%------------------------------------------------------------------------------
% File     : ITP212^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Assertions 00821_023885
% 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    : 0024_Assertions_00821_023885 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 8189 (4065 unt;1148 typ;   0 def)
%            Number of atoms       : 17733 (8364 equ;   0 cnn)
%            Maximal formula atoms :   48 (   2 avg)
%            Number of connectives : 61771 (2026   ~; 277   |; 978   &;52149   @)
%                                         (   0 <=>;6341  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   25 (   5 avg)
%            Number of types       :  105 ( 104 usr)
%            Number of type conns  : 3567 (3567   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1047 (1044 usr;  88 con; 0-6 aty)
%            Number of variables   : 15926 (1145   ^;14550   !; 231   ?;15926   :)
% 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 15:05:32.761
%------------------------------------------------------------------------------
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
    produc8551481072490612790e_term: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    set_Pr7564037697178849041atural: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    set_Pr4811707699266497531nteger: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
    option6357759511663192854e_term: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
    produc6653097349344004940it_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
    set_Pr5094982260447487303t_unit: $tType ).

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

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

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

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

thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    filter1242075044329608583at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_J,type,
    set_he8644174446132254637t_unit: $tType ).

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

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

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

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

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
    set_set_Product_unit: $tType ).

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

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

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

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

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

thf(ty_n_t__Predicate__Opred_It__Product____Type__Ounit_J,type,
    pred_Product_unit: $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__Onatural_J,type,
    set_Code_natural: $tType ).

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

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

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

thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    set_set_o: $tType ).

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

thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $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__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__Typerep__Otyperep,type,
    typerep: $tType ).

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

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

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

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

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

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

% Explicit typings (1044)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

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

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

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

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

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

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

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

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

thf(sy_c_Assertions_Ois__pure__assn,type,
    is_pure_assn: assn > $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_Oproper,type,
    proper: ( produc3658429121746597890et_nat > $o ) > $o ).

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

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

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

thf(sy_c_Assertions_OrelH,type,
    relH: set_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ).

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

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

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

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

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

thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
    bNF_Cardinal_cone: set_Pr5094982260447487303t_unit ).

thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set_Product_prod_o_o ).

thf(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
    bNF_Cardinal_czero_o: set_Product_prod_o_o ).

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

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Product____Type__Ounit,type,
    bNF_Ca7083678892733797993t_unit: set_Product_unit > set_Pr5094982260447487303t_unit ).

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

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_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_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).

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

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001_Eo_001_Eo,type,
    bNF_We2654380646378065620so_o_o: set_Pr1932065953672099015od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We5258908940166488438at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordLess_001_Eo_001t__Nat__Onat,type,
    bNF_We8182288985678559134_o_nat: set_Pr457366540195662369at_nat ).

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

thf(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Ointeger,type,
    gbinom8545251970709558553nteger: code_integer > nat > code_integer ).

thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: int > nat > int ).

thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: nat > nat > nat ).

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,
    bit_and_not_num: num > num > option_num ).

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_Oor__not__num__neg,type,
    bit_or_not_num_neg: num > num > num ).

thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Onatural,type,
    bit_se2773287842338716102atural: code_natural > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
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thf(sy_c_Int_Ointrel,type,
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thf(sy_c_Int_Opcr__int,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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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__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__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
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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,
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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_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_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osemilattice_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices_Osemilattice_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osemilattice_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osemilattice_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_Lattices_Osemilattice__neutr_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Int__Oint,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_It__Product____Type__Ounit_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__Nat__Onat,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_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
    sup_sup_o: $o > $o > $o ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Assertions__Oassn_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Code____Numeral__Onatural_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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    sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

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

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

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
    lattic8718645017227715691in_int: set_int > int ).

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

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

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Assertions__Oassn,type,
    lattic7708394068118247271o_assn: ( $o > assn ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Int__Oint,type,
    lattic2773365557947402859_o_int: ( $o > int ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Nat__Onat,type,
    lattic2775856028456453135_o_nat: ( $o > nat ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Assertions__Oassn,type,
    lattic3162799611691135855t_assn: ( int > assn ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Int__Oint,type,
    lattic8443796201974363763nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Nat__Onat,type,
    lattic8446286672483414039nt_nat: ( int > nat ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Assertions__Oassn,type,
    lattic2187264146484958483t_assn: ( nat > assn ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001_Eo,type,
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thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Assertions__Oassn,type,
    lattic47131356835913163n_assn: set_assn > assn ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Int__Oint,type,
    lattic5235898064620869839in_int: set_int > int ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
    lattic5238388535129920115in_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic3014633134055518761et_nat: set_set_nat > set_nat ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_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_Lattices__Big_Osemilattice__sup__class_OSup__fin_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001_Eo,type,
    lattic1508158080041050831_fin_o: set_o > $o ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Int__Oint,type,
    lattic1091506334969745077in_int: set_int > int ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
    lattic1093996805478795353in_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic3835124923745554447et_nat: set_set_nat > set_nat ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_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_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: int > list_int > list_int ).

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

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

thf(sy_c_List_Oupto__rel,type,
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thf(sy_c_Misc_Opairself_001t__Int__Oint_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
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thf(sy_c_Misc_Opairself__rel_001t__Int__Oint_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
    pairse6848479906795794847e_term: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o ).

thf(sy_c_Multiset_Oadd__mset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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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,
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thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Onatural,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: nat > int ).

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

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

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

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

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

thf(sy_c_Nat__Bijection_Oprod__decode,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_Osum__decode,type,
    nat_sum_decode: nat > sum_sum_nat_nat ).

thf(sy_c_Num_OBitM,type,
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thf(sy_c_Num_Oinc,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
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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_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
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thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: num > num > int ).

thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Rat__Orat,type,
    neg_numeral_sub_rat: num > num > rat ).

thf(sy_c_Num_Onum_OBit0,type,
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thf(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

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__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,
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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,
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thf(sy_c_Num_Oring__1__class_Oiszero_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oring__1__class_Oiszero_001t__Int__Oint,type,
    ring_1_iszero_int: int > $o ).

thf(sy_c_Num_Oring__1__class_Oiszero_001t__Rat__Orat,type,
    ring_1_iszero_rat: rat > $o ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

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__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_v_A,type,
    a: assn ).

thf(sy_v_B,type,
    b: assn ).

thf(sy_v_C,type,
    c: assn ).

% Relevant facts (7011)
thf(fact_0_ent__iffI,axiom,
    ! [A: assn,B: assn] :
      ( ( entails @ A @ B )
     => ( ( entails @ B @ A )
       => ( A = B ) ) ) ).

% ent_iffI
thf(fact_1_ent__refl,axiom,
    ! [P: assn] : ( entails @ P @ P ) ).

% ent_refl
thf(fact_2_ent__disjE,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ( entails @ A @ C )
     => ( ( entails @ B @ C )
       => ( entails @ ( sup_sup_assn @ A @ B ) @ C ) ) ) ).

% ent_disjE
thf(fact_3_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ Q )
     => ( ( entails @ Q @ R )
       => ( entails @ P @ R ) ) ) ).

% ent_trans
thf(fact_4_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ P @ R ) ) ).

% ent_disjI1
thf(fact_5_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ Q @ R ) ) ).

% ent_disjI2
thf(fact_6_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q2: assn] :
      ( ( entails @ P @ P2 )
     => ( ( entails @ Q @ Q2 )
       => ( entails @ ( times_times_assn @ P @ Q ) @ ( times_times_assn @ P2 @ Q2 ) ) ) ) ).

% ent_star_mono
thf(fact_7_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P3: assn,Q3: assn] : ( times_times_assn @ Q3 @ P3 ) ) ) ).

% assn_times_comm
thf(fact_8_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P @ Q ) @ R )
      = ( times_times_assn @ P @ ( times_times_assn @ Q @ R ) ) ) ).

% assn_times_assoc
thf(fact_9_ent__disjI1__direct,axiom,
    ! [A: assn,B: assn] : ( entails @ A @ ( sup_sup_assn @ A @ B ) ) ).

% ent_disjI1_direct
thf(fact_10_ent__disjI2__direct,axiom,
    ! [B: assn,A: assn] : ( entails @ B @ ( sup_sup_assn @ A @ B ) ) ).

% ent_disjI2_direct
thf(fact_11_sup_Oidem,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_12_sup_Oidem,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_13_sup_Oidem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_14_sup__idem,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ X )
      = X ) ).

% sup_idem
thf(fact_15_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_16_sup__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_17_sup_Oleft__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( sup_sup_assn @ A2 @ ( sup_sup_assn @ A2 @ B2 ) )
      = ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_18_sup_Oleft__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_19_sup_Oleft__idem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
      = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_20_sup__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% sup_left_idem
thf(fact_21_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_22_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_23_sup_Oright__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A2 @ B2 ) @ B2 )
      = ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_24_sup_Oright__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_25_sup_Oright__idem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ B2 )
      = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_26_ent__mp,axiom,
    ! [P: assn,Q: assn] : ( entails @ ( times_times_assn @ P @ ( wand_assn @ P @ Q ) ) @ Q ) ).

% ent_mp
thf(fact_27_ent__wandI,axiom,
    ! [Q: assn,P: assn,R: assn] :
      ( ( entails @ ( times_times_assn @ Q @ P ) @ R )
     => ( entails @ P @ ( wand_assn @ Q @ R ) ) ) ).

% ent_wandI
thf(fact_28_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ Q )
      = ( B2
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_29_is__pure__assn__starI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( is_pure_assn @ A2 )
     => ( ( is_pure_assn @ B2 )
       => ( is_pure_assn @ ( times_times_assn @ A2 @ B2 ) ) ) ) ).

% is_pure_assn_starI
thf(fact_30_inf__sup__aci_I8_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_31_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_32_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_33_inf__sup__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_34_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_35_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
      = ( sup_su5525570899277871387at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_36_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( ( pure_assn @ P )
        = ( pure_assn @ Q ) )
      = ( P = Q ) ) ).

% pure_assn_eq_conv
thf(fact_37_merge__pure__star,axiom,
    ! [A2: $o,B2: $o] :
      ( ( times_times_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
      = ( pure_assn
        @ ( A2
          & B2 ) ) ) ).

% merge_pure_star
thf(fact_38_merge__pure__or,axiom,
    ! [A2: $o,B2: $o] :
      ( ( sup_sup_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
      = ( pure_assn
        @ ( A2
          | B2 ) ) ) ).

% merge_pure_or
thf(fact_39_is__pure__assn__pure,axiom,
    ! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).

% is_pure_assn_pure
thf(fact_40_is__pure__assnE,axiom,
    ! [A2: assn] :
      ( ( is_pure_assn @ A2 )
     => ~ ! [P4: $o] :
            ( A2
           != ( pure_assn @ P4 ) ) ) ).

% is_pure_assnE
thf(fact_41_is__pure__assn__def,axiom,
    ( is_pure_assn
    = ( ^ [A3: assn] :
        ? [P3: $o] :
          ( A3
          = ( pure_assn @ P3 ) ) ) ) ).

