339628137nteger
            @ ^ [X4: nat] : ( zero_n356916108424825756nteger @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri4939895301339042750nteger @ ( finite_card_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9473_sum__of__bool__eq,axiom,
    ! [A4: set_nat,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X4: nat] : ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri1316708129612266289at_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9474_sum__of__bool__eq,axiom,
    ! [A4: set_int,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups4538972089207619220nt_int
            @ ^ [X4: int] : ( zero_n2684676970156552555ol_int @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri1314217659103216013at_int @ ( finite_card_int @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9475_sum__of__bool__eq,axiom,
    ! [A4: set_list_nat,P2: list_nat > $o] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite8100373058378681591st_nat @ A4 )
       => ( ( groups4396056296759096172at_nat
            @ ^ [X4: list_nat] : ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) )
            @ A4 )
          = ( semiri1316708129612266289at_nat @ ( finite_card_list_nat @ ( inf_inf_set_list_nat @ A4 @ ( collect_list_nat @ P2 ) ) ) ) ) ) ) ).

% sum_of_bool_eq
thf(fact_9476_option_Osimps_I9_J,axiom,
    ! [F: num > num,X2: num] :
      ( ( map_option_num_num @ F @ ( some_num @ X2 ) )
      = ( some_num @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9477_option_Osimps_I9_J,axiom,
    ! [F: produc7388388658123137530it_nat > num,X2: produc7388388658123137530it_nat] :
      ( ( map_op6233829773304596897at_num @ F @ ( some_P2818173045054083285it_nat @ X2 ) )
      = ( some_num @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9478_option_Osimps_I9_J,axiom,
    ! [F: produc3260487557148687353it_nat > num,X2: produc3260487557148687353it_nat] :
      ( ( map_op5165227558541518752at_num @ F @ ( some_P7913643980934408916it_nat @ X2 ) )
      = ( some_num @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9479_option_Osimps_I9_J,axiom,
    ! [F: produc8664842809031399944it_nat > num,X2: produc8664842809031399944it_nat] :
      ( ( map_op2731889312448867759at_num @ F @ ( some_P1914260805536162275it_nat @ X2 ) )
      = ( some_num @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9480_option_Osimps_I9_J,axiom,
    ! [F: num > produc7388388658123137530it_nat,X2: num] :
      ( ( map_op2490657901765117089it_nat @ F @ ( some_num @ X2 ) )
      = ( some_P2818173045054083285it_nat @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9481_option_Osimps_I9_J,axiom,
    ! [F: num > produc3260487557148687353it_nat,X2: num] :
      ( ( map_op7586128837645442720it_nat @ F @ ( some_num @ X2 ) )
      = ( some_P7913643980934408916it_nat @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9482_option_Osimps_I9_J,axiom,
    ! [F: num > produc8664842809031399944it_nat,X2: num] :
      ( ( map_op8118133509425879471it_nat @ F @ ( some_num @ X2 ) )
      = ( some_P1914260805536162275it_nat @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9483_option_Osimps_I9_J,axiom,
    ! [F: produc7388388658123137530it_nat > produc7388388658123137530it_nat,X2: produc7388388658123137530it_nat] :
      ( ( map_op8581638957091552385it_nat @ F @ ( some_P2818173045054083285it_nat @ X2 ) )
      = ( some_P2818173045054083285it_nat @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9484_option_Osimps_I9_J,axiom,
    ! [F: produc7388388658123137530it_nat > produc3260487557148687353it_nat,X2: produc7388388658123137530it_nat] :
      ( ( map_op4453737856117102208it_nat @ F @ ( some_P2818173045054083285it_nat @ X2 ) )
      = ( some_P7913643980934408916it_nat @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9485_option_Osimps_I9_J,axiom,
    ! [F: produc7388388658123137530it_nat > produc8664842809031399944it_nat,X2: produc7388388658123137530it_nat] :
      ( ( map_op8924890681001947535it_nat @ F @ ( some_P2818173045054083285it_nat @ X2 ) )
      = ( some_P1914260805536162275it_nat @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_9486_map__option__cong,axiom,
    ! [X: option_num,Y: option_num,F: num > num,G: num > num] :
      ( ( X = Y )
     => ( ! [A3: num] :
            ( ( Y
              = ( some_num @ A3 ) )
           => ( ( F @ A3 )
              = ( G @ A3 ) ) )
       => ( ( map_option_num_num @ F @ X )
          = ( map_option_num_num @ G @ Y ) ) ) ) ).

% map_option_cong
thf(fact_9487_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_9488_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_9489_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y ) ) ) ).

% of_nat_max
thf(fact_9490_map__option_Ocomp,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
      ( ( comp_o9011007105448239796nteger @ ( map_op9146685272036663823nteger @ F ) @ ( map_op3669829223712506439nteger @ G ) )
      = ( map_op9146685272036663823nteger @ ( comp_C1593894019821074884nteger @ F @ G ) ) ) ).

% map_option.comp
thf(fact_9491_map__option_Ocomp,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer] :
      ( ( comp_o7188567781464924208nteger @ ( map_op3563471509957180299nteger @ F ) @ ( map_op370346339404370627nteger @ G ) )
      = ( map_op9146685272036663823nteger @ ( comp_C8797469213163452608nteger @ F @ G ) ) ) ).

% map_option.comp
thf(fact_9492_map__option_Ocomp,axiom,
    ! [F: int > nat,G: int > int] :
      ( ( comp_o4824269118391204861on_int @ ( map_option_int_nat @ F ) @ ( map_option_int_int @ G ) )
      = ( map_option_int_nat @ ( comp_int_nat_int @ F @ G ) ) ) ).

% map_option.comp
thf(fact_9493_map__option_Ocomp,axiom,
    ! [F: num > num,G: num > num] :
      ( ( comp_o8931257242830428707on_num @ ( map_option_num_num @ F ) @ ( map_option_num_num @ G ) )
      = ( map_option_num_num @ ( comp_num_num_num @ F @ G ) ) ) ).

% map_option.comp
thf(fact_9494_option_Omap__comp,axiom,
    ! [G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer,V: option_Code_integer] :
      ( ( map_op9146685272036663823nteger @ G @ ( map_op3669829223712506439nteger @ F @ V ) )
      = ( map_op9146685272036663823nteger @ ( comp_C1593894019821074884nteger @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_9495_option_Omap__comp,axiom,
    ! [G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,V: option_Code_integer] :
      ( ( map_op3563471509957180299nteger @ G @ ( map_op370346339404370627nteger @ F @ V ) )
      = ( map_op9146685272036663823nteger @ ( comp_C8797469213163452608nteger @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_9496_option_Omap__comp,axiom,
    ! [G: int > nat,F: int > int,V: option_int] :
      ( ( map_option_int_nat @ G @ ( map_option_int_int @ F @ V ) )
      = ( map_option_int_nat @ ( comp_int_nat_int @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_9497_option_Omap__comp,axiom,
    ! [G: num > num,F: num > num,V: option_num] :
      ( ( map_option_num_num @ G @ ( map_option_num_num @ F @ V ) )
      = ( map_option_num_num @ ( comp_num_num_num @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_9498_map__option_Ocompositionality,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,Option: option_Code_integer] :
      ( ( map_op9146685272036663823nteger @ F @ ( map_op3669829223712506439nteger @ G @ Option ) )
      = ( map_op9146685272036663823nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_9499_map__option_Ocompositionality,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,Option: option_Code_integer] :
      ( ( map_op3563471509957180299nteger @ F @ ( map_op370346339404370627nteger @ G @ Option ) )
      = ( map_op9146685272036663823nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_9500_map__option_Ocompositionality,axiom,
    ! [F: int > nat,G: int > int,Option: option_int] :
      ( ( map_option_int_nat @ F @ ( map_option_int_int @ G @ Option ) )
      = ( map_option_int_nat @ ( comp_int_nat_int @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_9501_map__option_Ocompositionality,axiom,
    ! [F: num > num,G: num > num,Option: option_num] :
      ( ( map_option_num_num @ F @ ( map_option_num_num @ G @ Option ) )
      = ( map_option_num_num @ ( comp_num_num_num @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_9502_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_9503_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_9504_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ B )
     => ( ord_less_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_9505_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( ( ord_less_num @ C2 @ B )
     => ( ord_less_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_9506_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C2 @ B )
     => ( ord_less_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_9507_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_int @ C2 @ B )
     => ( ord_less_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_9508_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ A )
     => ( ord_less_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_9509_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( ( ord_less_num @ C2 @ A )
     => ( ord_less_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_9510_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C2 @ A )
     => ( ord_less_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_9511_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ A )
     => ( ord_less_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_9512_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( A5
            = ( ord_max_rat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_9513_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( A5
            = ( ord_max_num @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_9514_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( A5
            = ( ord_max_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_9515_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( A5
            = ( ord_max_int @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_9516_max_Ostrict__boundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_9517_max_Ostrict__boundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C2 ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_9518_max_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_9519_max_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_9520_less__max__iff__disj,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z3 @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z3 @ X )
        | ( ord_less_rat @ Z3 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_9521_less__max__iff__disj,axiom,
    ! [Z3: num,X: num,Y: num] :
      ( ( ord_less_num @ Z3 @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z3 @ X )
        | ( ord_less_num @ Z3 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_9522_less__max__iff__disj,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z3 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z3 @ X )
        | ( ord_less_nat @ Z3 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_9523_less__max__iff__disj,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ Z3 @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z3 @ X )
        | ( ord_less_int @ Z3 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_9524_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_9525_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_9526_sup__max,axiom,
    sup_sup_int = ord_max_int ).

% sup_max
thf(fact_9527_sup__max,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_max
thf(fact_9528_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
thf(fact_9529_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_9530_option_Omap__ident,axiom,
    ! [T6: option_num] :
      ( ( map_option_num_num
        @ ^ [X4: num] : X4
        @ T6 )
      = T6 ) ).

% option.map_ident
thf(fact_9531_max_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C2 @ B )
     => ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_9532_max_OcoboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C2 @ B )
     => ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_9533_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_9534_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ B )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_9535_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_9536_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C2 @ A )
     => ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_9537_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_9538_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_9539_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_max_rat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_9540_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_max_num @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_9541_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_max_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_9542_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_max_int @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_9543_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_max_rat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_9544_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_max_num @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_9545_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_max_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_9546_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_max_int @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_9547_le__max__iff__disj,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z3 @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z3 @ X )
        | ( ord_less_eq_rat @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_9548_le__max__iff__disj,axiom,
    ! [Z3: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z3 @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z3 @ X )
        | ( ord_less_eq_num @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_9549_le__max__iff__disj,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z3 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z3 @ X )
        | ( ord_less_eq_nat @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_9550_le__max__iff__disj,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z3 @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z3 @ X )
        | ( ord_less_eq_int @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_9551_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_9552_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_9553_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_9554_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_9555_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_9556_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_9557_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_9558_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_9559_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( A5
          = ( ord_max_rat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_9560_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( A5
          = ( ord_max_num @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_9561_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( A5
          = ( ord_max_nat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_9562_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( A5
          = ( ord_max_int @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_9563_max_OboundedI,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_9564_max_OboundedI,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_9565_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_9566_max_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_9567_max_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_9568_max_OboundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_9569_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_9570_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_9571_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_9572_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_9573_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_9574_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_9575_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_9576_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_9577_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_9578_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_9579_max_Omono,axiom,
    ! [C2: rat,A: rat,D2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ( ord_less_eq_rat @ D2 @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C2 @ D2 ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_9580_max_Omono,axiom,
    ! [C2: num,A: num,D2: num,B: num] :
      ( ( ord_less_eq_num @ C2 @ A )
     => ( ( ord_less_eq_num @ D2 @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C2 @ D2 ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_9581_max_Omono,axiom,
    ! [C2: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C2 @ D2 ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_9582_max_Omono,axiom,
    ! [C2: int,A: int,D2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ( ord_less_eq_int @ D2 @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C2 @ D2 ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_9583_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_max_set_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_9584_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_max_rat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_9585_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_max_num @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_9586_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_9587_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_9588_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_max_set_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_9589_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_max_rat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_9590_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_max_num @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_9591_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_9592_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_9593_max__def,axiom,
    ( ord_max_set_int
    = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_9594_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_9595_max__def,axiom,
    ( ord_max_num
    = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_9596_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_9597_max__def,axiom,
    ( ord_max_int
    = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_9598_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z3 ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_9599_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z3 ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_9600_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_9601_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z3 )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z3 ) @ ( plus_plus_rat @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_9602_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z3 )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z3 ) @ ( plus_plus_int @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_9603_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z3 )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z3 ) @ ( plus_plus_nat @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_9604_option_Osimps_I8_J,axiom,
    ! [F: num > num] :
      ( ( map_option_num_num @ F @ none_num )
      = none_num ) ).