% is_pure_assn_def
thf(fact_42_sup__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_43_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_44_sup__left__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
      = ( sup_su5525570899277871387at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_45_sup_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( sup_sup_assn @ B2 @ ( sup_sup_assn @ A2 @ C2 ) )
      = ( sup_sup_assn @ A2 @ ( sup_sup_assn @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_46_sup_Oleft__commute,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) )
      = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_47_sup_Oleft__commute,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A2 @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).

% sup.left_commute
thf(fact_48_sup__commute,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y2: assn] : ( sup_sup_assn @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_49_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_50_sup__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_51_sup_Ocommute,axiom,
    ( sup_sup_assn
    = ( ^ [A3: assn,B3: assn] : ( sup_sup_assn @ B3 @ A3 ) ) ) ).

% sup.commute
thf(fact_52_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).

% sup.commute
thf(fact_53_sup_Ocommute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B3 @ A3 ) ) ) ).

% sup.commute
thf(fact_54_sup__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_55_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_56_sup__assoc,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
      = ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_57_sup_Oassoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A2 @ B2 ) @ C2 )
      = ( sup_sup_assn @ A2 @ ( sup_sup_assn @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_58_sup_Oassoc,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
      = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_59_sup_Oassoc,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).

% sup.assoc
thf(fact_60_inf__sup__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y2: assn] : ( sup_sup_assn @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_61_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_62_inf__sup__aci_I5_J,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_63_inf__sup__aci_I6_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_64_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_65_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
      = ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_66_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] :
      ( ( entails @ P @ ( times_times_assn @ Q @ ( pure_assn @ B2 ) ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B2 )
        & ( entails @ P @ Q ) ) ) ).

% ent_pure_post_iff
thf(fact_67_boolean__algebra__cancel_Osup2,axiom,
    ! [B: assn,K: assn,B2: assn,A2: assn] :
      ( ( B
        = ( sup_sup_assn @ K @ B2 ) )
     => ( ( sup_sup_assn @ A2 @ B )
        = ( sup_sup_assn @ K @ ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_68_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
      ( ( B
        = ( sup_sup_set_nat @ K @ B2 ) )
     => ( ( sup_sup_set_nat @ A2 @ B )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_69_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( B
        = ( sup_su5525570899277871387at_nat @ K @ B2 ) )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B )
        = ( sup_su5525570899277871387at_nat @ K @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_70_boolean__algebra__cancel_Osup1,axiom,
    ! [A: assn,K: assn,A2: assn,B2: assn] :
      ( ( A
        = ( sup_sup_assn @ K @ A2 ) )
     => ( ( sup_sup_assn @ A @ B2 )
        = ( sup_sup_assn @ K @ ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_71_boolean__algebra__cancel_Osup1,axiom,
    ! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
      ( ( A
        = ( sup_sup_set_nat @ K @ A2 ) )
     => ( ( sup_sup_set_nat @ A @ B2 )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_72_boolean__algebra__cancel_Osup1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( A
        = ( sup_su5525570899277871387at_nat @ K @ A2 ) )
     => ( ( sup_su5525570899277871387at_nat @ A @ B2 )
        = ( sup_su5525570899277871387at_nat @ K @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_73_ent__pure__pre__iff__sng,axiom,
    ! [B2: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B2 ) @ Q )
      = ( B2
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_74_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ B2 @ ( times_3573771949741848930nteger @ A2 @ C2 ) )
      = ( times_3573771949741848930nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_75_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( times_times_assn @ B2 @ ( times_times_assn @ A2 @ C2 ) )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_76_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( times_times_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_77_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C2 ) )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_78_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( times_times_int @ B2 @ ( times_times_int @ A2 @ C2 ) )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_79_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_3573771949741848930nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( times_3573771949741848930nteger @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_80_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A3: assn,B3: assn] : ( times_times_assn @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_81_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_82_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_83_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_84_mult_Oassoc,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
      = ( times_3573771949741848930nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_85_mult_Oassoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_86_mult_Oassoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_87_mult_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_88_mult_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_89_mem__Collect__eq,axiom,
    ! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
    ! [A2: $o,P: $o > $o] :
      ( ( member_o @ A2 @ ( collect_o @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
    ! [A2: int,P: int > $o] :
      ( ( member_int @ A2 @ ( collect_int @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( member6576561426505652726_nat_o @ A2 @ ( collec939566748876313656_nat_o @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_94_Collect__mem__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
    ! [A: set_o] :
      ( ( collect_o
        @ ^ [X2: $o] : ( member_o @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
    ! [A: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( collec939566748876313656_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_99_Collect__cong,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec939566748876313656_nat_o @ P )
        = ( collec939566748876313656_nat_o @ Q ) ) ) ).

% Collect_cong
thf(fact_100_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_101_mult_Oright__commute,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ B2 ) ) ).

% mult.right_commute
thf(fact_102_mult_Oright__commute,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
      = ( times_times_assn @ ( times_times_assn @ A2 @ C2 ) @ B2 ) ) ).

% mult.right_commute
thf(fact_103_mult_Oright__commute,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 )
      = ( times_times_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ).

% mult.right_commute
thf(fact_104_mult_Oright__commute,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
      = ( times_times_nat @ ( times_times_nat @ A2 @ C2 ) @ B2 ) ) ).

% mult.right_commute
thf(fact_105_mult_Oright__commute,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
      = ( times_times_int @ ( times_times_int @ A2 @ C2 ) @ B2 ) ) ).

% mult.right_commute
thf(fact_106_mult_Oright__assoc,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
      = ( times_3573771949741848930nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% mult.right_assoc
thf(fact_107_mult_Oright__assoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% mult.right_assoc
thf(fact_108_mult_Oright__assoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% mult.right_assoc
thf(fact_109_mult_Oright__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% mult.right_assoc
thf(fact_110_mult_Oright__assoc,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% mult.right_assoc
thf(fact_111_mod__pure__star__dist,axiom,
    ! [P: assn,B2: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        & B2 ) ) ).

% mod_pure_star_dist
thf(fact_112_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_113_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_114_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( rep_assn @ X )
        = ( rep_assn @ Y ) )
      = ( X = Y ) ) ).

% Rep_assn_inject
thf(fact_115_mult_Oright__neutral,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
      = A2 ) ).

% mult.right_neutral
thf(fact_116_mult_Oright__neutral,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% mult.right_neutral
thf(fact_117_mult_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.right_neutral
thf(fact_118_mult_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.right_neutral
thf(fact_119_mult_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.right_neutral
thf(fact_120_mult__1,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
      = A2 ) ).

% mult_1
thf(fact_121_mult__1,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% mult_1
thf(fact_122_mult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_123_mult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_124_mult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% mult_1
thf(fact_125_mod__or__dist,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        | ( rep_assn @ Q @ H2 ) ) ) ).

% mod_or_dist
thf(fact_126_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: $o] :
      ( ( entails @ P @ ( pure_assn @ B2 ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B2 )
        & ( entails @ P @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_127_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
      = ( X = one_one_Code_integer ) ) ).

% one_reorient
thf(fact_128_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
      = ( X = one_one_assn ) ) ).

% one_reorient
thf(fact_129_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_130_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_131_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_132_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_133_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_134_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_135_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_136_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_137_mult_Ocomm__neutral,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
      = A2 ) ).

% mult.comm_neutral
thf(fact_138_mult_Ocomm__neutral,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% mult.comm_neutral
thf(fact_139_mult_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_140_mult_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_141_mult_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.comm_neutral
thf(fact_142_ent__fwd,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat,Q: assn] :
      ( ( rep_assn @ P @ H2 )
     => ( ( entails @ P @ Q )
       => ( rep_assn @ Q @ H2 ) ) ) ).

% ent_fwd
thf(fact_143_entailsD,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( ( entails @ P @ Q )
     => ( ( rep_assn @ P @ H2 )
       => ( rep_assn @ Q @ H2 ) ) ) ).

% entailsD
thf(fact_144_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => ( rep_assn @ Q @ H3 ) )
     => ( entails @ P @ Q ) ) ).

% entailsI
thf(fact_145_entails__def,axiom,
    ( entails
    = ( ^ [P3: assn,Q3: assn] :
        ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P3 @ H )
         => ( rep_assn @ Q3 @ H ) ) ) ) ).

% entails_def
thf(fact_146_mod__starD,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A @ H1 )
          & ( rep_assn @ B @ H22 ) ) ) ).

% mod_starD
thf(fact_147_mod__starE,axiom,
    ! [A2: assn,B2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A2 @ B2 ) @ H2 )
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A2 @ X_1 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B2 @ H_2 ) ) ) ).

% mod_starE
thf(fact_148_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_149_is__pure__assn__basic__simps_I2_J,axiom,
    is_pure_assn @ one_one_assn ).

% is_pure_assn_basic_simps(2)
thf(fact_150_mult_Osafe__commute,axiom,
    ! [X: code_integer,Y: code_integer,A2: code_integer,B2: code_integer] :
      ( ( syntax380677951125133566nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ A2 )
     => ( ( times_3573771949741848930nteger @ A2 @ B2 )
        = ( times_3573771949741848930nteger @ B2 @ A2 ) ) ) ).

% mult.safe_commute
thf(fact_151_mult_Osafe__commute,axiom,
    ! [X: assn,Y: assn,A2: assn,B2: assn] :
      ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A2 )
     => ( ( times_times_assn @ A2 @ B2 )
        = ( times_times_assn @ B2 @ A2 ) ) ) ).

% mult.safe_commute
thf(fact_152_mult_Osafe__commute,axiom,
    ! [X: rat,Y: rat,A2: rat,B2: rat] :
      ( ( syntax3730441303064801268at_rat @ ( times_times_rat @ X @ Y ) @ A2 )
     => ( ( times_times_rat @ A2 @ B2 )
        = ( times_times_rat @ B2 @ A2 ) ) ) ).

% mult.safe_commute
thf(fact_153_mult_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A2: nat,B2: nat] :
      ( ( syntax4682126007086162916at_nat @ ( times_times_nat @ X @ Y ) @ A2 )
     => ( ( times_times_nat @ A2 @ B2 )
        = ( times_times_nat @ B2 @ A2 ) ) ) ).

% mult.safe_commute
thf(fact_154_mult_Osafe__commute,axiom,
    ! [X: int,Y: int,A2: int,B2: int] :
      ( ( syntax5678989248478167196nt_int @ ( times_times_int @ X @ Y ) @ A2 )
     => ( ( times_times_int @ A2 @ B2 )
        = ( times_times_int @ B2 @ A2 ) ) ) ).

% mult.safe_commute
thf(fact_155_mult_Omonoid__axioms,axiom,
    monoid_Code_integer @ times_3573771949741848930nteger @ one_one_Code_integer ).

% mult.monoid_axioms
thf(fact_156_mult_Omonoid__axioms,axiom,
    monoid_assn @ times_times_assn @ one_one_assn ).

% mult.monoid_axioms
thf(fact_157_mult_Omonoid__axioms,axiom,
    monoid_rat @ times_times_rat @ one_one_rat ).

% mult.monoid_axioms
thf(fact_158_mult_Omonoid__axioms,axiom,
    monoid_nat @ times_times_nat @ one_one_nat ).

% mult.monoid_axioms
thf(fact_159_mult_Omonoid__axioms,axiom,
    monoid_int @ times_times_int @ one_one_int ).

% mult.monoid_axioms
thf(fact_160_ent__false__iff,axiom,
    ! [P: assn] :
      ( ( entails @ P @ bot_bot_assn )
      = ( ! [H: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H ) ) ) ).

% ent_false_iff
thf(fact_161_mult_Ocomm__monoid__axioms,axiom,
    comm_m1654649897431166824nteger @ times_3573771949741848930nteger @ one_one_Code_integer ).

% mult.comm_monoid_axioms
thf(fact_162_mult_Ocomm__monoid__axioms,axiom,
    comm_monoid_assn @ times_times_assn @ one_one_assn ).

% mult.comm_monoid_axioms
thf(fact_163_mult_Ocomm__monoid__axioms,axiom,
    comm_monoid_rat @ times_times_rat @ one_one_rat ).

% mult.comm_monoid_axioms
thf(fact_164_mult_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ times_times_nat @ one_one_nat ).

% mult.comm_monoid_axioms
thf(fact_165_mult_Ocomm__monoid__axioms,axiom,
    comm_monoid_int @ times_times_int @ one_one_int ).

% mult.comm_monoid_axioms
thf(fact_166_mod__star__trueE,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 )
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H4 ) ) ).

% mod_star_trueE
thf(fact_167_mod__star__trueI,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H2 )
     => ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 ) ) ).

% mod_star_trueI
thf(fact_168_Rep__assn__inverse,axiom,
    ! [X: assn] :
      ( ( abs_assn @ ( rep_assn @ X ) )
      = X ) ).

% Rep_assn_inverse
thf(fact_169_wand__assn__def,axiom,
    ( wand_assn
    = ( ^ [P3: assn,Q3: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P3 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).

% wand_assn_def
thf(fact_170_prod_Ocomm__monoid__list__set__axioms,axiom,
    groups7165323350619936122nteger @ times_3573771949741848930nteger @ one_one_Code_integer ).

% prod.comm_monoid_list_set_axioms
thf(fact_171_prod_Ocomm__monoid__list__set__axioms,axiom,
    groups5099259868496839749t_assn @ times_times_assn @ one_one_assn ).

% prod.comm_monoid_list_set_axioms
thf(fact_172_prod_Ocomm__monoid__list__set__axioms,axiom,
    groups2612948464029442805et_rat @ times_times_rat @ one_one_rat ).

% prod.comm_monoid_list_set_axioms
thf(fact_173_prod_Ocomm__monoid__list__set__axioms,axiom,
    groups3248078524115938541et_nat @ times_times_nat @ one_one_nat ).

% prod.comm_monoid_list_set_axioms
thf(fact_174_prod_Ocomm__monoid__list__set__axioms,axiom,
    groups3245588053606888265et_int @ times_times_int @ one_one_int ).

% prod.comm_monoid_list_set_axioms
thf(fact_175_mult_Oac__operator__axioms,axiom,
    syntax414191934074104506nteger @ times_3573771949741848930nteger ).

% mult.ac_operator_axioms
thf(fact_176_mult_Oac__operator__axioms,axiom,
    syntax3673046151410588037r_assn @ times_times_assn ).

% mult.ac_operator_axioms
thf(fact_177_mult_Oac__operator__axioms,axiom,
    syntax8165409007564224565or_rat @ times_times_rat ).

% mult.ac_operator_axioms
thf(fact_178_mult_Oac__operator__axioms,axiom,
    syntax8800539067650720301or_nat @ times_times_nat ).

% mult.ac_operator_axioms
thf(fact_179_mult_Oac__operator__axioms,axiom,
    syntax8798048597141670025or_int @ times_times_int ).

% mult.ac_operator_axioms
thf(fact_180_prod__list_Ocomm__monoid__list__axioms,axiom,
    groups2567640417189887108nteger @ times_3573771949741848930nteger @ one_one_Code_integer ).

% prod_list.comm_monoid_list_axioms
thf(fact_181_prod__list_Ocomm__monoid__list__axioms,axiom,
    groups9150563453463953231t_assn @ times_times_assn @ one_one_assn ).

% prod_list.comm_monoid_list_axioms
thf(fact_182_prod__list_Ocomm__monoid__list__axioms,axiom,
    groups593308882113688063st_rat @ times_times_rat @ one_one_rat ).

% prod_list.comm_monoid_list_axioms
thf(fact_183_prod__list_Ocomm__monoid__list__axioms,axiom,
    groups1228438942200183799st_nat @ times_times_nat @ one_one_nat ).

% prod_list.comm_monoid_list_axioms
thf(fact_184_prod__list_Ocomm__monoid__list__axioms,axiom,
    groups1225948471691133523st_int @ times_times_int @ one_one_int ).

% prod_list.comm_monoid_list_axioms
thf(fact_185_sup__bot_Oright__neutral,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ A2 @ bot_bot_assn )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_186_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ bot_bo228742789529271731at_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_187_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_188_sup__bot_Oright__neutral,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_189_sup__bot_Oright__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_190_sup__bot_Oright__neutral,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_191_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_assn )
        & ( B2 = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_192_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_193_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_194_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_195_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_196_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_197_sup__bot_Oleft__neutral,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_198_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_199_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_200_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_201_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_202_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_203_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ( sup_sup_assn @ A2 @ B2 )
        = bot_bot_assn )
      = ( ( A2 = bot_bot_assn )
        & ( B2 = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_204_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = bot_bo228742789529271731at_nat )
      = ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_205_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A2 @ B2 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A2 = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_206_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( sup_sup_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_207_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_208_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( sup_sup_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_209_sup__eq__bot__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( sup_sup_assn @ X @ Y )
        = bot_bot_assn )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
thf(fact_210_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_211_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_212_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_213_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_214_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_215_bot__eq__sup__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ X @ Y ) )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
thf(fact_216_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_217_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_218_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_219_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_220_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_221_sup__bot__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% sup_bot_right
thf(fact_222_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
      = X ) ).

% sup_bot_right
thf(fact_223_sup__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% sup_bot_right
thf(fact_224_sup__bot__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% sup_bot_right
thf(fact_225_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_226_sup__bot__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% sup_bot_right
thf(fact_227_sup__bot__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% sup_bot_left
thf(fact_228_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_229_sup__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_230_sup__bot__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X )
      = X ) ).

% sup_bot_left
thf(fact_231_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_232_sup__bot__left,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X )
      = X ) ).

% sup_bot_left
thf(fact_233_sup__top__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ top_to6833984726390702231at_nat )
      = top_to6833984726390702231at_nat ) ).

% sup_top_right
thf(fact_234_sup__top__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ top_top_assn )
      = top_top_assn ) ).

% sup_top_right
thf(fact_235_sup__top__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ top_top_set_o )
      = top_top_set_o ) ).

% sup_top_right
thf(fact_236_sup__top__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ top_top_set_nat )
      = top_top_set_nat ) ).

% sup_top_right
thf(fact_237_sup__top__right,axiom,
    ! [X: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
      = top_to1996260823553986621t_unit ) ).

% sup_top_right
thf(fact_238_sup__top__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ top_to6833984726390702231at_nat @ X )
      = top_to6833984726390702231at_nat ) ).

% sup_top_left
thf(fact_239_sup__top__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ top_top_assn @ X )
      = top_top_assn ) ).

% sup_top_left
thf(fact_240_sup__top__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ top_top_set_o @ X )
      = top_top_set_o ) ).

% sup_top_left
thf(fact_241_sup__top__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X )
      = top_top_set_nat ) ).

% sup_top_left
thf(fact_242_sup__top__left,axiom,
    ! [X: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
      = top_to1996260823553986621t_unit ) ).

% sup_top_left
thf(fact_243_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ top_to6833984726390702231at_nat )
      = top_to6833984726390702231at_nat ) ).

% boolean_algebra.disj_one_right
thf(fact_244_boolean__algebra_Odisj__one__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ top_top_assn )
      = top_top_assn ) ).

% boolean_algebra.disj_one_right
thf(fact_245_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ top_top_set_o )
      = top_top_set_o ) ).

% boolean_algebra.disj_one_right
thf(fact_246_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ top_top_set_nat )
      = top_top_set_nat ) ).

% boolean_algebra.disj_one_right
thf(fact_247_boolean__algebra_Odisj__one__right,axiom,
    ! [X: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
      = top_to1996260823553986621t_unit ) ).

% boolean_algebra.disj_one_right
thf(fact_248_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ top_to6833984726390702231at_nat @ X )
      = top_to6833984726390702231at_nat ) ).

% boolean_algebra.disj_one_left
thf(fact_249_boolean__algebra_Odisj__one__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ top_top_assn @ X )
      = top_top_assn ) ).

% boolean_algebra.disj_one_left
thf(fact_250_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ top_top_set_o @ X )
      = top_top_set_o ) ).

% boolean_algebra.disj_one_left
thf(fact_251_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X )
      = top_top_set_nat ) ).

% boolean_algebra.disj_one_left
thf(fact_252_boolean__algebra_Odisj__one__left,axiom,
    ! [X: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
      = top_to1996260823553986621t_unit ) ).

% boolean_algebra.disj_one_left
thf(fact_253_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_254_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_255_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_256_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = bot_bot_assn )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_257_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_258_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_259_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_260_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_261_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_assn @ sup_sup_assn @ bot_bot_assn ).

% sup_bot.comm_monoid_axioms
thf(fact_262_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_m2457627138559323900at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat ).

% sup_bot.comm_monoid_axioms
thf(fact_263_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_m5709050211238513046at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat ).

% sup_bot.comm_monoid_axioms
thf(fact_264_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_o @ sup_sup_set_o @ bot_bot_set_o ).

% sup_bot.comm_monoid_axioms
thf(fact_265_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).

% sup_bot.comm_monoid_axioms
thf(fact_266_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_int @ sup_sup_set_int @ bot_bot_set_int ).

% sup_bot.comm_monoid_axioms
thf(fact_267_sup__bot_Omonoid__axioms,axiom,
    monoid_assn @ sup_sup_assn @ bot_bot_assn ).

% sup_bot.monoid_axioms
thf(fact_268_sup__bot_Omonoid__axioms,axiom,
    monoid534952638934301128at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat ).

% sup_bot.monoid_axioms
thf(fact_269_sup__bot_Omonoid__axioms,axiom,
    monoid1519094961268053602at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat ).

% sup_bot.monoid_axioms
thf(fact_270_sup__bot_Omonoid__axioms,axiom,
    monoid_set_o @ sup_sup_set_o @ bot_bot_set_o ).

% sup_bot.monoid_axioms
thf(fact_271_sup__bot_Omonoid__axioms,axiom,
    monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).

% sup_bot.monoid_axioms
thf(fact_272_sup__bot_Omonoid__axioms,axiom,
    monoid_set_int @ sup_sup_set_int @ bot_bot_set_int ).

% sup_bot.monoid_axioms
thf(fact_273_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_274_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_275_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_276_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_277_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_278_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_279_mod__false,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ~ ( rep_assn @ bot_bot_assn @ H2 ) ).