% option.simps(8)
thf(fact_9605_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_9606_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_9607_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_9608_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_9609_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_9610_n__subsets,axiom,
    ! [A4: set_Pr958786334691620121nt_int,K2: nat] :
      ( ( finite2998713641127702882nt_int @ A4 )
     => ( ( finite4053189226111446337nt_int
          @ ( collec5210948495886036740nt_int
            @ ^ [B6: set_Pr958786334691620121nt_int] :
                ( ( ord_le2843351958646193337nt_int @ B6 @ A4 )
                & ( ( finite6756421564338198497nt_int @ B6 )
                  = K2 ) ) ) )
        = ( binomial @ ( finite6756421564338198497nt_int @ A4 ) @ K2 ) ) ) ).

% n_subsets
thf(fact_9611_n__subsets,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite2364142230527598318st_nat
          @ ( collect_set_list_nat
            @ ^ [B6: set_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ B6 @ A4 )
                & ( ( finite_card_list_nat @ B6 )
                  = K2 ) ) ) )
        = ( binomial @ ( finite_card_list_nat @ A4 ) @ K2 ) ) ) ).

% n_subsets
thf(fact_9612_n__subsets,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite1149291290879098388et_nat
          @ ( collect_set_set_nat
            @ ^ [B6: set_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ B6 @ A4 )
                & ( ( finite_card_set_nat @ B6 )
                  = K2 ) ) ) )
        = ( binomial @ ( finite_card_set_nat @ A4 ) @ K2 ) ) ) ).

% n_subsets
thf(fact_9613_n__subsets,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [B6: set_nat] :
                ( ( ord_less_eq_set_nat @ B6 @ A4 )
                & ( ( finite_card_nat @ B6 )
                  = K2 ) ) ) )
        = ( binomial @ ( finite_card_nat @ A4 ) @ K2 ) ) ) ).

% n_subsets
thf(fact_9614_n__subsets,axiom,
    ! [A4: set_Code_integer,K2: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite1340570857867686763nteger
          @ ( collec574505750873337192nteger
            @ ^ [B6: set_Code_integer] :
                ( ( ord_le7084787975880047091nteger @ B6 @ A4 )
                & ( ( finite4902975817058060853nteger @ B6 )
                  = K2 ) ) ) )
        = ( binomial @ ( finite4902975817058060853nteger @ A4 ) @ K2 ) ) ) ).

% n_subsets
thf(fact_9615_n__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,K2: nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite4356350796350151305at_nat
          @ ( collec5514110066124741708at_nat
            @ ^ [B6: set_Pr1261947904930325089at_nat] :
                ( ( ord_le3146513528884898305at_nat @ B6 @ A4 )
                & ( ( finite711546835091564841at_nat @ B6 )
                  = K2 ) ) ) )
        = ( binomial @ ( finite711546835091564841at_nat @ A4 ) @ K2 ) ) ) ).

% n_subsets
thf(fact_9616_n__subsets,axiom,
    ! [A4: set_int,K2: nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_card_set_int
          @ ( collect_set_int
            @ ^ [B6: set_int] :
                ( ( ord_less_eq_set_int @ B6 @ A4 )
                & ( ( finite_card_int @ B6 )
                  = K2 ) ) ) )
        = ( binomial @ ( finite_card_int @ A4 ) @ K2 ) ) ) ).

% n_subsets
thf(fact_9617_sum_Onon__neutral_H,axiom,
    ! [G: $o > code_integer,I5: set_o] :
      ( ( groups1402912129352969042nteger @ G
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_z3403309356797280102nteger ) ) ) )
      = ( groups1402912129352969042nteger @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9618_sum_Onon__neutral_H,axiom,
    ! [G: nat > code_integer,I5: set_nat] :
      ( ( groups555127423416065298nteger @ G
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_z3403309356797280102nteger ) ) ) )
      = ( groups555127423416065298nteger @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9619_sum_Onon__neutral_H,axiom,
    ! [G: int > code_integer,I5: set_int] :
      ( ( groups926780983652909934nteger @ G
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_z3403309356797280102nteger ) ) ) )
      = ( groups926780983652909934nteger @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9620_sum_Onon__neutral_H,axiom,
    ! [G: $o > rat,I5: set_o] :
      ( ( groups3921277224699582669_o_rat @ G
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_zero_rat ) ) ) )
      = ( groups3921277224699582669_o_rat @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9621_sum_Onon__neutral_H,axiom,
    ! [G: nat > rat,I5: set_nat] :
      ( ( groups1351286907653491341at_rat @ G
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_zero_rat ) ) ) )
      = ( groups1351286907653491341at_rat @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9622_sum_Onon__neutral_H,axiom,
    ! [G: int > rat,I5: set_int] :
      ( ( groups2350640619554545897nt_rat @ G
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_zero_rat ) ) ) )
      = ( groups2350640619554545897nt_rat @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9623_sum_Onon__neutral_H,axiom,
    ! [G: $o > nat,I5: set_o] :
      ( ( groups4556407284786078405_o_nat @ G
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_zero_nat ) ) ) )
      = ( groups4556407284786078405_o_nat @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9624_sum_Onon__neutral_H,axiom,
    ! [G: nat > nat,I5: set_nat] :
      ( ( groups1986416967739987077at_nat @ G
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_zero_nat ) ) ) )
      = ( groups1986416967739987077at_nat @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9625_sum_Onon__neutral_H,axiom,
    ! [G: int > nat,I5: set_int] :
      ( ( groups2985770679641041633nt_nat @ G
        @ ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_zero_nat ) ) ) )
      = ( groups2985770679641041633nt_nat @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9626_sum_Onon__neutral_H,axiom,
    ! [G: $o > int,I5: set_o] :
      ( ( groups4553916814277028129_o_int @ G
        @ ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ I5 )
              & ( ( G @ X4 )
               != zero_zero_int ) ) ) )
      = ( groups4553916814277028129_o_int @ G @ I5 ) ) ).

% sum.non_neutral'
thf(fact_9627_option_Osize__gen__o__map,axiom,
    ! [F: int > nat,G: int > int] :
      ( ( comp_o5073597352248849069on_int @ ( size_option_int @ F ) @ ( map_option_int_int @ G ) )
      = ( size_option_int @ ( comp_int_nat_int @ F @ G ) ) ) ).

% option.size_gen_o_map
thf(fact_9628_option_Osize__gen__o__map,axiom,
    ! [F: num > nat,G: num > num] :
      ( ( comp_o6878144249584144265on_num @ ( size_option_num @ F ) @ ( map_option_num_num @ G ) )
      = ( size_option_num @ ( comp_num_nat_num @ F @ G ) ) ) ).