% mod_false
thf(fact_280_ent__true,axiom,
    ! [P: assn] : ( entails @ P @ top_top_assn ) ).

% ent_true
thf(fact_281_ent__false,axiom,
    ! [P: assn] : ( entails @ bot_bot_assn @ P ) ).

% ent_false
thf(fact_282_is__pure__assn__basic__simps_I1_J,axiom,
    is_pure_assn @ bot_bot_assn ).

% is_pure_assn_basic_simps(1)
thf(fact_283_Abs__assn__eqI_I2_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( P @ H3 )
          = ( rep_assn @ Pr @ H3 ) )
     => ( Pr
        = ( abs_assn @ P ) ) ) ).

% Abs_assn_eqI(2)
thf(fact_284_Abs__assn__eqI_I1_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( P @ H3 )
          = ( rep_assn @ Pr @ H3 ) )
     => ( ( abs_assn @ P )
        = Pr ) ) ).

% Abs_assn_eqI(1)
thf(fact_285_times__assn__def,axiom,
    ( times_times_assn
    = ( ^ [P3: assn,Q3: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P3 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).

% times_assn_def
thf(fact_286_one__assn__def,axiom,
    ( one_one_assn
    = ( abs_assn @ one_assn_raw ) ) ).

% one_assn_def
thf(fact_287_pure__assn__def,axiom,
    ( pure_assn
    = ( ^ [B3: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B3 ) ) ) ) ).

% pure_assn_def
thf(fact_288_mod__true,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ top_top_assn @ H2 )
      = ( in_range @ H2 ) ) ).

% mod_true
thf(fact_289_prod__list_Omonoid__list__axioms,axiom,
    groups5100778238730182072nteger @ times_3573771949741848930nteger @ one_one_Code_integer ).

% prod_list.monoid_list_axioms
thf(fact_290_prod__list_Omonoid__list__axioms,axiom,
    groups2340186759238748291t_assn @ times_times_assn @ one_one_assn ).

% prod_list.monoid_list_axioms
thf(fact_291_prod__list_Omonoid__list__axioms,axiom,
    groups7826448149055759155st_rat @ times_times_rat @ one_one_rat ).

% prod_list.monoid_list_axioms
thf(fact_292_prod__list_Omonoid__list__axioms,axiom,
    groups8461578209142254891st_nat @ times_times_nat @ one_one_nat ).

% prod_list.monoid_list_axioms
thf(fact_293_prod__list_Omonoid__list__axioms,axiom,
    groups8459087738633204615st_int @ times_times_int @ one_one_int ).

% prod_list.monoid_list_axioms
thf(fact_294_models__in__range,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H2 )
     => ( in_range @ H2 ) ) ).

% models_in_range
thf(fact_295_top__assn__def,axiom,
    ( top_top_assn
    = ( abs_assn @ in_range ) ) ).

% top_assn_def
thf(fact_296_pure__assn__proper,axiom,
    ! [B2: $o] : ( proper @ ( pure_a825153325127701367it_nat @ B2 ) ) ).

% pure_assn_proper
thf(fact_297_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila5388846958306562758r_assn @ sup_sup_assn @ bot_bot_assn ).

% sup_bot.semilattice_neutr_axioms
thf(fact_298_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila3675781407871293737at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat ).

% sup_bot.semilattice_neutr_axioms
thf(fact_299_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila858173770281612099at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat ).

% sup_bot.semilattice_neutr_axioms
thf(fact_300_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila1065760912024005978_set_o @ sup_sup_set_o @ bot_bot_set_o ).

% sup_bot.semilattice_neutr_axioms
thf(fact_301_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila1241773964035338532et_nat @ sup_sup_set_nat @ bot_bot_set_nat ).

% sup_bot.semilattice_neutr_axioms
thf(fact_302_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila6287294981380917632et_int @ sup_sup_set_int @ bot_bot_set_int ).

% sup_bot.semilattice_neutr_axioms
thf(fact_303_mod__h__bot__iff_I7_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        | ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(7)
thf(fact_304_mod__h__bot__iff_I1_J,axiom,
    ! [B2: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = B2 ) ).

% mod_h_bot_iff(1)
thf(fact_305_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_306_in__range__dist__union,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat,As2: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As @ As2 ) ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
        & ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ).

% in_range_dist_union
thf(fact_307_bool__assn__proper_I1_J,axiom,
    proper @ in_range ).

% bool_assn_proper(1)
thf(fact_308_wand__proper,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] : ( proper @ ( wand_raw @ P @ Q ) ) ).

% wand_proper
thf(fact_309_one__assn__proper,axiom,
    proper @ one_assn_raw ).

% one_assn_proper
thf(fact_310_times__assn__proper,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper @ ( times_assn_raw @ P @ Q ) ) ) ) ).

% times_assn_proper
thf(fact_311_in__range__empty,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% in_range_empty
thf(fact_312_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ).

% one_assn_raw.cases
thf(fact_313_properD1,axiom,
    ! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( proper @ P )
     => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).

% properD1
thf(fact_314_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H5 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_315_Rep__assn,axiom,
    ! [X: assn] : ( member6576561426505652726_nat_o @ ( rep_assn @ X ) @ ( collec939566748876313656_nat_o @ proper ) ) ).

% Rep_assn
thf(fact_316_Rep__assn__cases,axiom,
    ! [Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ~ ! [X3: assn] :
            ( Y
           != ( rep_assn @ X3 ) ) ) ).

% Rep_assn_cases
thf(fact_317_Rep__assn__induct,axiom,
    ! [Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ! [X3: assn] : ( P @ ( rep_assn @ X3 ) )
       => ( P @ Y ) ) ) ).

% Rep_assn_induct
thf(fact_318_mult_Osemigroup__axioms,axiom,
    semigr7036097719740861967nteger @ times_3573771949741848930nteger ).

% mult.semigroup_axioms
thf(fact_319_mult_Osemigroup__axioms,axiom,
    semigroup_assn @ times_times_assn ).

% mult.semigroup_axioms
thf(fact_320_mult_Osemigroup__axioms,axiom,
    semigroup_rat @ times_times_rat ).

% mult.semigroup_axioms
thf(fact_321_mult_Osemigroup__axioms,axiom,
    semigroup_nat @ times_times_nat ).

% mult.semigroup_axioms
thf(fact_322_mult_Osemigroup__axioms,axiom,
    semigroup_int @ times_times_int ).

% mult.semigroup_axioms
thf(fact_323_sup_Osemigroup__axioms,axiom,
    semigroup_assn @ sup_sup_assn ).

% sup.semigroup_axioms
thf(fact_324_sup_Osemigroup__axioms,axiom,
    semigroup_set_nat @ sup_sup_set_nat ).

% sup.semigroup_axioms
thf(fact_325_sup_Osemigroup__axioms,axiom,
    semigr6257743987242979861at_nat @ sup_su5525570899277871387at_nat ).

% sup.semigroup_axioms
thf(fact_326_pure__assn__raw_Osimps,axiom,
    ! [B2: $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( pure_a825153325127701367it_nat @ B2 @ ( produc7507926704131184380et_nat @ H2 @ As ) )
      = ( ( As = bot_bot_set_nat )
        & B2 ) ) ).

% pure_assn_raw.simps
thf(fact_327_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
        = Y )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( Y
              = ( ~ ( ( As3 = bot_bot_set_nat )
                    & X ) ) ) ) ) ).

% pure_assn_raw.elims(1)
thf(fact_328_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ~ ( ( As3 = bot_bot_set_nat )
                & X ) ) ) ).

% pure_assn_raw.elims(2)
thf(fact_329_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( ( As3 = bot_bot_set_nat )
              & X ) ) ) ).

% pure_assn_raw.elims(3)
thf(fact_330_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y3: produc3658429121746597890et_nat > $o] :
          ( ( X
            = ( abs_assn @ Y3 ) )
         => ~ ( member6576561426505652726_nat_o @ Y3 @ ( collec939566748876313656_nat_o @ proper ) ) ) ).

% Abs_assn_cases
thf(fact_331_Abs__assn__induct,axiom,
    ! [P: assn > $o,X: assn] :
      ( ! [Y3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ Y3 @ ( collec939566748876313656_nat_o @ proper ) )
         => ( P @ ( abs_assn @ Y3 ) ) )
     => ( P @ X ) ) ).

% Abs_assn_induct
thf(fact_332_Abs__assn__inject,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ X @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
       => ( ( ( abs_assn @ X )
            = ( abs_assn @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_assn_inject
thf(fact_333_one__assn__raw_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H2 @ As ) )
      = ( As = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
thf(fact_334_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( Y
              = ( As3 != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
thf(fact_335_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( As3 != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
thf(fact_336_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( As3 = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
thf(fact_337_semilattice__neutr_Oaxioms_I2_J,axiom,
    ! [F: nat > nat > nat,Z: nat] :
      ( ( semila9081495762789891438tr_nat @ F @ Z )
     => ( comm_monoid_nat @ F @ Z ) ) ).

% semilattice_neutr.axioms(2)
thf(fact_338_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_339_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_340_Abs__assn__inverse,axiom,
    ! [Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ( rep_assn @ ( abs_assn @ Y ) )
        = Y ) ) ).

% Abs_assn_inverse
thf(fact_341_empty__iff,axiom,
    ! [C2: produc3658429121746597890et_nat > $o] :
      ~ ( member6576561426505652726_nat_o @ C2 @ bot_bo7824918357723582233_nat_o ) ).

% empty_iff
thf(fact_342_empty__iff,axiom,
    ! [C2: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ).

% empty_iff
thf(fact_343_empty__iff,axiom,
    ! [C2: $o] :
      ~ ( member_o @ C2 @ bot_bot_set_o ) ).

% empty_iff
thf(fact_344_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_345_empty__iff,axiom,
    ! [C2: int] :
      ~ ( member_int @ C2 @ bot_bot_set_int ) ).

% empty_iff
thf(fact_346_all__not__in__conv,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( ! [X2: produc3658429121746597890et_nat > $o] :
            ~ ( member6576561426505652726_nat_o @ X2 @ A ) )
      = ( A = bot_bo7824918357723582233_nat_o ) ) ).

% all_not_in_conv
thf(fact_347_all__not__in__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ! [X2: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X2 @ A ) )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
thf(fact_348_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ( ! [X2: $o] :
            ~ ( member_o @ X2 @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_349_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X2: nat] :
            ~ ( member_nat @ X2 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_350_all__not__in__conv,axiom,
    ! [A: set_int] :
      ( ( ! [X2: int] :
            ~ ( member_int @ X2 @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_351_Collect__empty__eq,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ( collec939566748876313656_nat_o @ P )
        = bot_bo7824918357723582233_nat_o )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_352_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: product_prod_nat_nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_353_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_354_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_355_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_356_empty__Collect__eq,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( collec939566748876313656_nat_o @ P ) )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_357_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X2: product_prod_nat_nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_358_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X2: $o] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_359_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_360_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X2: int] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_361_Un__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A @ B )
        = bot_bo228742789529271731at_nat )
      = ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
thf(fact_362_Un__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
thf(fact_363_Un__empty,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_364_Un__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_365_Un__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% Un_empty
thf(fact_366_prod_Oinject,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X1 @ X22 )
        = ( produc8603105652947943368nteger @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_367_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ X1 @ X22 )
        = ( produc5001842942810119800et_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_368_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ X1 @ X22 )
        = ( produc2245416461498447860et_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_369_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y22: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_370_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y22: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_371_old_Oprod_Oinject,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A2 @ B2 )
        = ( produc8603105652947943368nteger @ A4 @ B4 ) )
      = ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
thf(fact_372_old_Oprod_Oinject,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A2 @ B2 )
        = ( produc5001842942810119800et_nat @ A4 @ B4 ) )
      = ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
thf(fact_373_old_Oprod_Oinject,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A2 @ B2 )
        = ( produc2245416461498447860et_nat @ A4 @ B4 ) )
      = ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
thf(fact_374_old_Oprod_Oinject,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A2 @ B2 )
        = ( produc5700946648718959541nt_int @ A4 @ B4 ) )
      = ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
thf(fact_375_old_Oprod_Oinject,axiom,
    ! [A2: int > option6357759511663192854e_term,B2: product_prod_int_int,A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A2 @ B2 )
        = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
      = ( ( A2 = A4 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
thf(fact_376_mod__star__conv,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
      = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
            ( ( H2
              = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As22 ) ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( rep_assn @ A @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
            & ( rep_assn @ B @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).

% mod_star_conv
thf(fact_377_star__assnI,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,As2: set_nat] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
     => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
       => ( ( ( inf_inf_set_nat @ As @ As2 )
            = bot_bot_set_nat )
         => ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As @ As2 ) ) ) ) ) ) ).

% star_assnI
thf(fact_378_proper__iff,axiom,
    ! [P: produc3658429121746597890et_nat > $o,As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( proper @ P )
     => ( ( relH @ As @ H2 @ H5 )
       => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
         => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
            = ( P @ ( produc7507926704131184380et_nat @ H5 @ As ) ) ) ) ) ) ).

% proper_iff
thf(fact_379_proper__def,axiom,
    ( proper
    = ( ^ [P3: produc3658429121746597890et_nat > $o] :
        ! [H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit,As4: set_nat] :
          ( ( ( P3 @ ( produc7507926704131184380et_nat @ H @ As4 ) )
           => ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) ) )
          & ( ( ( P3 @ ( produc7507926704131184380et_nat @ H @ As4 ) )
              & ( relH @ As4 @ H @ H6 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As4 ) ) )
           => ( P3 @ ( produc7507926704131184380et_nat @ H6 @ As4 ) ) ) ) ) ) ).

% proper_def
thf(fact_380_inf_Oidem,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_381_inf_Oidem,axiom,
    ! [A2: assn] :
      ( ( inf_inf_assn @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_382_inf__idem,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_383_inf__idem,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ X )
      = X ) ).

% inf_idem
thf(fact_384_inf_Oleft__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
      = ( inf_inf_set_nat @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_385_inf_Oleft__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( inf_inf_assn @ A2 @ ( inf_inf_assn @ A2 @ B2 ) )
      = ( inf_inf_assn @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_386_inf__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_387_inf__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_left_idem
thf(fact_388_inf_Oright__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 )
      = ( inf_inf_set_nat @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_389_inf_Oright__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A2 @ B2 ) @ B2 )
      = ( inf_inf_assn @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_390_inf__right__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_391_inf__right__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Y )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_right_idem
thf(fact_392_inf__bot__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% inf_bot_left
thf(fact_393_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
thf(fact_394_inf__bot__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% inf_bot_left
thf(fact_395_inf__bot__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_396_inf__bot__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% inf_bot_left
thf(fact_397_inf__bot__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% inf_bot_right
thf(fact_398_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
thf(fact_399_inf__bot__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% inf_bot_right
thf(fact_400_inf__bot__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_401_inf__bot__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% inf_bot_right
thf(fact_402_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% boolean_algebra.conj_zero_left
thf(fact_403_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_404_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_left
thf(fact_405_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_406_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_left
thf(fact_407_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% boolean_algebra.conj_zero_right
thf(fact_408_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_409_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_right
thf(fact_410_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_411_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_right
thf(fact_412_inf__top__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ top_top_assn @ X )
      = X ) ).

% inf_top_left
thf(fact_413_inf__top__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ top_top_set_o @ X )
      = X ) ).

% inf_top_left
thf(fact_414_inf__top__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ X )
      = X ) ).

% inf_top_left
thf(fact_415_inf__top__left,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ X )
      = X ) ).

% inf_top_left
thf(fact_416_inf__top__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ top_top_assn )
      = X ) ).

% inf_top_right
thf(fact_417_inf__top__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ top_top_set_o )
      = X ) ).

% inf_top_right
thf(fact_418_inf__top__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ top_top_set_nat )
      = X ) ).

% inf_top_right
thf(fact_419_inf__top__right,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
      = X ) ).

% inf_top_right
thf(fact_420_inf__eq__top__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( inf_inf_assn @ X @ Y )
        = top_top_assn )
      = ( ( X = top_top_assn )
        & ( Y = top_top_assn ) ) ) ).

% inf_eq_top_iff
thf(fact_421_inf__eq__top__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( inf_inf_set_o @ X @ Y )
        = top_top_set_o )
      = ( ( X = top_top_set_o )
        & ( Y = top_top_set_o ) ) ) ).

% inf_eq_top_iff
thf(fact_422_inf__eq__top__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y )
        = top_top_set_nat )
      = ( ( X = top_top_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% inf_eq_top_iff
thf(fact_423_inf__eq__top__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ X @ Y )
        = top_to1996260823553986621t_unit )
      = ( ( X = top_to1996260823553986621t_unit )
        & ( Y = top_to1996260823553986621t_unit ) ) ) ).

% inf_eq_top_iff
thf(fact_424_top__eq__inf__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( top_top_assn
        = ( inf_inf_assn @ X @ Y ) )
      = ( ( X = top_top_assn )
        & ( Y = top_top_assn ) ) ) ).

% top_eq_inf_iff
thf(fact_425_top__eq__inf__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( top_top_set_o
        = ( inf_inf_set_o @ X @ Y ) )
      = ( ( X = top_top_set_o )
        & ( Y = top_top_set_o ) ) ) ).

% top_eq_inf_iff
thf(fact_426_top__eq__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ X @ Y ) )
      = ( ( X = top_top_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% top_eq_inf_iff
thf(fact_427_top__eq__inf__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( top_to1996260823553986621t_unit
        = ( inf_in4660618365625256667t_unit @ X @ Y ) )
      = ( ( X = top_to1996260823553986621t_unit )
        & ( Y = top_to1996260823553986621t_unit ) ) ) ).

% top_eq_inf_iff
thf(fact_428_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ( inf_inf_assn @ A2 @ B2 )
        = top_top_assn )
      = ( ( A2 = top_top_assn )
        & ( B2 = top_top_assn ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_429_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B2 )
        = top_top_set_o )
      = ( ( A2 = top_top_set_o )
        & ( B2 = top_top_set_o ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_430_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B2 )
        = top_top_set_nat )
      = ( ( A2 = top_top_set_nat )
        & ( B2 = top_top_set_nat ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_431_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_Product_unit,B2: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ A2 @ B2 )
        = top_to1996260823553986621t_unit )
      = ( ( A2 = top_to1996260823553986621t_unit )
        & ( B2 = top_to1996260823553986621t_unit ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_432_inf__top_Oleft__neutral,axiom,
    ! [A2: assn] :
      ( ( inf_inf_assn @ top_top_assn @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_433_inf__top_Oleft__neutral,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ top_top_set_o @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_434_inf__top_Oleft__neutral,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_435_inf__top_Oleft__neutral,axiom,
    ! [A2: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_436_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( top_top_assn
        = ( inf_inf_assn @ A2 @ B2 ) )
      = ( ( A2 = top_top_assn )
        & ( B2 = top_top_assn ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_437_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( top_top_set_o
        = ( inf_inf_set_o @ A2 @ B2 ) )
      = ( ( A2 = top_top_set_o )
        & ( B2 = top_top_set_o ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_438_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ A2 @ B2 ) )
      = ( ( A2 = top_top_set_nat )
        & ( B2 = top_top_set_nat ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_439_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_Product_unit,B2: set_Product_unit] :
      ( ( top_to1996260823553986621t_unit
        = ( inf_in4660618365625256667t_unit @ A2 @ B2 ) )
      = ( ( A2 = top_to1996260823553986621t_unit )
        & ( B2 = top_to1996260823553986621t_unit ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_440_inf__top_Oright__neutral,axiom,
    ! [A2: assn] :
      ( ( inf_inf_assn @ A2 @ top_top_assn )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_441_inf__top_Oright__neutral,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ A2 @ top_top_set_o )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_442_inf__top_Oright__neutral,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_443_inf__top_Oright__neutral,axiom,
    ! [A2: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ A2 @ top_to1996260823553986621t_unit )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_444_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_445_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_446_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_447_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_448_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_449_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_450_relH__dist__union,axiom,
    ! [As: set_nat,As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ ( sup_sup_set_nat @ As @ As2 ) @ H2 @ H5 )
      = ( ( relH @ As @ H2 @ H5 )
        & ( relH @ As2 @ H2 @ H5 ) ) ) ).

% relH_dist_union
thf(fact_451_inf__sup__aci_I4_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_452_inf__sup__aci_I4_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_453_inf__sup__aci_I3_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_454_inf__sup__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_455_inf__sup__aci_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_456_inf__sup__aci_I2_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_457_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_458_inf__sup__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X2: assn,Y2: assn] : ( inf_inf_assn @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_459_inf_Oassoc,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
      = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).

% inf.assoc
thf(fact_460_inf_Oassoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C2 )
      = ( inf_inf_assn @ A2 @ ( inf_inf_assn @ B2 @ C2 ) ) ) ).

% inf.assoc
thf(fact_461_inf__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_462_inf__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_463_inf_Ocommute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).

% inf.commute
thf(fact_464_inf_Ocommute,axiom,
    ( inf_inf_assn
    = ( ^ [A3: assn,B3: assn] : ( inf_inf_assn @ B3 @ A3 ) ) ) ).

% inf.commute
thf(fact_465_inf__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X2 ) ) ) ).

% inf_commute
thf(fact_466_inf__commute,axiom,
    ( inf_inf_assn
    = ( ^ [X2: assn,Y2: assn] : ( inf_inf_assn @ Y2 @ X2 ) ) ) ).

% inf_commute
thf(fact_467_inf_Oleft__commute,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C2 ) )
      = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).

% inf.left_commute
thf(fact_468_inf_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( inf_inf_assn @ B2 @ ( inf_inf_assn @ A2 @ C2 ) )
      = ( inf_inf_assn @ A2 @ ( inf_inf_assn @ B2 @ C2 ) ) ) ).

% inf.left_commute
thf(fact_469_inf__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_470_inf__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_471_relH__trans,axiom,
    ! [As: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H12 @ H23 )
     => ( ( relH @ As @ H23 @ H32 )
       => ( relH @ As @ H12 @ H32 ) ) ) ).