% option.size_gen_o_map
thf(fact_9629_card__subset__eq,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B5 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
       => ( ( ( finite_card_list_nat @ A4 )
            = ( finite_card_list_nat @ B5 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_subset_eq
thf(fact_9630_card__subset__eq,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
       => ( ( ( finite_card_set_nat @ A4 )
            = ( finite_card_set_nat @ B5 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_subset_eq
thf(fact_9631_card__subset__eq,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ( ( finite_card_nat @ A4 )
            = ( finite_card_nat @ B5 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_subset_eq
thf(fact_9632_card__subset__eq,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ( ( finite4902975817058060853nteger @ A4 )
            = ( finite4902975817058060853nteger @ B5 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_subset_eq
thf(fact_9633_card__subset__eq,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
       => ( ( ( finite711546835091564841at_nat @ A4 )
            = ( finite711546835091564841at_nat @ B5 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_subset_eq
thf(fact_9634_card__subset__eq,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ( ( finite_card_int @ A4 )
            = ( finite_card_int @ B5 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_subset_eq
thf(fact_9635_infinite__arbitrarily__large,axiom,
    ! [A4: set_list_nat,N2: nat] :
      ( ~ ( finite8100373058378681591st_nat @ A4 )
     => ? [B9: set_list_nat] :
          ( ( finite8100373058378681591st_nat @ B9 )
          & ( ( finite_card_list_nat @ B9 )
            = N2 )
          & ( ord_le6045566169113846134st_nat @ B9 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9636_infinite__arbitrarily__large,axiom,
    ! [A4: set_set_nat,N2: nat] :
      ( ~ ( finite1152437895449049373et_nat @ A4 )
     => ? [B9: set_set_nat] :
          ( ( finite1152437895449049373et_nat @ B9 )
          & ( ( finite_card_set_nat @ B9 )
            = N2 )
          & ( ord_le6893508408891458716et_nat @ B9 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9637_infinite__arbitrarily__large,axiom,
    ! [A4: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ? [B9: set_nat] :
          ( ( finite_finite_nat @ B9 )
          & ( ( finite_card_nat @ B9 )
            = N2 )
          & ( ord_less_eq_set_nat @ B9 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9638_infinite__arbitrarily__large,axiom,
    ! [A4: set_Code_integer,N2: nat] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ? [B9: set_Code_integer] :
          ( ( finite6017078050557962740nteger @ B9 )
          & ( ( finite4902975817058060853nteger @ B9 )
            = N2 )
          & ( ord_le7084787975880047091nteger @ B9 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9639_infinite__arbitrarily__large,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ~ ( finite6177210948735845034at_nat @ A4 )
     => ? [B9: set_Pr1261947904930325089at_nat] :
          ( ( finite6177210948735845034at_nat @ B9 )
          & ( ( finite711546835091564841at_nat @ B9 )
            = N2 )
          & ( ord_le3146513528884898305at_nat @ B9 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9640_infinite__arbitrarily__large,axiom,
    ! [A4: set_int,N2: nat] :
      ( ~ ( finite_finite_int @ A4 )
     => ? [B9: set_int] :
          ( ( finite_finite_int @ B9 )
          & ( ( finite_card_int @ B9 )
            = N2 )
          & ( ord_less_eq_set_int @ B9 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_9641_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_9642_card__le__if__inj__on__rel,axiom,
    ! [B5: set_o,A4: set_o,R2: $o > $o > $o] :
      ( ( finite_finite_o @ B5 )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A4 )
           => ? [B11: $o] :
                ( ( member_o @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: $o,A24: $o,B3: $o] :
              ( ( member_o @ A13 @ A4 )
             => ( ( member_o @ A24 @ A4 )
               => ( ( member_o @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A4 ) @ ( finite_card_o @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9643_card__le__if__inj__on__rel,axiom,
    ! [B5: set_o,A4: set_nat,R2: nat > $o > $o] :
      ( ( finite_finite_o @ B5 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ? [B11: $o] :
                ( ( member_o @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: nat,A24: nat,B3: $o] :
              ( ( member_nat @ A13 @ A4 )
             => ( ( member_nat @ A24 @ A4 )
               => ( ( member_o @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_o @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9644_card__le__if__inj__on__rel,axiom,
    ! [B5: set_o,A4: set_int,R2: int > $o > $o] :
      ( ( finite_finite_o @ B5 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ? [B11: $o] :
                ( ( member_o @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: int,A24: int,B3: $o] :
              ( ( member_int @ A13 @ A4 )
             => ( ( member_int @ A24 @ A4 )
               => ( ( member_o @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_o @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9645_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_o,R2: $o > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A4 )
           => ? [B11: nat] :
                ( ( member_nat @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: $o,A24: $o,B3: nat] :
              ( ( member_o @ A13 @ A4 )
             => ( ( member_o @ A24 @ A4 )
               => ( ( member_nat @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9646_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_nat,R2: nat > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ? [B11: nat] :
                ( ( member_nat @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: nat,A24: nat,B3: nat] :
              ( ( member_nat @ A13 @ A4 )
             => ( ( member_nat @ A24 @ A4 )
               => ( ( member_nat @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9647_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_int,R2: int > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ? [B11: nat] :
                ( ( member_nat @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: int,A24: int,B3: nat] :
              ( ( member_int @ A13 @ A4 )
             => ( ( member_int @ A24 @ A4 )
               => ( ( member_nat @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9648_card__le__if__inj__on__rel,axiom,
    ! [B5: set_int,A4: set_o,R2: $o > int > $o] :
      ( ( finite_finite_int @ B5 )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A4 )
           => ? [B11: int] :
                ( ( member_int @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: $o,A24: $o,B3: int] :
              ( ( member_o @ A13 @ A4 )
             => ( ( member_o @ A24 @ A4 )
               => ( ( member_int @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9649_card__le__if__inj__on__rel,axiom,
    ! [B5: set_int,A4: set_nat,R2: nat > int > $o] :
      ( ( finite_finite_int @ B5 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ? [B11: int] :
                ( ( member_int @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: nat,A24: nat,B3: int] :
              ( ( member_nat @ A13 @ A4 )
             => ( ( member_nat @ A24 @ A4 )
               => ( ( member_int @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9650_card__le__if__inj__on__rel,axiom,
    ! [B5: set_int,A4: set_int,R2: int > int > $o] :
      ( ( finite_finite_int @ B5 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ? [B11: int] :
                ( ( member_int @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: int,A24: int,B3: int] :
              ( ( member_int @ A13 @ A4 )
             => ( ( member_int @ A24 @ A4 )
               => ( ( member_int @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9651_card__le__if__inj__on__rel,axiom,
    ! [B5: set_Code_integer,A4: set_o,R2: $o > code_integer > $o] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A4 )
           => ? [B11: code_integer] :
                ( ( member_Code_integer @ B11 @ B5 )
                & ( R2 @ A3 @ B11 ) ) )
       => ( ! [A13: $o,A24: $o,B3: code_integer] :
              ( ( member_o @ A13 @ A4 )
             => ( ( member_o @ A24 @ A4 )
               => ( ( member_Code_integer @ B3 @ B5 )
                 => ( ( R2 @ A13 @ B3 )
                   => ( ( R2 @ A24 @ B3 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A4 ) @ ( finite4902975817058060853nteger @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_9652_card__insert__le,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ A4 ) @ ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_9653_card__insert__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_9654_card__insert__le,axiom,
    ! [A4: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A4 ) @ ( finite_card_o @ ( insert_o @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_9655_card__insert__le,axiom,
    ! [A4: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( insert_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_9656_card__insert__le,axiom,
    ! [A4: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ ( insert_int @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_9657_card__insert__le,axiom,
    ! [A4: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_9658_card__insert__le,axiom,
    ! [A4: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_9659_sum__multicount__gen,axiom,
    ! [S2: set_o,T6: set_o,R3: $o > $o > $o,K2: $o > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_o @ T6 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T6 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I: $o] :
                        ( ( member_o @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I: $o] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J: $o] :
                        ( ( member_o @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9660_sum__multicount__gen,axiom,
    ! [S2: set_o,T6: set_int,R3: $o > int > $o,K2: int > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T6 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I: $o] :
                        ( ( member_o @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I: $o] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9661_sum__multicount__gen,axiom,
    ! [S2: set_o,T6: set_Code_integer,R3: $o > code_integer > $o,K2: code_integer > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T6 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I: $o] :
                        ( ( member_o @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I: $o] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J: code_integer] :
                        ( ( member_Code_integer @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups7237345082560585321er_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9662_sum__multicount__gen,axiom,
    ! [S2: set_int,T6: set_o,R3: int > $o > $o,K2: $o > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_o @ T6 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T6 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J: $o] :
                        ( ( member_o @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9663_sum__multicount__gen,axiom,
    ! [S2: set_int,T6: set_int,R3: int > int > $o,K2: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T6 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9664_sum__multicount__gen,axiom,
    ! [S2: set_int,T6: set_Code_integer,R3: int > code_integer > $o,K2: code_integer > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T6 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J: code_integer] :
                        ( ( member_Code_integer @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups7237345082560585321er_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9665_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T6: set_o,R3: code_integer > $o > $o,K2: $o > nat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite_finite_o @ T6 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T6 )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I: code_integer] :
                        ( ( member_Code_integer @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I: code_integer] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J: $o] :
                        ( ( member_o @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9666_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T6: set_int,R3: code_integer > int > $o,K2: int > nat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T6 )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I: code_integer] :
                        ( ( member_Code_integer @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I: code_integer] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9667_sum__multicount__gen,axiom,
    ! [S2: set_Code_integer,T6: set_Code_integer,R3: code_integer > code_integer > $o,K2: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T6 )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I: code_integer] :
                        ( ( member_Code_integer @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I: code_integer] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J: code_integer] :
                        ( ( member_Code_integer @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups7237345082560585321er_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9668_sum__multicount__gen,axiom,
    ! [S2: set_o,T6: set_nat,R3: $o > nat > $o,K2: nat > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_nat @ T6 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T6 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I: $o] :
                        ( ( member_o @ I @ S2 )
                        & ( R3 @ I @ X3 ) ) ) )
                = ( K2 @ X3 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I: $o] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T6 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups3542108847815614940at_nat @ K2 @ T6 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_9669_card__eq__sum,axiom,
    ( finite_card_int
    = ( groups4541462559716669496nt_nat
      @ ^ [X4: int] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_9670_card__eq__sum,axiom,
    ( finite_card_list_nat
    = ( groups4396056296759096172at_nat
      @ ^ [X4: list_nat] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_9671_card__eq__sum,axiom,
    ( finite_card_set_nat
    = ( groups8294997508430121362at_nat
      @ ^ [X4: set_nat] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_9672_card__eq__sum,axiom,
    ( finite_card_nat
    = ( groups3542108847815614940at_nat
      @ ^ [X4: nat] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_9673_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_9674_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_9675_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > rat,H: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups1351286907653491341at_rat
          @ ^ [I: nat] : ( plus_plus_rat @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_rat @ ( groups1351286907653491341at_rat @ G @ I5 ) @ ( groups1351286907653491341at_rat @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9676_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > rat,H: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups2350640619554545897nt_rat
          @ ^ [I: int] : ( plus_plus_rat @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_rat @ ( groups2350640619554545897nt_rat @ G @ I5 ) @ ( groups2350640619554545897nt_rat @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9677_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > rat,H: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( groups8878813951405302554er_rat
          @ ^ [I: code_integer] : ( plus_plus_rat @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_rat @ ( groups8878813951405302554er_rat @ G @ I5 ) @ ( groups8878813951405302554er_rat @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9678_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > nat,H: nat > nat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups1986416967739987077at_nat
          @ ^ [I: nat] : ( plus_plus_nat @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I5 ) @ ( groups1986416967739987077at_nat @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9679_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > nat,H: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups2985770679641041633nt_nat
          @ ^ [I: int] : ( plus_plus_nat @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_nat @ ( groups2985770679641041633nt_nat @ G @ I5 ) @ ( groups2985770679641041633nt_nat @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9680_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > nat,H: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( groups290571974637022482er_nat
          @ ^ [I: code_integer] : ( plus_plus_nat @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_nat @ ( groups290571974637022482er_nat @ G @ I5 ) @ ( groups290571974637022482er_nat @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9681_sum_Odistrib__triv_H,axiom,
    ! [I5: set_nat,G: nat > int,H: nat > int] :
      ( ( finite_finite_nat @ I5 )
     => ( ( groups1983926497230936801at_int
          @ ^ [I: nat] : ( plus_plus_int @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_int @ ( groups1983926497230936801at_int @ G @ I5 ) @ ( groups1983926497230936801at_int @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9682_sum_Odistrib__triv_H,axiom,
    ! [I5: set_int,G: int > int,H: int > int] :
      ( ( finite_finite_int @ I5 )
     => ( ( groups2983280209131991357nt_int
          @ ^ [I: int] : ( plus_plus_int @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_int @ ( groups2983280209131991357nt_int @ G @ I5 ) @ ( groups2983280209131991357nt_int @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9683_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Code_integer,G: code_integer > int,H: code_integer > int] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( groups288081504127972206er_int
          @ ^ [I: code_integer] : ( plus_plus_int @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_int @ ( groups288081504127972206er_int @ G @ I5 ) @ ( groups288081504127972206er_int @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9684_sum_Odistrib__triv_H,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat,H: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ I5 )
     => ( ( groups3328076802468863542at_rat
          @ ^ [I: product_prod_nat_nat] : ( plus_plus_rat @ ( G @ I ) @ ( H @ I ) )
          @ I5 )
        = ( plus_plus_rat @ ( groups3328076802468863542at_rat @ G @ I5 ) @ ( groups3328076802468863542at_rat @ H @ I5 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_9685_card__ge__0__finite,axiom,
    ! [A4: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A4 ) )
     => ( finite8100373058378681591st_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_9686_card__ge__0__finite,axiom,
    ! [A4: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A4 ) )
     => ( finite1152437895449049373et_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_9687_card__ge__0__finite,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
     => ( finite_finite_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_9688_card__ge__0__finite,axiom,
    ! [A4: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
     => ( finite_finite_int @ A4 ) ) ).

% card_ge_0_finite
thf(fact_9689_card__ge__0__finite,axiom,
    ! [A4: set_Code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite4902975817058060853nteger @ A4 ) )
     => ( finite6017078050557962740nteger @ A4 ) ) ).

% card_ge_0_finite
thf(fact_9690_card__ge__0__finite,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A4 ) )
     => ( finite6177210948735845034at_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_9691_finite__if__finite__subsets__card__bdd,axiom,
    ! [F5: set_list_nat,C4: nat] :
      ( ! [G5: set_list_nat] :
          ( ( ord_le6045566169113846134st_nat @ G5 @ F5 )
         => ( ( finite8100373058378681591st_nat @ G5 )
           => ( ord_less_eq_nat @ ( finite_card_list_nat @ G5 ) @ C4 ) ) )
     => ( ( finite8100373058378681591st_nat @ F5 )
        & ( ord_less_eq_nat @ ( finite_card_list_nat @ F5 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9692_finite__if__finite__subsets__card__bdd,axiom,
    ! [F5: set_set_nat,C4: nat] :
      ( ! [G5: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ G5 @ F5 )
         => ( ( finite1152437895449049373et_nat @ G5 )
           => ( ord_less_eq_nat @ ( finite_card_set_nat @ G5 ) @ C4 ) ) )
     => ( ( finite1152437895449049373et_nat @ F5 )
        & ( ord_less_eq_nat @ ( finite_card_set_nat @ F5 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9693_finite__if__finite__subsets__card__bdd,axiom,
    ! [F5: set_nat,C4: nat] :
      ( ! [G5: set_nat] :
          ( ( ord_less_eq_set_nat @ G5 @ F5 )
         => ( ( finite_finite_nat @ G5 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G5 ) @ C4 ) ) )
     => ( ( finite_finite_nat @ F5 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F5 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9694_finite__if__finite__subsets__card__bdd,axiom,
    ! [F5: set_Code_integer,C4: nat] :
      ( ! [G5: set_Code_integer] :
          ( ( ord_le7084787975880047091nteger @ G5 @ F5 )
         => ( ( finite6017078050557962740nteger @ G5 )
           => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ G5 ) @ C4 ) ) )
     => ( ( finite6017078050557962740nteger @ F5 )
        & ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ F5 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9695_finite__if__finite__subsets__card__bdd,axiom,
    ! [F5: set_Pr1261947904930325089at_nat,C4: nat] :
      ( ! [G5: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ G5 @ F5 )
         => ( ( finite6177210948735845034at_nat @ G5 )
           => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ G5 ) @ C4 ) ) )
     => ( ( finite6177210948735845034at_nat @ F5 )
        & ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ F5 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9696_finite__if__finite__subsets__card__bdd,axiom,
    ! [F5: set_int,C4: nat] :
      ( ! [G5: set_int] :
          ( ( ord_less_eq_set_int @ G5 @ F5 )
         => ( ( finite_finite_int @ G5 )
           => ( ord_less_eq_nat @ ( finite_card_int @ G5 ) @ C4 ) ) )
     => ( ( finite_finite_int @ F5 )
        & ( ord_less_eq_nat @ ( finite_card_int @ F5 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_9697_card__seteq,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B5 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B5 ) @ ( finite_card_list_nat @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_9698_card__seteq,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B5 ) @ ( finite_card_set_nat @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_9699_card__seteq,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_9700_card__seteq,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ B5 ) @ ( finite4902975817058060853nteger @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_9701_card__seteq,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ B5 ) @ ( finite711546835091564841at_nat @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_9702_card__seteq,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ B5 ) @ ( finite_card_int @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_9703_card__mono,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B5 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) ) ) ) ).

% card_mono
thf(fact_9704_card__mono,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) ) ) ) ).

% card_mono
thf(fact_9705_card__mono,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ).

% card_mono
thf(fact_9706_card__mono,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B5 ) ) ) ) ).

% card_mono
thf(fact_9707_card__mono,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) ) ) ) ).

% card_mono
thf(fact_9708_card__mono,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ).