% relH_trans
thf(fact_472_relH__sym,axiom,
    ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H2 @ H5 )
     => ( relH @ As @ H5 @ H2 ) ) ).

% relH_sym
thf(fact_473_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
      ( ( A
        = ( inf_inf_set_nat @ K @ A2 ) )
     => ( ( inf_inf_set_nat @ A @ B2 )
        = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_474_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: assn,K: assn,A2: assn,B2: assn] :
      ( ( A
        = ( inf_inf_assn @ K @ A2 ) )
     => ( ( inf_inf_assn @ A @ B2 )
        = ( inf_inf_assn @ K @ ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_475_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
      ( ( B
        = ( inf_inf_set_nat @ K @ B2 ) )
     => ( ( inf_inf_set_nat @ A2 @ B )
        = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_476_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: assn,K: assn,B2: assn,A2: assn] :
      ( ( B
        = ( inf_inf_assn @ K @ B2 ) )
     => ( ( inf_inf_assn @ A2 @ B )
        = ( inf_inf_assn @ K @ ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_477_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P4: produc3658429121746597890et_nat > $o,Q4: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P4 @ ( produc5001842942810119800et_nat @ Q4 @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ).

% times_assn_raw.cases
thf(fact_478_disjoint__iff__not__equal,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
           => ! [Y2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y2 @ B )
               => ( X2 != Y2 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_479_disjoint__iff__not__equal,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ( ( member_o @ X2 @ A )
           => ! [Y2: $o] :
                ( ( member_o @ Y2 @ B )
               => ( X2 = ~ Y2 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_480_disjoint__iff__not__equal,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ! [Y2: nat] :
                ( ( member_nat @ Y2 @ B )
               => ( X2 != Y2 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_481_disjoint__iff__not__equal,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A )
           => ! [Y2: int] :
                ( ( member_int @ Y2 @ B )
               => ( X2 != Y2 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_482_Int__empty__right,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_483_Int__empty__right,axiom,
    ! [A: set_o] :
      ( ( inf_inf_set_o @ A @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_484_Int__empty__right,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_485_Int__empty__right,axiom,
    ! [A: set_int] :
      ( ( inf_inf_set_int @ A @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% Int_empty_right
thf(fact_486_Int__empty__left,axiom,
    ! [B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_487_Int__empty__left,axiom,
    ! [B: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_488_Int__empty__left,axiom,
    ! [B: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_489_Int__empty__left,axiom,
    ! [B: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ B )
      = bot_bot_set_int ) ).

% Int_empty_left
thf(fact_490_disjoint__iff,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ A @ B )
        = bot_bo7824918357723582233_nat_o )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X2 @ A )
           => ~ ( member6576561426505652726_nat_o @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_491_disjoint__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
           => ~ ( member8440522571783428010at_nat @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_492_disjoint__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ( ( member_o @ X2 @ A )
           => ~ ( member_o @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_493_disjoint__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ~ ( member_nat @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_494_disjoint__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A )
           => ~ ( member_int @ X2 @ B ) ) ) ) ).

% disjoint_iff
thf(fact_495_Int__emptyI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A )
         => ~ ( member6576561426505652726_nat_o @ X3 @ B ) )
     => ( ( inf_in1906310914598751387_nat_o @ A @ B )
        = bot_bo7824918357723582233_nat_o ) ) ).

% Int_emptyI
thf(fact_496_Int__emptyI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A )
         => ~ ( member8440522571783428010at_nat @ X3 @ B ) )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_497_Int__emptyI,axiom,
    ! [A: set_o,B: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A )
         => ~ ( member_o @ X3 @ B ) )
     => ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_498_Int__emptyI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ~ ( member_nat @ X3 @ B ) )
     => ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_499_Int__emptyI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A )
         => ~ ( member_int @ X3 @ B ) )
     => ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int ) ) ).

% Int_emptyI
thf(fact_500_boolean__algebra_Oconj__one__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ top_top_assn )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_501_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ top_top_set_o )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_502_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ top_top_set_nat )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_503_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_504_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_505_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_506_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_507_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_508_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_509_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_510_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_511_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_512_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z ) @ X )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ X ) @ ( inf_in7913087082777306421at_nat @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_513_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_514_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_515_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ Z ) @ X )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ X ) @ ( sup_su5525570899277871387at_nat @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_516_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] :
          ( ( inf_inf_assn @ X3 @ ( sup_sup_assn @ Y3 @ Z2 ) )
          = ( sup_sup_assn @ ( inf_inf_assn @ X3 @ Y3 ) @ ( inf_inf_assn @ X3 @ Z2 ) ) )
     => ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
        = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_517_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
          ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
          = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) )
     => ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_518_distrib__imp1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
          ( ( inf_in7913087082777306421at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ Y3 @ Z2 ) )
          = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X3 @ Y3 ) @ ( inf_in7913087082777306421at_nat @ X3 @ Z2 ) ) )
     => ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) )
        = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_519_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] :
          ( ( sup_sup_assn @ X3 @ ( inf_inf_assn @ Y3 @ Z2 ) )
          = ( inf_inf_assn @ ( sup_sup_assn @ X3 @ Y3 ) @ ( sup_sup_assn @ X3 @ Z2 ) ) )
     => ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
        = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_520_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
          ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
          = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) )
     => ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
        = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_521_distrib__imp2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ X3 @ ( inf_in7913087082777306421at_nat @ Y3 @ Z2 ) )
          = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X3 @ Y3 ) @ ( sup_su5525570899277871387at_nat @ X3 @ Z2 ) ) )
     => ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
        = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_522_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_523_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_524_inf__sup__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_525_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_526_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_527_inf__sup__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z ) @ X )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ X ) @ ( inf_in7913087082777306421at_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_528_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_529_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_530_sup__inf__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_531_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_532_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_533_sup__inf__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ Y @ Z ) @ X )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ Y @ X ) @ ( sup_su5525570899277871387at_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_534_inf_Osemigroup__axioms,axiom,
    semigroup_set_nat @ inf_inf_set_nat ).

% inf.semigroup_axioms
thf(fact_535_inf_Osemigroup__axioms,axiom,
    semigroup_assn @ inf_inf_assn ).

% inf.semigroup_axioms
thf(fact_536_inf__top_Ocomm__monoid__axioms,axiom,
    comm_monoid_assn @ inf_inf_assn @ top_top_assn ).

% inf_top.comm_monoid_axioms
thf(fact_537_inf__top_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_o @ inf_inf_set_o @ top_top_set_o ).

% inf_top.comm_monoid_axioms
thf(fact_538_inf__top_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_nat @ inf_inf_set_nat @ top_top_set_nat ).

% inf_top.comm_monoid_axioms
thf(fact_539_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m7591727228438362018t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit ).

% inf_top.comm_monoid_axioms
thf(fact_540_inf__top_Omonoid__axioms,axiom,
    monoid_assn @ inf_inf_assn @ top_top_assn ).

% inf_top.monoid_axioms
thf(fact_541_inf__top_Omonoid__axioms,axiom,
    monoid_set_o @ inf_inf_set_o @ top_top_set_o ).

% inf_top.monoid_axioms
thf(fact_542_inf__top_Omonoid__axioms,axiom,
    monoid_set_nat @ inf_inf_set_nat @ top_top_set_nat ).

% inf_top.monoid_axioms
thf(fact_543_inf__top_Omonoid__axioms,axiom,
    monoid202360481972076142t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit ).

% inf_top.monoid_axioms
thf(fact_544_inf__top_Osemilattice__neutr__axioms,axiom,
    semila5388846958306562758r_assn @ inf_inf_assn @ top_top_assn ).

% inf_top.semilattice_neutr_axioms
thf(fact_545_inf__top_Osemilattice__neutr__axioms,axiom,
    semila1065760912024005978_set_o @ inf_inf_set_o @ top_top_set_o ).

% inf_top.semilattice_neutr_axioms
thf(fact_546_inf__top_Osemilattice__neutr__axioms,axiom,
    semila1241773964035338532et_nat @ inf_inf_set_nat @ top_top_set_nat ).

% inf_top.semilattice_neutr_axioms
thf(fact_547_inf__top_Osemilattice__neutr__axioms,axiom,
    semila7638931295954862799t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit ).

% inf_top.semilattice_neutr_axioms
thf(fact_548_wand__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( wand_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H2 @ As ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
        & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
            ( ( ( ( inf_inf_set_nat @ As @ As5 )
                = bot_bot_set_nat )
              & ( relH @ As @ H2 @ H6 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
              & ( P @ ( produc7507926704131184380et_nat @ H6 @ As5 ) ) )
           => ( Q @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ).

% wand_raw.simps
thf(fact_549_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 )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( Y
              = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
                    & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As3 @ As5 )
                            = bot_bot_set_nat )
                          & ( relH @ As3 @ H3 @ H6 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As3 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H6 @ As5 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As3 @ As5 ) ) ) ) ) ) ) ) ) ).

% wand_raw.elims(1)
thf(fact_550_wand__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
                & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                    ( ( ( ( inf_inf_set_nat @ As3 @ As6 )
                        = bot_bot_set_nat )
                      & ( relH @ As3 @ H3 @ H7 )
                      & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As3 ) )
                      & ( X @ ( produc7507926704131184380et_nat @ H7 @ As6 ) ) )
                   => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As3 @ As6 ) ) ) ) ) ) ) ).

% wand_raw.elims(2)
thf(fact_551_wand__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
              & ! [H4: heap_e7401611519738050253t_unit,As7: set_nat] :
                  ( ( ( ( inf_inf_set_nat @ As3 @ As7 )
                      = bot_bot_set_nat )
                    & ( relH @ As3 @ H3 @ H4 )
                    & ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
                    & ( X @ ( produc7507926704131184380et_nat @ H4 @ As7 ) ) )
                 => ( Xa @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As3 @ As7 ) ) ) ) ) ) ) ).

% wand_raw.elims(3)
thf(fact_552_mod__relH,axiom,
    ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,P: assn] :
      ( ( relH @ As @ H2 @ H5 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H5 @ As ) ) ) ) ).

% mod_relH
thf(fact_553_relH__refl,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
     => ( relH @ As @ H2 @ H2 ) ) ).

% relH_refl
thf(fact_554_relH__in__rangeI_I1_J,axiom,
    ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H2 @ H5 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ).

% relH_in_rangeI(1)
thf(fact_555_relH__in__rangeI_I2_J,axiom,
    ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As @ H2 @ H5 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) ) ) ).

% relH_in_rangeI(2)
thf(fact_556_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ A2 @ X )
        = bot_bo228742789529271731at_nat )
     => ( ( ( sup_su5525570899277871387at_nat @ A2 @ X )
          = top_to6833984726390702231at_nat )
       => ( ( ( inf_in7913087082777306421at_nat @ A2 @ Y )
            = bot_bo228742789529271731at_nat )
         => ( ( ( sup_su5525570899277871387at_nat @ A2 @ Y )
              = top_to6833984726390702231at_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_557_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A2 @ X )
        = bot_bo2099793752762293965at_nat )
     => ( ( ( sup_su6327502436637775413at_nat @ A2 @ X )
          = top_to4669805908274784177at_nat )
       => ( ( ( inf_in2572325071724192079at_nat @ A2 @ Y )
            = bot_bo2099793752762293965at_nat )
         => ( ( ( sup_su6327502436637775413at_nat @ A2 @ Y )
              = top_to4669805908274784177at_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_558_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_int,X: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ A2 @ X )
        = bot_bot_set_int )
     => ( ( ( sup_sup_set_int @ A2 @ X )
          = top_top_set_int )
       => ( ( ( inf_inf_set_int @ A2 @ Y )
            = bot_bot_set_int )
         => ( ( ( sup_sup_set_int @ A2 @ Y )
              = top_top_set_int )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_559_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: assn,X: assn,Y: assn] :
      ( ( ( inf_inf_assn @ A2 @ X )
        = bot_bot_assn )
     => ( ( ( sup_sup_assn @ A2 @ X )
          = top_top_assn )
       => ( ( ( inf_inf_assn @ A2 @ Y )
            = bot_bot_assn )
         => ( ( ( sup_sup_assn @ A2 @ Y )
              = top_top_assn )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_560_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_o,X: set_o,Y: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ X )
        = bot_bot_set_o )
     => ( ( ( sup_sup_set_o @ A2 @ X )
          = top_top_set_o )
       => ( ( ( inf_inf_set_o @ A2 @ Y )
            = bot_bot_set_o )
         => ( ( ( sup_sup_set_o @ A2 @ Y )
              = top_top_set_o )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_561_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_nat,X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ X )
        = bot_bot_set_nat )
     => ( ( ( sup_sup_set_nat @ A2 @ X )
          = top_top_set_nat )
       => ( ( ( inf_inf_set_nat @ A2 @ Y )
            = bot_bot_set_nat )
         => ( ( ( sup_sup_set_nat @ A2 @ Y )
              = top_top_set_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_562_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_Product_unit,X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ A2 @ X )
        = bot_bo3957492148770167129t_unit )
     => ( ( ( sup_su793286257634532545t_unit @ A2 @ X )
          = top_to1996260823553986621t_unit )
       => ( ( ( inf_in4660618365625256667t_unit @ A2 @ Y )
            = bot_bo3957492148770167129t_unit )
         => ( ( ( sup_su793286257634532545t_unit @ A2 @ Y )
              = top_to1996260823553986621t_unit )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_563_prod__induct4,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat > $o,C3: heap_e7401611519738050253t_unit,D: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A5 @ ( produc5001842942810119800et_nat @ B5 @ ( produc7507926704131184380et_nat @ C3 @ D ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_564_prod__induct3,axiom,
    ! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
      ( ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: code_integer,C3: code_integer] : ( P @ ( produc8603105652947943368nteger @ A5 @ ( produc1086072967326762835nteger @ B5 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_565_prod__induct3,axiom,
    ! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
      ( ! [A5: produc3658429121746597890et_nat > $o,B5: heap_e7401611519738050253t_unit,C3: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A5 @ ( produc7507926704131184380et_nat @ B5 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_566_prod__induct3,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat > $o,C3: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A5 @ ( produc5001842942810119800et_nat @ B5 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_567_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
      ( ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: int,C3: int] : ( P @ ( produc5700946648718959541nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_568_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
      ( ! [A5: int > option6357759511663192854e_term,B5: int,C3: int] : ( P @ ( produc4305682042979456191nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_569_prod__cases4,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat > $o,C3: heap_e7401611519738050253t_unit,D: set_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A5 @ ( produc5001842942810119800et_nat @ B5 @ ( produc7507926704131184380et_nat @ C3 @ D ) ) ) ) ).

% prod_cases4
thf(fact_570_prod__cases3,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: code_integer,C3: code_integer] :
          ( Y
         != ( produc8603105652947943368nteger @ A5 @ ( produc1086072967326762835nteger @ B5 @ C3 ) ) ) ).

% prod_cases3
thf(fact_571_prod__cases3,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A5: produc3658429121746597890et_nat > $o,B5: heap_e7401611519738050253t_unit,C3: set_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A5 @ ( produc7507926704131184380et_nat @ B5 @ C3 ) ) ) ).

% prod_cases3
thf(fact_572_prod__cases3,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat > $o,C3: produc3658429121746597890et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A5 @ ( produc5001842942810119800et_nat @ B5 @ C3 ) ) ) ).

% prod_cases3
thf(fact_573_prod__cases3,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: int,C3: int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C3 ) ) ) ).

% prod_cases3
thf(fact_574_prod__cases3,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A5: int > option6357759511663192854e_term,B5: int,C3: int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C3 ) ) ) ).

% prod_cases3
thf(fact_575_Pair__inject,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A2 @ B2 )
        = ( produc8603105652947943368nteger @ A4 @ B4 ) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
thf(fact_576_Pair__inject,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,A4: produc3658429121746597890et_nat > $o,B4: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A2 @ B2 )
        = ( produc5001842942810119800et_nat @ A4 @ B4 ) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
thf(fact_577_Pair__inject,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,A4: produc3658429121746597890et_nat > $o,B4: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A2 @ B2 )
        = ( produc2245416461498447860et_nat @ A4 @ B4 ) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
thf(fact_578_Pair__inject,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A2 @ B2 )
        = ( produc5700946648718959541nt_int @ A4 @ B4 ) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
thf(fact_579_Pair__inject,axiom,
    ! [A2: int > option6357759511663192854e_term,B2: product_prod_int_int,A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A2 @ B2 )
        = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
thf(fact_580_prod__cases,axiom,
    ! [P: produc1908205239877642774nteger > $o,P5: produc1908205239877642774nteger] :
      ( ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_581_prod__cases,axiom,
    ! [P: produc3925858234332021118et_nat > $o,P5: produc3925858234332021118et_nat] :
      ( ! [A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_582_prod__cases,axiom,
    ! [P: produc2732055786443039994et_nat > $o,P5: produc2732055786443039994et_nat] :
      ( ! [A5: produc3658429121746597890et_nat > $o,B5: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_583_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P5: produc2285326912895808259nt_int] :
      ( ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_584_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P5: produc7773217078559923341nt_int] :
      ( ! [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_585_surj__pair,axiom,
    ! [P5: produc1908205239877642774nteger] :
    ? [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
      ( P5
      = ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_586_surj__pair,axiom,
    ! [P5: produc3925858234332021118et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
      ( P5
      = ( produc5001842942810119800et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_587_surj__pair,axiom,
    ! [P5: produc2732055786443039994et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
      ( P5
      = ( produc2245416461498447860et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_588_surj__pair,axiom,
    ! [P5: produc2285326912895808259nt_int] :
    ? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P5
      = ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_589_surj__pair,axiom,
    ! [P5: produc7773217078559923341nt_int] :
    ? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P5
      = ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_590_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( Y
         != ( produc8603105652947943368nteger @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_591_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_592_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A5: produc3658429121746597890et_nat > $o,B5: produc3925858234332021118et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_593_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_594_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_595_Un__empty__left,axiom,
    ! [B: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_596_Un__empty__left,axiom,
    ! [B: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_597_Un__empty__left,axiom,
    ! [B: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B )
      = B ) ).

% Un_empty_left
thf(fact_598_Un__empty__left,axiom,
    ! [B: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_599_Un__empty__left,axiom,
    ! [B: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B )
      = B ) ).

% Un_empty_left
thf(fact_600_Un__empty__right,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ bot_bo228742789529271731at_nat )
      = A ) ).

% Un_empty_right
thf(fact_601_Un__empty__right,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A @ bot_bo2099793752762293965at_nat )
      = A ) ).

% Un_empty_right
thf(fact_602_Un__empty__right,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% Un_empty_right
thf(fact_603_Un__empty__right,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Un_empty_right
thf(fact_604_Un__empty__right,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% Un_empty_right
thf(fact_605_empty__not__UNIV,axiom,
    bot_bo2099793752762293965at_nat != top_to4669805908274784177at_nat ).

% empty_not_UNIV
thf(fact_606_empty__not__UNIV,axiom,
    bot_bot_set_int != top_top_set_int ).

% empty_not_UNIV
thf(fact_607_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
thf(fact_608_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_609_empty__not__UNIV,axiom,
    bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).

% empty_not_UNIV
thf(fact_610_bot__set__def,axiom,
    ( bot_bo7824918357723582233_nat_o
    = ( collec939566748876313656_nat_o @ bot_bo7963750851167320836at_o_o ) ) ).

% bot_set_def
thf(fact_611_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_612_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_613_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_614_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_615_ex__in__conv,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( ? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
      = ( A != bot_bo7824918357723582233_nat_o ) ) ).

% ex_in_conv
thf(fact_616_ex__in__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ? [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A ) )
      = ( A != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
thf(fact_617_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
      = ( A != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_618_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_619_ex__in__conv,axiom,
    ! [A: set_int] :
      ( ( ? [X2: int] : ( member_int @ X2 @ A ) )
      = ( A != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_620_equals0I,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ! [Y3: produc3658429121746597890et_nat > $o] :
          ~ ( member6576561426505652726_nat_o @ Y3 @ A )
     => ( A = bot_bo7824918357723582233_nat_o ) ) ).

% equals0I
thf(fact_621_equals0I,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ Y3 @ A )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
thf(fact_622_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A )
     => ( A = bot_bot_set_o ) ) ).

% equals0I
thf(fact_623_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_624_equals0I,axiom,
    ! [A: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int @ Y3 @ A )
     => ( A = bot_bot_set_int ) ) ).

% equals0I
thf(fact_625_equals0D,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
      ( ( A = bot_bo7824918357723582233_nat_o )
     => ~ ( member6576561426505652726_nat_o @ A2 @ A ) ) ).

% equals0D
thf(fact_626_equals0D,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( A = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_627_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A ) ) ).

% equals0D
thf(fact_628_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_629_equals0D,axiom,
    ! [A: set_int,A2: int] :
      ( ( A = bot_bot_set_int )
     => ~ ( member_int @ A2 @ A ) ) ).

% equals0D
thf(fact_630_emptyE,axiom,
    ! [A2: produc3658429121746597890et_nat > $o] :
      ~ ( member6576561426505652726_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ).

% emptyE
thf(fact_631_emptyE,axiom,
    ! [A2: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).

% emptyE
thf(fact_632_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_633_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_634_emptyE,axiom,
    ! [A2: int] :
      ~ ( member_int @ A2 @ bot_bot_set_int ) ).

% emptyE
thf(fact_635_wand__assnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat,Q: assn,R: assn] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
     => ( ! [H4: heap_e7401611519738050253t_unit,As7: set_nat] :
            ( ( ( inf_inf_set_nat @ As @ As7 )
              = bot_bot_set_nat )
           => ( ( relH @ As @ H2 @ H4 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
               => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H4 @ As7 ) )
                 => ( rep_assn @ R @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As @ As7 ) ) ) ) ) ) )
       => ( rep_assn @ ( wand_assn @ Q @ R ) @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).