% card_mono
thf(fact_9709_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_list_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_list_nat @ S ) )
     => ~ ! [T8: set_list_nat] :
            ( ( ord_le6045566169113846134st_nat @ T8 @ S )
           => ( ( ( finite_card_list_nat @ T8 )
                = N2 )
             => ~ ( finite8100373058378681591st_nat @ T8 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9710_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_set_nat @ S ) )
     => ~ ! [T8: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ T8 @ S )
           => ( ( ( finite_card_set_nat @ T8 )
                = N2 )
             => ~ ( finite1152437895449049373et_nat @ T8 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9711_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
     => ~ ! [T8: set_nat] :
            ( ( ord_less_eq_set_nat @ T8 @ S )
           => ( ( ( finite_card_nat @ T8 )
                = N2 )
             => ~ ( finite_finite_nat @ T8 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9712_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Code_integer] :
      ( ( ord_less_eq_nat @ N2 @ ( finite4902975817058060853nteger @ S ) )
     => ~ ! [T8: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ T8 @ S )
           => ( ( ( finite4902975817058060853nteger @ T8 )
                = N2 )
             => ~ ( finite6017078050557962740nteger @ T8 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9713_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite711546835091564841at_nat @ S ) )
     => ~ ! [T8: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ T8 @ S )
           => ( ( ( finite711546835091564841at_nat @ T8 )
                = N2 )
             => ~ ( finite6177210948735845034at_nat @ T8 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9714_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_int] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ S ) )
     => ~ ! [T8: set_int] :
            ( ( ord_less_eq_set_int @ T8 @ S )
           => ( ( ( finite_card_int @ T8 )
                = N2 )
             => ~ ( finite_finite_int @ T8 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_9715_card__less__sym__Diff,axiom,
    ! [A4: set_list_nat,B5: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite8100373058378681591st_nat @ B5 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9716_card__less__sym__Diff,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite1152437895449049373et_nat @ B5 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9717_card__less__sym__Diff,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9718_card__less__sym__Diff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ord_less_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9719_card__less__sym__Diff,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( ord_less_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B5 ) )
         => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9720_card__less__sym__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) )
         => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_9721_card__le__sym__Diff,axiom,
    ! [A4: set_list_nat,B5: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite8100373058378681591st_nat @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9722_card__le__sym__Diff,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite1152437895449049373et_nat @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9723_card__le__sym__Diff,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9724_card__le__sym__Diff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9725_card__le__sym__Diff,axiom,
    ! [A4: set_Code_integer,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B5 ) )
         => ( ord_less_eq_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A4 @ B5 ) ) @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9726_card__le__sym__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_9727_psubset__card__mono,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B5 )
     => ( ( ord_le1190675801316882794st_nat @ A4 @ B5 )
       => ( ord_less_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) ) ) ) ).

% psubset_card_mono
thf(fact_9728_psubset__card__mono,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( ord_less_set_set_nat @ A4 @ B5 )
       => ( ord_less_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) ) ) ) ).

% psubset_card_mono
thf(fact_9729_psubset__card__mono,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_set_nat @ A4 @ B5 )
       => ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ).

% psubset_card_mono
thf(fact_9730_psubset__card__mono,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_set_int @ A4 @ B5 )
       => ( ord_less_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ).

% psubset_card_mono
thf(fact_9731_psubset__card__mono,axiom,
    ! [B5: set_Code_integer,A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le1307284697595431911nteger @ A4 @ B5 )
       => ( ord_less_nat @ ( finite4902975817058060853nteger @ A4 ) @ ( finite4902975817058060853nteger @ B5 ) ) ) ) ).

% psubset_card_mono
thf(fact_9732_psubset__card__mono,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
       => ( ord_less_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) ) ) ) ).

% psubset_card_mono
thf(fact_9733_card__Un__le,axiom,
    ! [A4: set_int,B5: set_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ).

% card_Un_le
thf(fact_9734_card__Un__le,axiom,
    ! [A4: set_list_nat,B5: set_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_9735_card__Un__le,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_9736_card__Un__le,axiom,
    ! [A4: set_Pr8551490117392284871at_nat,B5: set_Pr8551490117392284871at_nat] : ( ord_less_eq_nat @ ( finite7007076676225009423at_nat @ ( sup_su3035147773818789531at_nat @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite7007076676225009423at_nat @ A4 ) @ ( finite7007076676225009423at_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_9737_card__Un__le,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A4 ) @ ( finite3771342082235030671at_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_9738_card__Un__le,axiom,
    ! [A4: set_nat,B5: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_9739_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_9740_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_9741_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_9742_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_9743_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) )
     => ( A4
        = ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9744_Max_Osemilattice__order__set__axioms,axiom,
    ( lattic5374021519246970238et_rat @ ord_max_rat
    @ ^ [X4: rat,Y4: rat] : ( ord_less_eq_rat @ Y4 @ X4 )
    @ ^ [X4: rat,Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ) ).

% Max.semilattice_order_set_axioms
thf(fact_9745_Max_Osemilattice__order__set__axioms,axiom,
    ( lattic2566483365489244608et_num @ ord_max_num
    @ ^ [X4: num,Y4: num] : ( ord_less_eq_num @ Y4 @ X4 )
    @ ^ [X4: num,Y4: num] : ( ord_less_num @ Y4 @ X4 ) ) ).

% Max.semilattice_order_set_axioms
thf(fact_9746_Max_Osemilattice__order__set__axioms,axiom,
    ( lattic6009151579333465974et_nat @ ord_max_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 ) ) ).

% Max.semilattice_order_set_axioms
thf(fact_9747_Max_Osemilattice__order__set__axioms,axiom,
    ( lattic6006661108824415698et_int @ ord_max_int
    @ ^ [X4: int,Y4: int] : ( ord_less_eq_int @ Y4 @ X4 )
    @ ^ [X4: int,Y4: int] : ( ord_less_int @ Y4 @ X4 ) ) ).

% Max.semilattice_order_set_axioms
thf(fact_9748_sum__Suc,axiom,
    ! [F: int > nat,A4: set_int] :
      ( ( groups4541462559716669496nt_nat
        @ ^ [X4: int] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( finite_card_int @ A4 ) ) ) ).

% sum_Suc
thf(fact_9749_sum__Suc,axiom,
    ! [F: list_nat > nat,A4: set_list_nat] :
      ( ( groups4396056296759096172at_nat
        @ ^ [X4: list_nat] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups4396056296759096172at_nat @ F @ A4 ) @ ( finite_card_list_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_9750_sum__Suc,axiom,
    ! [F: set_nat > nat,A4: set_set_nat] :
      ( ( groups8294997508430121362at_nat
        @ ^ [X4: set_nat] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( finite_card_set_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_9751_sum__Suc,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( suc @ ( F @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( finite_card_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_9752_sum__multicount,axiom,
    ! [S: set_o,T7: set_o,R3: $o > $o > $o,K2: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_o @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T7 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I: $o] :
                        ( ( member_o @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I: $o] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J: $o] :
                        ( ( member_o @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite_card_o @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9753_sum__multicount,axiom,
    ! [S: set_o,T7: set_nat,R3: $o > nat > $o,K2: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_nat @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T7 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I: $o] :
                        ( ( member_o @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I: $o] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite_card_nat @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9754_sum__multicount,axiom,
    ! [S: set_o,T7: set_int,R3: $o > int > $o,K2: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_int @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T7 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I: $o] :
                        ( ( member_o @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I: $o] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite_card_int @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9755_sum__multicount,axiom,
    ! [S: set_o,T7: set_Code_integer,R3: $o > code_integer > $o,K2: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite6017078050557962740nteger @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T7 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I: $o] :
                        ( ( member_o @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I: $o] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J: code_integer] :
                        ( ( member_Code_integer @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite4902975817058060853nteger @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9756_sum__multicount,axiom,
    ! [S: set_int,T7: set_o,R3: int > $o > $o,K2: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_o @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T7 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J: $o] :
                        ( ( member_o @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite_card_o @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9757_sum__multicount,axiom,
    ! [S: set_int,T7: set_nat,R3: int > nat > $o,K2: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T7 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite_card_nat @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9758_sum__multicount,axiom,
    ! [S: set_int,T7: set_int,R3: int > int > $o,K2: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T7 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite_card_int @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9759_sum__multicount,axiom,
    ! [S: set_int,T7: set_Code_integer,R3: int > code_integer > $o,K2: nat] :
      ( ( finite_finite_int @ S )
     => ( ( finite6017078050557962740nteger @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T7 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [J: code_integer] :
                        ( ( member_Code_integer @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite4902975817058060853nteger @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9760_sum__multicount,axiom,
    ! [S: set_Code_integer,T7: set_o,R3: code_integer > $o > $o,K2: nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_o @ T7 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ T7 )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I: code_integer] :
                        ( ( member_Code_integer @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I: code_integer] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J: $o] :
                        ( ( member_o @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite_card_o @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9761_sum__multicount,axiom,
    ! [S: set_Code_integer,T7: set_nat,R3: code_integer > nat > $o,K2: nat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_nat @ T7 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T7 )
             => ( ( finite4902975817058060853nteger
                  @ ( collect_Code_integer
                    @ ^ [I: code_integer] :
                        ( ( member_Code_integer @ I @ S )
                        & ( R3 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups7237345082560585321er_nat
              @ ^ [I: code_integer] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T7 )
                        & ( R3 @ I @ J ) ) ) )
              @ S )
            = ( times_times_nat @ K2 @ ( finite_card_nat @ T7 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_9762_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > code_integer] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z3403309356797280102nteger ) )
       => ( ( groups3803682039294397627nteger @ G @ S )
          = ( groups3803682039294397627nteger @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9763_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > multis2468970476368604999at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z1048942125864253310at_nat ) )
       => ( ( groups8361914993108267113at_nat @ G @ S )
          = ( groups8361914993108267113at_nat @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9764_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ( groups3328076802468863542at_rat @ G @ S )
          = ( groups3328076802468863542at_rat @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9765_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ( groups3963206862555359278at_nat @ G @ S )
          = ( groups3963206862555359278at_nat @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9766_sum_Omono__neutral__left_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > int] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ( groups3960716392046309002at_int @ G @ S )
          = ( groups3960716392046309002at_int @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9767_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,G: int > code_integer] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z3403309356797280102nteger ) )
       => ( ( groups926780983652909934nteger @ G @ S )
          = ( groups926780983652909934nteger @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9768_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,G: int > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z1048942125864253310at_nat ) )
       => ( ( groups6906599089918680438at_nat @ G @ S )
          = ( groups6906599089918680438at_nat @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9769_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ( groups2350640619554545897nt_rat @ G @ S )
          = ( groups2350640619554545897nt_rat @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9770_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ( groups2985770679641041633nt_nat @ G @ S )
          = ( groups2985770679641041633nt_nat @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9771_sum_Omono__neutral__left_H,axiom,
    ! [S: set_int,T7: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ( groups2983280209131991357nt_int @ G @ S )
          = ( groups2983280209131991357nt_int @ G @ T7 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_9772_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > code_integer] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z3403309356797280102nteger ) )
       => ( ( groups3803682039294397627nteger @ G @ T7 )
          = ( groups3803682039294397627nteger @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9773_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > multis2468970476368604999at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z1048942125864253310at_nat ) )
       => ( ( groups8361914993108267113at_nat @ G @ T7 )
          = ( groups8361914993108267113at_nat @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9774_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ( groups3328076802468863542at_rat @ G @ T7 )
          = ( groups3328076802468863542at_rat @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9775_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ( groups3963206862555359278at_nat @ G @ T7 )
          = ( groups3963206862555359278at_nat @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9776_sum_Omono__neutral__right_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T7: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > int] :
      ( ( ord_le3146513528884898305at_nat @ S @ T7 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ( groups3960716392046309002at_int @ G @ T7 )
          = ( groups3960716392046309002at_int @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9777_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > code_integer] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z3403309356797280102nteger ) )
       => ( ( groups926780983652909934nteger @ G @ T7 )
          = ( groups926780983652909934nteger @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9778_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z1048942125864253310at_nat ) )
       => ( ( groups6906599089918680438at_nat @ G @ T7 )
          = ( groups6906599089918680438at_nat @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9779_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ( groups2350640619554545897nt_rat @ G @ T7 )
          = ( groups2350640619554545897nt_rat @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9780_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ( groups2985770679641041633nt_nat @ G @ T7 )
          = ( groups2985770679641041633nt_nat @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9781_sum_Omono__neutral__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ( groups2983280209131991357nt_int @ G @ T7 )
          = ( groups2983280209131991357nt_int @ G @ S ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_9782_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_o,T7: set_o,H: $o > code_integer,G: $o > code_integer] :
      ( ( ord_less_eq_set_o @ S @ T7 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ ( minus_minus_set_o @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_z3403309356797280102nteger ) )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups1402912129352969042nteger @ G @ S )
            = ( groups1402912129352969042nteger @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9783_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_nat,T7: set_nat,H: nat > code_integer,G: nat > code_integer] :
      ( ( ord_less_eq_set_nat @ S @ T7 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ ( minus_minus_set_nat @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_z3403309356797280102nteger ) )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups555127423416065298nteger @ G @ S )
            = ( groups555127423416065298nteger @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9784_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_o,T7: set_o,H: $o > rat,G: $o > rat] :
      ( ( ord_less_eq_set_o @ S @ T7 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ ( minus_minus_set_o @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_zero_rat ) )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups3921277224699582669_o_rat @ G @ S )
            = ( groups3921277224699582669_o_rat @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9785_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_nat,T7: set_nat,H: nat > rat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ S @ T7 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ ( minus_minus_set_nat @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_zero_rat ) )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups1351286907653491341at_rat @ G @ S )
            = ( groups1351286907653491341at_rat @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9786_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_o,T7: set_o,H: $o > nat,G: $o > nat] :
      ( ( ord_less_eq_set_o @ S @ T7 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ ( minus_minus_set_o @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_zero_nat ) )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups4556407284786078405_o_nat @ G @ S )
            = ( groups4556407284786078405_o_nat @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9787_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_nat,T7: set_nat,H: nat > nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S @ T7 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ ( minus_minus_set_nat @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_zero_nat ) )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups1986416967739987077at_nat @ G @ S )
            = ( groups1986416967739987077at_nat @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9788_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_o,T7: set_o,H: $o > int,G: $o > int] :
      ( ( ord_less_eq_set_o @ S @ T7 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ ( minus_minus_set_o @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_zero_int ) )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups4553916814277028129_o_int @ G @ S )
            = ( groups4553916814277028129_o_int @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9789_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_nat,T7: set_nat,H: nat > int,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S @ T7 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ ( minus_minus_set_nat @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_zero_int ) )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups1983926497230936801at_int @ G @ S )
            = ( groups1983926497230936801at_int @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9790_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_int,T7: set_int,H: int > code_integer,G: int > code_integer] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_z3403309356797280102nteger ) )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups926780983652909934nteger @ G @ S )
            = ( groups926780983652909934nteger @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9791_sum_Omono__neutral__cong__left_H,axiom,
    ! [S: set_int,T7: set_int,H: int > rat,G: int > rat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( H @ I3 )
              = zero_zero_rat ) )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups2350640619554545897nt_rat @ G @ S )
            = ( groups2350640619554545897nt_rat @ H @ T7 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_9792_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > code_integer,H: int > code_integer] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z3403309356797280102nteger ) )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups926780983652909934nteger @ G @ T7 )
            = ( groups926780983652909934nteger @ H @ S ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_9793_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > multis2468970476368604999at_nat,H: int > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_z1048942125864253310at_nat ) )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups6906599089918680438at_nat @ G @ T7 )
            = ( groups6906599089918680438at_nat @ H @ S ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_9794_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > rat,H: int > rat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_rat ) )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups2350640619554545897nt_rat @ G @ T7 )
            = ( groups2350640619554545897nt_rat @ H @ S ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_9795_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > nat,H: int > nat] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_nat ) )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups2985770679641041633nt_nat @ G @ T7 )
            = ( groups2985770679641041633nt_nat @ H @ S ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_9796_sum_Omono__neutral__cong__right_H,axiom,
    ! [S: set_int,T7: set_int,G: int > int,H: int > int] :
      ( ( ord_less_eq_set_int @ S @ T7 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S ) )
           => ( ( G @ X3 )
              = zero_zero_int ) )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ S )
             => ( ( G @ X3 )
                = ( H @ X3 ) ) )
         => ( ( groups2983280209131991357nt_int @ G @ T7 )
            = ( groups2983280209131991357nt_int @ H @ S ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_9797_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_9798_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_9799_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 )
                                      @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
                                      @ ( 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_9800_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 )
                                    @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9801_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_9802_and__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
        @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(9)
thf(fact_9803_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 )
                                        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
                                        @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
thf(fact_9804_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_9805_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ? [R4: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R4 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
          & ! [N10: nat] :
              ( ( ord_less_nat @ N10 @ ( finite_card_nat @ S ) )
             => ( member_nat @ ( R4 @ N10 ) @ S ) ) ) ) ).

% finite_enumerate
thf(fact_9806_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9807_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L @ U ) ) ).

% finite_greaterThanLessThan_integer
thf(fact_9808_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J2 ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9809_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_9810_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J2: nat,I2: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J2 @ ( suc @ I2 ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J2 ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I2 @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9811_split__seed__def,axiom,
    ( split_seed
    = ( ^ [S5: produc7822875418678951345atural] :
          ( produc8282080750456430313atural
          @ ^ [V2: code_natural,W3: code_natural] :
              ( produc8282080750456430313atural
              @ ^ [V4: 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 @ V4 @ ( 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 @ S5 ) ) )
          @ S5 ) ) ) ).

% split_seed_def
thf(fact_9812_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J2: nat,I2: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J2 @ I2 ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J2 ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I2 @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9813_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
              @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U3 ) ) ) )
          @ Xa
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_9814_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_9815_mod__emp,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ one_one_assn @ H )
      = ( ( produc8586169260539613262et_nat @ H )
        = bot_bot_set_nat ) ) ).

% mod_emp
thf(fact_9816_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_9817_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J2 ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_9818_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
            @ ^ [U3: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) )
        @ Xa
        @ X ) ) ).

% less_int.abs_eq
thf(fact_9819_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
            @ ^ [U3: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) )
        @ Xa
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_9820_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
              @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) )
          @ Xa
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_9821_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
              @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U3 ) ) )
          @ Xa
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_9822_rat__minus__code,axiom,
    ! [P3: rat,Q6: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P3 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_minus_code
thf(fact_9823_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L @ U ) ) ).

% finite_greaterThanAtMost_integer
thf(fact_9824_drop__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N2 @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9825_drop__bit__negative__int__iff,axiom,
    ! [N2: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9826_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q8: rat,R5: rat] : ( plus_plus_rat @ Q8 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).

% diff_rat_def
thf(fact_9827_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9828_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9829_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q6: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P3 @ Q6 ) )
     => ( ord_less_int @ zero_zero_int @ Q6 ) ) ).

% quotient_of_denom_pos
thf(fact_9830_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P7: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P7 ) ) ) ) ).

% rat_floor_code
thf(fact_9831_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_9832_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_9833_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P7: rat,Q8: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D5: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ C3 @ B4 ) )
              @ ( quotient_of @ Q8 ) )
          @ ( quotient_of @ P7 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9834_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P7: rat,Q8: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D5: int] : ( ord_less_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ C3 @ B4 ) )
              @ ( quotient_of @ Q8 ) )
          @ ( quotient_of @ P7 ) ) ) ) ).

% rat_less_code
thf(fact_9835_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_9836_rat__divide__code,axiom,
    ! [P3: rat,Q6: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P3 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ C3 @ B4 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_divide_code
thf(fact_9837_rat__times__code,axiom,
    ! [P3: rat,Q6: rat] :
      ( ( quotient_of @ ( times_times_rat @ P3 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C3 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_times_code
thf(fact_9838_rat__plus__code,axiom,
    ! [P3: rat,Q6: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P3 @ Q6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D5 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D5 ) ) )
            @ ( quotient_of @ Q6 ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_plus_code
thf(fact_9839_rat__inverse__code,axiom,
    ! [P3: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,B4: 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 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
        @ ( quotient_of @ P3 ) ) ) ).

% rat_inverse_code
thf(fact_9840_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Xa2: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y4: nat,Z: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U3: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y4 @ V2 ) @ ( plus_plus_nat @ U3 @ Z ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa2 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9841_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Xa2: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y4: nat,Z: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U3: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y4 @ V2 ) @ ( plus_plus_nat @ U3 @ Z ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa2 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9842_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_9843_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_9844_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_9845_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X4: rat] :
          ( the_int
          @ ^ [Z: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 )
              & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9846_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_9847_range,axiom,
    ! [K2: code_natural,S2: produc7822875418678951345atural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ K2 )
     => ( ord_le5570908160329646204atural @ ( produc497848011034438852atural @ ( range @ K2 @ S2 ) ) @ K2 ) ) ).

% range
thf(fact_9848_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_9849_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
            @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U3 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9850_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
            @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9851_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
            @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U3 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9852_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_9853_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_9854_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_9855_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U4: int,V5: int] :
      ( ( plus_plus_int @ ( times_times_int @ U4 @ X ) @ ( times_times_int @ V5 @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% bezout_int
thf(fact_9856_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_9857_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_9858_gcd__cases__int,axiom,
    ! [X: int,Y: int,P2: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( P2 @ ( gcd_gcd_int @ X @ Y ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y @ zero_zero_int )
           => ( P2 @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y )
             => ( P2 @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y @ zero_zero_int )
               => ( P2 @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
           => ( P2 @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9859_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_9860_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_9861_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_9862_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_9863_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_9864_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_9865_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_9866_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y3: nat] :
          ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9867_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_9868_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_9869_bezw__aux,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).

% bezw_aux
thf(fact_9870_upto_Opsimps,axiom,
    ! [I2: int,J2: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
     => ( ( ( ord_less_eq_int @ I2 @ J2 )
         => ( ( upto @ I2 @ J2 )
            = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ I2 @ J2 )
         => ( ( upto @ I2 @ J2 )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_9871_upto__Nil,axiom,
    ! [I2: int,J2: int] :
      ( ( ( upto @ I2 @ J2 )
        = nil_int )
      = ( ord_less_int @ J2 @ I2 ) ) ).

% upto_Nil
thf(fact_9872_upto__Nil2,axiom,
    ! [I2: int,J2: int] :
      ( ( nil_int
        = ( upto @ I2 @ J2 ) )
      = ( ord_less_int @ J2 @ I2 ) ) ).

% upto_Nil2
thf(fact_9873_upto__empty,axiom,
    ! [J2: int,I2: int] :
      ( ( ord_less_int @ J2 @ I2 )
     => ( ( upto @ I2 @ J2 )
        = nil_int ) ) ).

% upto_empty
thf(fact_9874_nth__upto,axiom,
    ! [I2: int,K2: nat,J2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K2 ) ) @ J2 )
     => ( ( nth_int @ ( upto @ I2 @ J2 ) @ K2 )
        = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K2 ) ) ) ) ).

% nth_upto
thf(fact_9875_length__upto,axiom,
    ! [I2: int,J2: int] :
      ( ( size_size_list_int @ ( upto @ I2 @ J2 ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J2 @ I2 ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_9876_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_9877_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_9878_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_9879_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_9880_sorted__wrt__upto,axiom,
    ! [I2: int,J2: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I2 @ J2 ) ) ).

% sorted_wrt_upto
thf(fact_9881_sorted__upto,axiom,
    ! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).

% sorted_upto
thf(fact_9882_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I: int,J: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_9883_upto__rec1,axiom,
    ! [I2: int,J2: int] :
      ( ( ord_less_eq_int @ I2 @ J2 )
     => ( ( upto @ I2 @ J2 )
        = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) ) ) ) ).

% upto_rec1
thf(fact_9884_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_9885_upto_Osimps,axiom,
    ( upto
    = ( ^ [I: int,J: int] : ( if_list_int @ ( ord_less_eq_int @ I @ J ) @ ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9886_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I: int,J: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ ( minus_minus_int @ J @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_9887_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_9888_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_9889_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_9890_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_9891_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_9892_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_9893_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M3: nat,N: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ) ).

% pred_nat_def
thf(fact_9894_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_9895_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_9896_upto__rec2,axiom,
    ! [I2: int,J2: int] :
      ( ( ord_less_eq_int @ I2 @ J2 )
     => ( ( upto @ I2 @ J2 )
        = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J2 @ one_one_int ) ) @ ( cons_int @ J2 @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_9897_less__rat,axiom,
    ! [B: int,D2: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D2 ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% less_rat
thf(fact_9898_le__rat,axiom,
    ! [B: int,D2: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D2 ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% le_rat
thf(fact_9899_add__rat,axiom,
    ! [B: int,D2: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D2 ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ C2 @ B ) ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).

% add_rat
thf(fact_9900_Rat__induct__pos,axiom,
    ! [P2: rat > $o,Q6: rat] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( P2 @ ( fract @ A3 @ B3 ) ) )
     => ( P2 @ Q6 ) ) ).

% Rat_induct_pos
thf(fact_9901_upto__split2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( ( ord_less_eq_int @ I2 @ J2 )
     => ( ( ord_less_eq_int @ J2 @ K2 )
       => ( ( upto @ I2 @ K2 )
          = ( append_int @ ( upto @ I2 @ J2 ) @ ( upto @ ( plus_plus_int @ J2 @ one_one_int ) @ K2 ) ) ) ) ) ).

% upto_split2
thf(fact_9902_upto__split1,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( ( ord_less_eq_int @ I2 @ J2 )
     => ( ( ord_less_eq_int @ J2 @ K2 )
       => ( ( upto @ I2 @ K2 )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J2 @ one_one_int ) ) @ ( upto @ J2 @ K2 ) ) ) ) ) ).

% upto_split1
thf(fact_9903_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_9904_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_9905_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_9906_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_9907_Fract__add__one,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( fract @ ( plus_plus_int @ M @ N2 ) @ N2 )
        = ( plus_plus_rat @ ( fract @ M @ N2 ) @ one_one_rat ) ) ) ).

% Fract_add_one
thf(fact_9908_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_9909_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_9910_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_9911_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_9912_upto__split3,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( ( ord_less_eq_int @ I2 @ J2 )
     => ( ( ord_less_eq_int @ J2 @ K2 )
       => ( ( upto @ I2 @ K2 )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J2 @ one_one_int ) ) @ ( cons_int @ J2 @ ( upto @ ( plus_plus_int @ J2 @ one_one_int ) @ K2 ) ) ) ) ) ) ).

% upto_split3
thf(fact_9913_sort__upto,axiom,
    ! [I2: int,J2: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X4: int] : X4
        @ ( upto @ I2 @ J2 ) )
      = ( upto @ I2 @ J2 ) ) ).

% sort_upto
thf(fact_9914_sort__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X4: nat] : X4
        @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sort_upt
thf(fact_9915_upt__0__eq__Nil__conv,axiom,
    ! [J2: nat] :
      ( ( ( upt @ zero_zero_nat @ J2 )
        = nil_nat )
      = ( J2 = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
thf(fact_9916_upt__merge,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
        & ( ord_less_eq_nat @ J2 @ K2 ) )
     => ( ( append_nat @ ( upt @ I2 @ J2 ) @ ( upt @ J2 @ K2 ) )
        = ( upt @ I2 @ K2 ) ) ) ).

% upt_merge
thf(fact_9917_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( upt @ I2 @ J2 )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9918_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( upt @ I2 @ J2 )
        = nil_nat )
      = ( ( J2 = zero_zero_nat )
        | ( ord_less_eq_nat @ J2 @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_9919_nth__upt,axiom,
    ! [I2: nat,K2: nat,J2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J2 )
     => ( ( nth_nat @ ( upt @ I2 @ J2 ) @ K2 )
        = ( plus_plus_nat @ I2 @ K2 ) ) ) ).

% nth_upt
thf(fact_9920_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_9921_upt__eq__append__conv,axiom,
    ! [I2: nat,J2: nat,Xs: list_nat,Ys3: list_nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ( upt @ I2 @ J2 )
          = ( append_nat @ Xs @ Ys3 ) )
        = ( ? [K3: nat] :
              ( ( ord_less_eq_nat @ I2 @ K3 )
              & ( ord_less_eq_nat @ K3 @ J2 )
              & ( ( upt @ I2 @ K3 )
                = Xs )
              & ( ( upt @ K3 @ J2 )
                = Ys3 ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_9922_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] :
      ( ( map_nat_nat
        @ ^ [I: nat] : ( plus_plus_nat @ I @ Ofs )
        @ ( upt @ A @ B ) )
      = ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_9923_upt__append,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( append_nat @ ( upt @ zero_zero_nat @ I2 ) @ ( upt @ I2 @ J2 ) )
        = ( upt @ zero_zero_nat @ J2 ) ) ) ).

% upt_append
thf(fact_9924_upt__add__eq__append,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J2 @ K2 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_9925_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).

% sorted_wrt_upt
thf(fact_9926_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I: nat] : ( plus_plus_nat @ I @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_9927_sorted__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).

% sorted_upt
thf(fact_9928_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ J2 )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) ) ) ).

% upt_conv_Cons
thf(fact_9929_upt__rec,axiom,
    ( upt
    = ( ^ [I: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I @ J ) @ ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9930_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_9931_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( suc @ J2 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_9932_upt__filter__extend,axiom,
    ! [U: nat,U5: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U5 )
     => ( ! [I3: nat] :
            ( ( ( ord_less_eq_nat @ U @ I3 )
              & ( ord_less_nat @ I3 @ U5 ) )
           => ~ ( P2 @ I3 ) )
       => ( ( filter_nat2 @ P2 @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat2 @ P2 @ ( upt @ zero_zero_nat @ U5 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_9933_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_9934_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_9935_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J2: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J2 )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J2 )
        & ( I2 = X )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J2 )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_9936_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_9937_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_9938_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_9939_hd__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( hd_nat @ ( upt @ I2 @ J2 ) )
        = I2 ) ) ).

% hd_upt
thf(fact_9940_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S4: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_9941_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S4: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_9942_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_9943_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_9944_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_9945_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_9946_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C2 @ Y )
       => ( ( image_nat_nat
            @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C2 ) @ ( minus_minus_nat @ Y @ C2 ) ) ) )
      & ( ~ ( ord_less_nat @ C2 @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9947_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9948_Sup__unit__def,axiom,
    ( comple4687483117567863418t_unit
    = ( ^ [Uu3: set_Product_unit] : product_Unity ) ) ).

% Sup_unit_def
thf(fact_9949_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_9950_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_9951_Inf__unit__def,axiom,
    ( comple2584293577114468500t_unit
    = ( ^ [Uu3: set_Product_unit] : product_Unity ) ) ).

% Inf_unit_def
thf(fact_9952_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_9953_Inf__int__def,axiom,
    ( complete_Inf_Inf_int
    = ( ^ [X11: set_int] : ( uminus_uminus_int @ ( complete_Sup_Sup_int @ ( image_int_int @ uminus_uminus_int @ X11 ) ) ) ) ) ).

% Inf_int_def
thf(fact_9954_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_9955_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_9956_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_9957_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_9958_mod__true,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ top_top_assn @ H )
      = ( in_range @ H ) ) ).

% mod_true
thf(fact_9959_UNIV__unit,axiom,
    ( top_to1996260823553986621t_unit
    = ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).

% UNIV_unit
thf(fact_9960_ent__true,axiom,
    ! [P2: assn] : ( entails @ P2 @ top_top_assn ) ).

% ent_true
thf(fact_9961_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
thf(fact_9962_mod__star__trueI,axiom,
    ! [P2: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P2 @ H )
     => ( rep_assn @ ( times_times_assn @ P2 @ top_top_assn ) @ H ) ) ).

% mod_star_trueI
thf(fact_9963_mod__star__trueE,axiom,
    ! [P2: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ top_top_assn ) @ H )
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P2 @ H4 ) ) ).

% mod_star_trueE
thf(fact_9964_top__assn__def,axiom,
    ( top_top_assn
    = ( abs_assn @ in_range ) ) ).

% top_assn_def
thf(fact_9965_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_9966_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_9967_mod__star__trueE_H,axiom,
    ! [P2: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ top_top_assn ) @ H )
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ( ( ( produc1824681642469235216et_nat @ H4 )
              = ( produc1824681642469235216et_nat @ H ) )
           => ( ( ord_less_eq_set_nat @ ( produc8586169260539613262et_nat @ H4 ) @ ( produc8586169260539613262et_nat @ H ) )
             => ~ ( rep_assn @ P2 @ H4 ) ) ) ) ).

% mod_star_trueE'
thf(fact_9968_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_9969_drop__upt,axiom,
    ! [M: nat,I2: nat,J2: nat] :
      ( ( drop_nat @ M @ ( upt @ I2 @ J2 ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M ) @ J2 ) ) ).

% drop_upt
thf(fact_9970_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_9971_last__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( last_nat @ ( upt @ I2 @ J2 ) )
        = ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).

% last_upt
thf(fact_9972_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_9973_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_9974_plus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
    @ plus_plus_int
    @ plus_plus_int ) ).

% plus_integer.rsp
thf(fact_9975_plus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    @ plus_plus_nat
    @ plus_plus_nat ) ).

% plus_natural.rsp
thf(fact_9976_sub_Orsp,axiom,
    ( bNF_re8402795839162346335um_int
    @ ^ [Y6: num,Z4: num] : Y6 = Z4
    @ ( bNF_re1822329894187522285nt_int
      @ ^ [Y6: num,Z4: num] : Y6 = Z4
      @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
    @ ^ [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_9977_dup_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
    @ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).

% dup.rsp
thf(fact_9978_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_9979_less__eq__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ord_less_eq_int
    @ ord_less_eq_int ) ).

% less_eq_integer.rsp
thf(fact_9980_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_9981_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_9982_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ pcr_rat )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_9983_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_9984_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_9985_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_9986_positive_Otransfer,axiom,
    ( bNF_re1494630372529172596at_o_o @ pcr_rat
    @ ^ [Y6: $o,Z4: $o] : Y6 = Z4
    @ ^ [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_9987_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U3 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_9988_positive__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( positive @ X )
     => ( ( positive @ Y )
       => ( positive @ ( plus_plus_rat @ X @ Y ) ) ) ) ).

% positive_add
thf(fact_9989_nat_Otransfer,axiom,
    ( bNF_re4555766996558763186at_nat @ pcr_int
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ nat2 ) ).

% nat.transfer
thf(fact_9990_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ pcr_int
    @ ^ [N: nat] : ( product_Pair_nat_nat @ N @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_9991_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] : ( positive @ ( minus_minus_rat @ Y4 @ X4 ) ) ) ) ).

% less_rat_def
thf(fact_9992_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_9993_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_9994_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_9995_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_9996_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_9997_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U3 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_9998_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_9999_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_10000_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M3: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N ) ) @ M3 ) ) ) ).

% prod_encode_def
thf(fact_10001_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_10002_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_10003_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_10004_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_10005_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_10006_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_10007_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_10008_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_10009_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_10010_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_10011_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_10012_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_10013_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_10014_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K2 @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K2 ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_10015_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U3 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y4 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V2 ) @ ( times_times_nat @ Y4 @ U3 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_10016_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_10017_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_10018_nat_Orsp,axiom,
    ( bNF_re8246922863344978751at_nat @ intrel
    @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).

% nat.rsp
thf(fact_10019_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V2: nat] :
              ( ( plus_plus_nat @ X4 @ V2 )
              = ( plus_plus_nat @ U3 @ Y4 ) ) ) ) ) ).

% intrel_def
thf(fact_10020_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) ) ) ) ).

% less_int.rsp
thf(fact_10021_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y6: $o,Z4: $o] : Y6 = Z4 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U3: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_10022_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y4 @ V2 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_10023_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U3 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U3: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ Y4 @ U3 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_10024_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10025_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_10026_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( order_mono_nat_nat
        @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K2 @ M3 ) @ M3 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10027_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_10028_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
            @ ^ [U3: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) ) ) ) ) ).

% less_int_def
thf(fact_10029_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
            @ ^ [U3: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V2 ) @ ( plus_plus_nat @ U3 @ Y4 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_10030_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_10031_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_10032_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_10033_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_10034_plus__integer__def,axiom,
    ( plus_p5714425477246183910nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ plus_plus_int ) ) ).

% plus_integer_def
thf(fact_10035_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_10036_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_10037_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_10038_Fract__def,axiom,
    ( fract
    = ( map_fu7831380289885515383nt_rat @ id_int @ ( map_fu3424225382358772769nt_rat @ id_int @ abs_Rat )
      @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) ) ) ) ).

% Fract_def
thf(fact_10039_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_10040_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_10041_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_10042_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_10043_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y6: int,Z4: int] : Y6 = Z4
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y6: int,Z4: int] : Y6 = Z4
      @ ratrel )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) ) ) ).

% Fract.rsp
thf(fact_10044_dup_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( code_dup @ X ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).

% dup.rep_eq
thf(fact_10045_dup_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_dup @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ X @ X ) ) ) ).

% dup.abs_eq
thf(fact_10046_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_10047_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_10048_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_10049_positive_Orsp,axiom,
    ( bNF_re8699439704749558557nt_o_o @ ratrel
    @ ^ [Y6: $o,Z4: $o] : Y6 = Z4
    @ ^ [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_10050_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_10051_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_10052_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M @ N2 ) ) @ one_one_Code_integer ) ) ).

% Code_Numeral.sub_code(8)
thf(fact_10053_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_10054_Code__Numeral_Osub__def,axiom,
    ( code_sub
    = ( map_fu6891787308814931657nteger @ id_num @ ( map_fu8638147718074629079nteger @ id_num @ code_integer_of_int )
      @ ^ [M3: num,N: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% Code_Numeral.sub_def
thf(fact_10055_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_10056_plus__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( plus_p4538020629002901425atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( plus_plus_nat @ Xa @ X ) ) ) ).

% plus_natural.abs_eq
thf(fact_10057_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_10058_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_10059_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_10060_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_10061_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_10062_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_10063_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_10064_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_10065_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_10066_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_10067_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_10068_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_10069_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_10070_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_10071_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_10072_inj__on__diff__nat,axiom,
    ! [N7: set_nat,K2: nat] :
      ( ! [N5: nat] :
          ( ( member_nat @ N5 @ N7 )
         => ( ord_less_eq_nat @ K2 @ N5 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K2 )
        @ N7 ) ) ).

% inj_on_diff_nat
thf(fact_10073_inj__Suc,axiom,
    ! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).

% inj_Suc
thf(fact_10074_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_10075_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_10076_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_10077_Rat__cases,axiom,
    ! [Q6: rat] :
      ~ ! [A3: int,B3: int] :
          ( ( Q6
            = ( fract @ A3 @ B3 ) )
         => ( ( ord_less_int @ zero_zero_int @ B3 )
           => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) ).

% Rat_cases
thf(fact_10078_Rat__induct,axiom,
    ! [P2: rat > $o,Q6: rat] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( ( algebr932160517623751201me_int @ A3 @ B3 )
           => ( P2 @ ( fract @ A3 @ B3 ) ) ) )
     => ( P2 @ Q6 ) ) ).

% Rat_induct
thf(fact_10079_Rat__cases__nonzero,axiom,
    ! [Q6: rat] :
      ( ! [A3: int,B3: int] :
          ( ( Q6
            = ( fract @ A3 @ B3 ) )
         => ( ( ord_less_int @ zero_zero_int @ B3 )
           => ( ( A3 != zero_zero_int )
             => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) )
     => ( Q6 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
thf(fact_10080_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X3: product_prod_int_int] :
      ( ( R2
        = ( 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 ) )
      & ! [Y5: product_prod_int_int] :
          ( ( ( R2
              = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
            & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
            & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
         => ( Y5 = X3 ) ) ) ).

% quotient_of_unique
thf(fact_10081_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_10082_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_10083_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_10084_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_10085_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_10086_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_10087_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_10088_Least__eq__0,axiom,
    ! [P2: nat > $o] :
      ( ( P2 @ zero_zero_nat )
     => ( ( ord_Least_nat @ P2 )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_10089_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_10090_Least__Suc2,axiom,
    ! [P2: nat > $o,N2: nat,Q2: nat > $o,M: nat] :
      ( ( P2 @ N2 )
     => ( ( Q2 @ M )
       => ( ~ ( P2 @ zero_zero_nat )
         => ( ! [K: nat] :
                ( ( P2 @ ( suc @ K ) )
                = ( Q2 @ K ) )
           => ( ( ord_Least_nat @ P2 )
              = ( suc @ ( ord_Least_nat @ Q2 ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_10091_Least__Suc,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ N2 )
     => ( ~ ( P2 @ zero_zero_nat )
       => ( ( ord_Least_nat @ P2 )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M3: nat] : ( P2 @ ( suc @ M3 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_10092_GreatestI__nat,axiom,
    ! [P2: nat > $o,K2: nat,B: nat] :
      ( ( P2 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_nat
thf(fact_10093_Greatest__le__nat,axiom,
    ! [P2: nat > $o,K2: nat,B: nat] :
      ( ( P2 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% Greatest_le_nat
thf(fact_10094_GreatestI__ex__nat,axiom,
    ! [P2: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P2 @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_10095_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_10096_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_10097_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_10098_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_10099_sqr_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N2 ) @ N2 ) ) ) ) ).

% sqr.simps(3)
thf(fact_10100_Rep__unit,axiom,
    ! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_10101_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_10102_Rep__unit__induct,axiom,
    ! [Y: $o,P2: $o > $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X3: product_unit] : ( P2 @ ( product_Rep_unit @ X3 ) )
       => ( P2 @ Y ) ) ) ).

% Rep_unit_induct
thf(fact_10103_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_10104_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_10105_Unity__def,axiom,
    ( product_Unity
    = ( product_Abs_unit @ $true ) ) ).

% Unity_def
thf(fact_10106_Rep__unit__inverse,axiom,
    ! [X: product_unit] :
      ( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
      = X ) ).

% Rep_unit_inverse
thf(fact_10107_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_10108_Abs__unit__induct,axiom,
    ! [P2: product_unit > $o,X: product_unit] :
      ( ! [Y3: $o] :
          ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P2 @ ( product_Abs_unit @ Y3 ) ) )
     => ( P2 @ X ) ) ).

% Abs_unit_induct
thf(fact_10109_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_10110_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_10111_plus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( plus_p4538020629002901425atural @ X @ Xa ) )
      = ( plus_plus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% plus_natural.rep_eq
thf(fact_10112_less__natural_Orep__eq,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [X4: code_natural,Xa2: code_natural] : ( ord_less_nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa2 ) ) ) ) ).

% less_natural.rep_eq
thf(fact_10113_less__eq__natural_Orep__eq,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [X4: code_natural,Xa2: code_natural] : ( ord_less_eq_nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa2 ) ) ) ) ).

% less_eq_natural.rep_eq
thf(fact_10114_plus__natural__def,axiom,
    ( plus_p4538020629002901425atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ plus_plus_nat ) ) ).

% plus_natural_def
thf(fact_10115_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_10116_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_10117_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_10118_eventually__sequentially__Suc,axiom,
    ! [P2: nat > $o] :
      ( ( eventually_nat
        @ ^ [I: nat] : ( P2 @ ( suc @ I ) )
        @ at_top_nat )
      = ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_10119_eventually__sequentially__seg,axiom,
    ! [P2: nat > $o,K2: nat] :
      ( ( eventually_nat
        @ ^ [N: nat] : ( P2 @ ( plus_plus_nat @ N @ K2 ) )
        @ at_top_nat )
      = ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10120_eventually__False__sequentially,axiom,
    ~ ( eventually_nat
      @ ^ [N: nat] : $false
      @ at_top_nat ) ).

% eventually_False_sequentially
thf(fact_10121_eventually__sequentially,axiom,
    ! [P2: nat > $o] :
      ( ( eventually_nat @ P2 @ at_top_nat )
      = ( ? [N8: nat] :
          ! [N: nat] :
            ( ( ord_less_eq_nat @ N8 @ N )
           => ( P2 @ N ) ) ) ) ).

% eventually_sequentially
thf(fact_10122_eventually__sequentiallyI,axiom,
    ! [C2: nat,P2: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C2 @ X3 )
         => ( P2 @ X3 ) )
     => ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_10123_le__sequentially,axiom,
    ! [F5: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
      = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F5 ) ) ) ).

% le_sequentially
thf(fact_10124_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_10125_pair__lessI2,axiom,
    ! [A: nat,B: nat,S2: nat,T6: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S2 @ T6 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T6 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_10126_total__pair__less,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A4 @ fun_pair_less ) ).

% total_pair_less
thf(fact_10127_trans__pair__less,axiom,
    trans_4347625901269045472at_nat @ fun_pair_less ).

% trans_pair_less
thf(fact_10128_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z3 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z3 ) ) ).

% pair_less_iff1
thf(fact_10129_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat @ fun_pair_less ).

% wf_pair_less
thf(fact_10130_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_10131_pair__lessI1,axiom,
    ! [A: nat,B: nat,S2: nat,T6: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T6 ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_10132_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_10133_pair__leqI2,axiom,
    ! [A: nat,B: nat,S2: nat,T6: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S2 @ T6 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T6 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_10134_pair__leq__def,axiom,
    ( fun_pair_leq
    = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).

% pair_leq_def
thf(fact_10135_smin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( ( X7 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_10136_min__strict__def,axiom,
    ( fun_min_strict
    = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).

% min_strict_def
thf(fact_10137_pair__leqI1,axiom,
    ! [A: nat,B: nat,S2: nat,T6: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T6 ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_10138_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_10139_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_10140_wmin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_10141_wmax__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X7 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X7 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_10142_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_10143_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_10144_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_10145_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X7 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_10146_max__strict__def,axiom,
    ( fun_max_strict
    = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).

% max_strict_def
thf(fact_10147_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_10148_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_10149_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_10150_smsI,axiom,
    ! [A4: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A4 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A4 ) @ ( plus_p7104986032573967614at_nat @ Z5 @ B5 ) ) @ ms_strict ) ) ).

% smsI
thf(fact_10151_wmsI,axiom,
    ! [A4: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
      ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A4 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
        | ( ( A4 = zero_z1048942125864253310at_nat )
          & ( B5 = zero_z1048942125864253310at_nat ) ) )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A4 ) @ ( plus_p7104986032573967614at_nat @ Z5 @ B5 ) ) @ ms_weak ) ) ).

% wmsI
thf(fact_10152_ms__weak__def,axiom,
    ( ms_weak
    = ( sup_su3035147773818789531at_nat @ ms_strict @ id_mul2649389997224486051at_nat ) ) ).

% ms_weak_def
thf(fact_10153_ms__weakI1,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat,A4: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z5 @ Z9 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A4 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A4 ) @ ( plus_p7104986032573967614at_nat @ Z9 @ B5 ) ) @ ms_weak ) ) ) ).

% ms_weakI1
thf(fact_10154_ms__strictI,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat,A4: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z5 @ Z9 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A4 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A4 ) @ ( plus_p7104986032573967614at_nat @ Z9 @ B5 ) ) @ ms_strict ) ) ) ).

% ms_strictI
thf(fact_10155_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 ) )
             => ! [Y10: multis2468970476368604999at_nat] :
                  ( ( A23
                    = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y3 @ zero_z1048942125864253310at_nat ) @ Y10 ) )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
                   => ~ ( pw_leq @ X14 @ Y10 ) ) ) ) ) ) ).

% pw_leq.cases
thf(fact_10156_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,Y11: 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 ) @ Y11 ) )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ fun_pair_leq )
              & ( pw_leq @ X11 @ Y11 ) ) ) ) ) ).

% pw_leq.simps
thf(fact_10157_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X7: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
     => ( ( pw_leq @ X7 @ Y7 )
       => ( pw_leq @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ X7 ) @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) @ Y7 ) ) ) ) ).

% pw_leq_step
thf(fact_10158_ms__weakI2,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z5 @ Z9 )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ zero_z1048942125864253310at_nat ) @ ( plus_p7104986032573967614at_nat @ Z9 @ zero_z1048942125864253310at_nat ) ) @ ms_weak ) ) ).

% ms_weakI2
thf(fact_10159_pw__leq__split,axiom,
    ! [X7: multis2468970476368604999at_nat,Y7: multis2468970476368604999at_nat] :
      ( ( pw_leq @ X7 @ Y7 )
     => ? [A9: multis2468970476368604999at_nat,B9: multis2468970476368604999at_nat,Z10: multis2468970476368604999at_nat] :
          ( ( X7
            = ( plus_p7104986032573967614at_nat @ A9 @ Z10 ) )
          & ( Y7
            = ( plus_p7104986032573967614at_nat @ B9 @ Z10 ) )
          & ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A9 ) @ ( set_ms8126754132646256062at_nat @ B9 ) ) @ fun_max_strict )
            | ( ( B9 = zero_z1048942125864253310at_nat )
              & ( A9 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
thf(fact_10160_bdd__above__nat,axiom,
    condit2214826472909112428ve_nat = finite_finite_nat ).

% bdd_above_nat
thf(fact_10161_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_10162_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_10163_natLeq__Card__order,axiom,
    bNF_Ca1281551314933786834on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Card_order
thf(fact_10164_natLeq__antisym,axiom,
    antisym_nat @ bNF_Ca8665028551170535155natLeq ).

% natLeq_antisym
thf(fact_10165_natLeq__natLess__Id,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( minus_1356011639430497352at_nat @ bNF_Ca8665028551170535155natLeq @ id_nat2 ) ) ).

% natLeq_natLess_Id
thf(fact_10166_natLeq__trans,axiom,
    trans_nat @ bNF_Ca8665028551170535155natLeq ).

% natLeq_trans
thf(fact_10167_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_10168_natLeq__Total,axiom,
    total_on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Total
thf(fact_10169_natLeq__Linear__order,axiom,
    order_4473980167227706203on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Linear_order
thf(fact_10170_natLeq__Preorder,axiom,
    order_4861654808422542329on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Preorder
thf(fact_10171_natLeq__Refl,axiom,
    refl_on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Refl
thf(fact_10172_Field__natLeq,axiom,
    ( ( field_nat @ bNF_Ca8665028551170535155natLeq )
    = top_top_set_nat ) ).

% Field_natLeq
thf(fact_10173_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_10174_natLeq__Well__order,axiom,
    order_2888998067076097458on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Well_order
thf(fact_10175_Restr__natLeq2,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
          @ ^ [Uu2: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ 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_natLeq2
thf(fact_10176_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
thf(fact_10177_card__of__nat,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_nat
thf(fact_10178_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_Field_natLeq
thf(fact_10179_eventually__prod__sequentially,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P2 @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N8: nat] :
          ! [M3: nat] :
            ( ( ord_less_eq_nat @ N8 @ M3 )
           => ! [N: nat] :
                ( ( ord_less_eq_nat @ N8 @ N )
               => ( P2 @ ( product_Pair_nat_nat @ N @ M3 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_10180_prod__decode__triangle__add,axiom,
    ! [K2: nat,M: nat] :
      ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K2 ) @ M ) )
      = ( nat_prod_decode_aux @ K2 @ M ) ) ).

% prod_decode_triangle_add
thf(fact_10181_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N5: nat] :
              ( ( X
                = ( suc @ N5 ) )
             => ( Y
               != ( produc2761476792215241774st_nat
                  @ ^ [X4: nat,Y4: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y4 ) )
                  @ ( nat_prod_decode @ N5 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_10182_list__decode_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( nat_list_decode @ ( suc @ N2 ) )
      = ( produc2761476792215241774st_nat
        @ ^ [X4: nat,Y4: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y4 ) )
        @ ( nat_prod_decode @ N2 ) ) ) ).

% list_decode.simps(2)
thf(fact_10183_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( accp_nat @ nat_list_decode_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N5: nat] :
                ( ( X
                  = ( suc @ N5 ) )
               => ( ( Y
                    = ( produc2761476792215241774st_nat
                      @ ^ [X4: nat,Y4: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y4 ) )
                      @ ( nat_prod_decode @ N5 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N5 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_10184_list__decode_Opsimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
     => ( ( nat_list_decode @ ( suc @ N2 ) )
        = ( produc2761476792215241774st_nat
          @ ^ [X4: nat,Y4: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y4 ) )
          @ ( nat_prod_decode @ N2 ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_10185_rat_Odomain,axiom,
    ( ( domain9213661015745956397nt_rat @ pcr_rat )
    = ( ^ [X4: product_prod_int_int] :
        ? [Y4: product_prod_int_int] :
          ( ( basic_4387203522000727145nt_int
            @ ^ [Y6: int,Z4: int] : Y6 = Z4
            @ ^ [Y6: int,Z4: int] : Y6 = Z4
            @ X4
            @ Y4 )
          & ( ratrel @ Y4 @ Y4 ) ) ) ) ).

% rat.domain
thf(fact_10186_int_Oid__abs__transfer,axiom,
    ( bNF_re7934895593101944656at_int
    @ ( basic_5328504652464829177at_nat
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4
      @ ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    @ pcr_int
    @ ^ [X4: product_prod_nat_nat] : X4
    @ abs_Integ ) ).

% int.id_abs_transfer
thf(fact_10187_rat_Odomain__par__left__total,axiom,
    ! [P6: product_prod_int_int > $o] :
      ( ( left_t3131394472396969446nt_int
        @ ( basic_4387203522000727145nt_int
          @ ^ [Y6: int,Z4: int] : Y6 = Z4
          @ ^ [Y6: int,Z4: int] : Y6 = Z4 ) )
     => ( ( bNF_re8699439704749558557nt_o_o
          @ ( basic_4387203522000727145nt_int
            @ ^ [Y6: int,Z4: int] : Y6 = Z4
            @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
          @ ^ [Y6: $o,Z4: $o] : Y6 = Z4
          @ P6
          @ ^ [X4: product_prod_int_int] : ( ratrel @ X4 @ X4 ) )
       => ( ( domain9213661015745956397nt_rat @ pcr_rat )
          = P6 ) ) ) ).

% rat.domain_par_left_total
thf(fact_10188_rat_Odomain__par,axiom,
    ! [DR1: int > $o,DR2: int > $o,P22: product_prod_int_int > $o] :
      ( ( ( domainp_int_int
          @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
        = DR1 )
     => ( ( ( domainp_int_int
            @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
          = DR2 )
       => ( ( bNF_re8699439704749558557nt_o_o
            @ ( basic_4387203522000727145nt_int
              @ ^ [Y6: int,Z4: int] : Y6 = Z4
              @ ^ [Y6: int,Z4: int] : Y6 = Z4 )
            @ ^ [Y6: $o,Z4: $o] : Y6 = Z4
            @ P22
            @ ^ [X4: product_prod_int_int] : ( ratrel @ X4 @ X4 ) )
         => ( ( domain9213661015745956397nt_rat @ pcr_rat )
            = ( inf_in3604695632404883862_int_o @ ( basic_1567116559311922317nt_int @ DR1 @ DR2 ) @ P22 ) ) ) ) ) ).

% rat.domain_par
thf(fact_10189_Abs__rat__cases,axiom,
    ! [X: rat] :
      ~ ! [Y3: set_Pr958786334691620121nt_int] :
          ( ( X
            = ( abs_rat @ Y3 ) )
         => ~ ( member2340774599025711042nt_int @ Y3
              @ ( collec5210948495886036740nt_int
                @ ^ [C3: set_Pr958786334691620121nt_int] :
                  ? [X4: product_prod_int_int] :
                    ( ( ratrel @ X4 @ X4 )
                    & ( C3
                      = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) ) ) ).

% Abs_rat_cases
thf(fact_10190_Abs__rat__induct,axiom,
    ! [P2: rat > $o,X: rat] :
      ( ! [Y3: set_Pr958786334691620121nt_int] :
          ( ( member2340774599025711042nt_int @ Y3
            @ ( collec5210948495886036740nt_int
              @ ^ [C3: set_Pr958786334691620121nt_int] :
                ? [X4: product_prod_int_int] :
                  ( ( ratrel @ X4 @ X4 )
                  & ( C3
                    = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) )
         => ( P2 @ ( abs_rat @ Y3 ) ) )
     => ( P2 @ X ) ) ).

% Abs_rat_induct
thf(fact_10191_Abs__rat__inject,axiom,
    ! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ X
        @ ( collec5210948495886036740nt_int
          @ ^ [C3: set_Pr958786334691620121nt_int] :
            ? [X4: product_prod_int_int] :
              ( ( ratrel @ X4 @ X4 )
              & ( C3
                = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) )
     => ( ( member2340774599025711042nt_int @ Y
          @ ( collec5210948495886036740nt_int
            @ ^ [C3: set_Pr958786334691620121nt_int] :
              ? [X4: product_prod_int_int] :
                ( ( ratrel @ X4 @ X4 )
                & ( C3
                  = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) )
       => ( ( ( abs_rat @ X )
            = ( abs_rat @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_rat_inject
thf(fact_10192_Abs__rat__inverse,axiom,
    ! [Y: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ Y
        @ ( collec5210948495886036740nt_int
          @ ^ [C3: set_Pr958786334691620121nt_int] :
            ? [X4: product_prod_int_int] :
              ( ( ratrel @ X4 @ X4 )
              & ( C3
                = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) )
     => ( ( rep_rat @ ( abs_rat @ Y ) )
        = Y ) ) ).

% Abs_rat_inverse
thf(fact_10193_type__definition__rat,axiom,
    ( type_d8554052265237484179nt_int @ rep_rat @ abs_rat
    @ ( collec5210948495886036740nt_int
      @ ^ [C3: set_Pr958786334691620121nt_int] :
        ? [X4: product_prod_int_int] :
          ( ( ratrel @ X4 @ X4 )
          & ( C3
            = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) ) ).

% type_definition_rat
thf(fact_10194_Rep__rat,axiom,
    ! [X: rat] :
      ( member2340774599025711042nt_int @ ( rep_rat @ X )
      @ ( collec5210948495886036740nt_int
        @ ^ [C3: set_Pr958786334691620121nt_int] :
          ? [X4: product_prod_int_int] :
            ( ( ratrel @ X4 @ X4 )
            & ( C3
              = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) ) ).

% Rep_rat
thf(fact_10195_Rep__rat__cases,axiom,
    ! [Y: set_Pr958786334691620121nt_int] :
      ( ( member2340774599025711042nt_int @ Y
        @ ( collec5210948495886036740nt_int
          @ ^ [C3: set_Pr958786334691620121nt_int] :
            ? [X4: product_prod_int_int] :
              ( ( ratrel @ X4 @ X4 )
              & ( C3
                = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) )
     => ~ ! [X3: rat] :
            ( Y
           != ( rep_rat @ X3 ) ) ) ).

% Rep_rat_cases
thf(fact_10196_Rep__rat__induct,axiom,
    ! [Y: set_Pr958786334691620121nt_int,P2: set_Pr958786334691620121nt_int > $o] :
      ( ( member2340774599025711042nt_int @ Y
        @ ( collec5210948495886036740nt_int
          @ ^ [C3: set_Pr958786334691620121nt_int] :
            ? [X4: product_prod_int_int] :
              ( ( ratrel @ X4 @ X4 )
              & ( C3
                = ( collec213857154873943460nt_int @ ( ratrel @ X4 ) ) ) ) ) )
     => ( ! [X3: rat] : ( P2 @ ( rep_rat @ X3 ) )
       => ( P2 @ Y ) ) ) ).

% Rep_rat_induct
thf(fact_10197_one__assn__def,axiom,
    ( one_one_assn
    = ( abs_assn @ one_assn_raw ) ) ).

% one_assn_def
thf(fact_10198_natLeq__Partial__order,axiom,
    order_5251275573222108571on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Partial_order
thf(fact_10199_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_10200_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( Y
              = ( As2 != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
thf(fact_10201_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( As2 != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
thf(fact_10202_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
           => ( As2 = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
thf(fact_10203_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
               => ( As2 = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
thf(fact_10204_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
               => ( As2 != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
thf(fact_10205_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 )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As2 ) )
             => ( ( Y
                  = ( As2 = bot_bot_set_nat ) )
               => ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ) ) ).

% one_assn_raw.pelims(1)
thf(fact_10206_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).

% integer_of_char_code

% Helper facts (59)
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__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__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_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: int > product_prod_int_int,Y: int > product_prod_int_int] :
      ( ( if_int2409958687428134794nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: int > product_prod_int_int,Y: int > product_prod_int_int] :
      ( ( if_int2409958687428134794nt_int @ $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__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_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option4065278094766928714it_nat,Y: option4065278094766928714it_nat] :
      ( ( if_opt7386294288321364996it_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option4065278094766928714it_nat,Y: option4065278094766928714it_nat] :
      ( ( if_opt7386294288321364996it_nat @ $true @ X @ Y )
      = X ) ).

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 ) ).

thf(help_If_2_1_If_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J_T,axiom,
    ! [X: produc7822875418678951345atural > produc5835291356934675326atural,Y: produc7822875418678951345atural > produc5835291356934675326atural] :
      ( ( if_Pro2760142802149639654atural @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_J_T,axiom,
    ! [X: produc7822875418678951345atural > produc5835291356934675326atural,Y: produc7822875418678951345atural > produc5835291356934675326atural] :
      ( ( if_Pro2760142802149639654atural @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__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: produc6653097349344004940it_nat > produc3260487557148687353it_nat,Y: produc6653097349344004940it_nat > produc3260487557148687353it_nat] :
      ( ( if_Pro3578090444564244254it_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__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: produc6653097349344004940it_nat > produc3260487557148687353it_nat,Y: produc6653097349344004940it_nat > produc3260487557148687353it_nat] :
      ( ( if_Pro3578090444564244254it_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc6653097349344004940it_nat > produc7388388658123137530it_nat,Y: produc6653097349344004940it_nat > produc7388388658123137530it_nat] :
      ( ( if_Pro4080778131203054751it_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: produc6653097349344004940it_nat > produc7388388658123137530it_nat,Y: produc6653097349344004940it_nat > produc7388388658123137530it_nat] :
      ( ( if_Pro4080778131203054751it_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: produc3658429121746597890et_nat > produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > produc3925858234332021118et_nat] :
      ( ( if_Pro6440128116348121305et_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: produc3658429121746597890et_nat > produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > produc3925858234332021118et_nat] :
      ( ( if_Pro6440128116348121305et_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001_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_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001_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_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J_T,axiom,
    ! [X: produc3925858234332021118et_nat > produc2732055786443039994et_nat,Y: produc3925858234332021118et_nat > produc2732055786443039994et_nat] :
      ( ( if_Pro5796790085669187537et_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_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_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J_T,axiom,
    ! [X: produc3925858234332021118et_nat > produc2732055786443039994et_nat,Y: produc3925858234332021118et_nat > produc2732055786443039994et_nat] :
      ( ( if_Pro5796790085669187537et_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    rep_assn @ ( q @ rg ) @ ( produc7507926704131184380et_nat @ h @ ( hoare_new_addrs @ h3 @ as @ h ) ) ).

%------------------------------------------------------------------------------