% wand_assnI
thf(fact_636_times__assn__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( times_assn_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H2 @ As ) )
      = ( ? [As1: set_nat,As22: set_nat] :
            ( ( As
              = ( sup_sup_set_nat @ As1 @ As22 ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( P @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
            & ( Q @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ).

% times_assn_raw.simps
thf(fact_637_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 )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( Y
              = ( ~ ? [As1: set_nat,As22: set_nat] :
                      ( ( As3
                        = ( sup_sup_set_nat @ As1 @ As22 ) )
                      & ( ( inf_inf_set_nat @ As1 @ As22 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H3 @ As1 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As22 ) ) ) ) ) ) ) ).

% times_assn_raw.elims(1)
thf(fact_638_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ~ ? [As12: set_nat,As23: set_nat] :
                  ( ( As3
                    = ( sup_sup_set_nat @ As12 @ As23 ) )
                  & ( ( inf_inf_set_nat @ As12 @ As23 )
                    = bot_bot_set_nat )
                  & ( X @ ( produc7507926704131184380et_nat @ H3 @ As12 ) )
                  & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As23 ) ) ) ) ) ).

% times_assn_raw.elims(2)
thf(fact_639_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ? [As13: set_nat,As24: set_nat] :
                ( ( As3
                  = ( sup_sup_set_nat @ As13 @ As24 ) )
                & ( ( inf_inf_set_nat @ As13 @ As24 )
                  = bot_bot_set_nat )
                & ( X @ ( produc7507926704131184380et_nat @ H3 @ As13 ) )
                & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As24 ) ) ) ) ) ).

% times_assn_raw.elims(3)
thf(fact_640_properI,axiom,
    ! [P: produc3658429121746597890et_nat > $o] :
      ( ! [As3: set_nat,H3: heap_e7401611519738050253t_unit] :
          ( ( P @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
         => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) )
     => ( ! [As3: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
            ( ( P @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( ( relH @ As3 @ H3 @ H4 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
               => ( P @ ( produc7507926704131184380et_nat @ H4 @ As3 ) ) ) ) )
       => ( proper @ P ) ) ) ).

% properI
thf(fact_641_properD2,axiom,
    ! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat,H5: heap_e7401611519738050253t_unit] :
      ( ( proper @ P )
     => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
       => ( ( relH @ As @ H2 @ H5 )
         => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
           => ( P @ ( produc7507926704131184380et_nat @ H5 @ As ) ) ) ) ) ) ).

% properD2
thf(fact_642_disjointI,axiom,
    ! [A2: set_Pr4532377907799695533_nat_o,B2: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A2 )
         => ~ ( member6576561426505652726_nat_o @ X3 @ B2 ) )
     => ( ( inf_in1906310914598751387_nat_o @ A2 @ B2 )
        = bot_bo7824918357723582233_nat_o ) ) ).

% disjointI
thf(fact_643_disjointI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A2 )
         => ~ ( member8440522571783428010at_nat @ X3 @ B2 ) )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
thf(fact_644_disjointI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A2 )
         => ~ ( member_o @ X3 @ B2 ) )
     => ( ( inf_inf_set_o @ A2 @ B2 )
        = bot_bot_set_o ) ) ).

% disjointI
thf(fact_645_disjointI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ~ ( member_nat @ X3 @ B2 ) )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = bot_bot_set_nat ) ) ).

% disjointI
thf(fact_646_disjointI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ~ ( member_int @ X3 @ B2 ) )
     => ( ( inf_inf_set_int @ A2 @ B2 )
        = bot_bot_set_int ) ) ).

% disjointI
thf(fact_647_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) )
               => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
                  & ! [H4: heap_e7401611519738050253t_unit,As7: set_nat] :
                      ( ( ( ( inf_inf_set_nat @ As3 @ As7 )
                          = bot_bot_set_nat )
                        & ( relH @ As3 @ H3 @ H4 )
                        & ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
                        & ( X @ ( produc7507926704131184380et_nat @ H4 @ As7 ) ) )
                     => ( Xa @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As3 @ As7 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
thf(fact_648_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) )
               => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As3 @ As6 )
                            = bot_bot_set_nat )
                          & ( relH @ As3 @ H3 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As3 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As6 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As3 @ As6 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
thf(fact_649_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( Y
                  = ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
                    & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As3 @ As5 )
                            = bot_bot_set_nat )
                          & ( relH @ As3 @ H3 @ H6 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As3 ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H6 @ As5 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As3 @ As5 ) ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(1)
thf(fact_650_Set_Ois__empty__def,axiom,
    ( is_emp1662574758705540307at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] : A6 = bot_bo2099793752762293965at_nat ) ) ).

% Set.is_empty_def
thf(fact_651_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A6: set_o] : A6 = bot_bot_set_o ) ) ).

% Set.is_empty_def
thf(fact_652_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A6: set_nat] : A6 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
thf(fact_653_Set_Ois__empty__def,axiom,
    ( is_empty_int
    = ( ^ [A6: set_int] : A6 = bot_bot_set_int ) ) ).

% Set.is_empty_def
thf(fact_654_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) )
               => ? [As13: set_nat,As24: set_nat] :
                    ( ( As3
                      = ( sup_sup_set_nat @ As13 @ As24 ) )
                    & ( ( inf_inf_set_nat @ As13 @ As24 )
                      = bot_bot_set_nat )
                    & ( X @ ( produc7507926704131184380et_nat @ H3 @ As13 ) )
                    & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As24 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
thf(fact_655_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) )
               => ~ ? [As12: set_nat,As23: set_nat] :
                      ( ( As3
                        = ( sup_sup_set_nat @ As12 @ As23 ) )
                      & ( ( inf_inf_set_nat @ As12 @ As23 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H3 @ As12 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As23 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
thf(fact_656_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( Y
                  = ( ? [As1: set_nat,As22: set_nat] :
                        ( ( As3
                          = ( sup_sup_set_nat @ As1 @ As22 ) )
                        & ( ( inf_inf_set_nat @ As1 @ As22 )
                          = bot_bot_set_nat )
                        & ( X @ ( produc7507926704131184380et_nat @ H3 @ As1 ) )
                        & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As22 ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(1)
thf(fact_657_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ X @ Y )
        = bot_bo228742789529271731at_nat )
     => ( ( ( sup_su5525570899277871387at_nat @ X @ Y )
          = top_to6833984726390702231at_nat )
       => ( ( uminus935396558254630718at_nat @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_658_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
     => ( ( ( sup_su6327502436637775413at_nat @ X @ Y )
          = top_to4669805908274784177at_nat )
       => ( ( uminus6524753893492686040at_nat @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_659_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ X @ Y )
        = bot_bot_set_int )
     => ( ( ( sup_sup_set_int @ X @ Y )
          = top_top_set_int )
       => ( ( uminus1532241313380277803et_int @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_660_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( inf_inf_set_o @ X @ Y )
        = bot_bot_set_o )
     => ( ( ( sup_sup_set_o @ X @ Y )
          = top_top_set_o )
       => ( ( uminus_uminus_set_o @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_661_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y )
        = bot_bot_set_nat )
     => ( ( ( sup_sup_set_nat @ X @ Y )
          = top_top_set_nat )
       => ( ( uminus5710092332889474511et_nat @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_662_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ X @ Y )
        = bot_bo3957492148770167129t_unit )
     => ( ( ( sup_su793286257634532545t_unit @ X @ Y )
          = top_to1996260823553986621t_unit )
       => ( ( uminus5944136376168626660t_unit @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_663_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( inf_inf_assn @ X @ Y )
        = bot_bot_assn )
     => ( ( ( sup_sup_assn @ X @ Y )
          = top_top_assn )
       => ( ( uminus_uminus_assn @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_664_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( uminus_uminus_assn @ X )
        = ( uminus_uminus_assn @ Y ) )
      = ( X = Y ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_665_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X: assn] :
      ( ( uminus_uminus_assn @ ( uminus_uminus_assn @ X ) )
      = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_666_neg__equal__iff__equal,axiom,
    ! [A2: int,B2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = ( uminus_uminus_int @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_667_neg__equal__iff__equal,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = ( uminus_uminus_rat @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_668_neg__equal__iff__equal,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_669_add_Oinverse__inverse,axiom,
    ! [A2: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_670_add_Oinverse__inverse,axiom,
    ! [A2: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_671_add_Oinverse__inverse,axiom,
    ! [A2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_672_merge__pure__and,axiom,
    ! [A2: $o,B2: $o] :
      ( ( inf_inf_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
      = ( pure_assn
        @ ( A2
          & B2 ) ) ) ).

% merge_pure_and
thf(fact_673_Compl__disjoint2,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A ) @ A )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint2
thf(fact_674_Compl__disjoint2,axiom,
    ! [A: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A ) @ A )
      = bot_bot_set_o ) ).

% Compl_disjoint2
thf(fact_675_Compl__disjoint2,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
      = bot_bot_set_nat ) ).

% Compl_disjoint2
thf(fact_676_Compl__disjoint2,axiom,
    ! [A: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A ) @ A )
      = bot_bot_set_int ) ).

% Compl_disjoint2
thf(fact_677_Compl__disjoint,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( uminus6524753893492686040at_nat @ A ) )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint
thf(fact_678_Compl__disjoint,axiom,
    ! [A: set_o] :
      ( ( inf_inf_set_o @ A @ ( uminus_uminus_set_o @ A ) )
      = bot_bot_set_o ) ).

% Compl_disjoint
thf(fact_679_Compl__disjoint,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
      = bot_bot_set_nat ) ).

% Compl_disjoint
thf(fact_680_Compl__disjoint,axiom,
    ! [A: set_int] :
      ( ( inf_inf_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
      = bot_bot_set_int ) ).

% Compl_disjoint
thf(fact_681_mod__not__dist,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( uminus_uminus_assn @ P ) @ H2 )
      = ( ( in_range @ H2 )
        & ~ ( rep_assn @ P @ H2 ) ) ) ).

% mod_not_dist
thf(fact_682_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat )
    = top_to4669805908274784177at_nat ) ).

% boolean_algebra.compl_zero
thf(fact_683_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
    = top_top_set_int ) ).

% boolean_algebra.compl_zero
thf(fact_684_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus_uminus_set_o @ bot_bot_set_o )
    = top_top_set_o ) ).

% boolean_algebra.compl_zero
thf(fact_685_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
    = top_top_set_nat ) ).

% boolean_algebra.compl_zero
thf(fact_686_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
    = top_to1996260823553986621t_unit ) ).

% boolean_algebra.compl_zero
thf(fact_687_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus_uminus_assn @ bot_bot_assn )
    = top_top_assn ) ).

% boolean_algebra.compl_zero
thf(fact_688_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus6524753893492686040at_nat @ top_to4669805908274784177at_nat )
    = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.compl_one
thf(fact_689_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus1532241313380277803et_int @ top_top_set_int )
    = bot_bot_set_int ) ).

% boolean_algebra.compl_one
thf(fact_690_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus_uminus_set_o @ top_top_set_o )
    = bot_bot_set_o ) ).

% boolean_algebra.compl_one
thf(fact_691_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
    = bot_bot_set_nat ) ).

% boolean_algebra.compl_one
thf(fact_692_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
    = bot_bo3957492148770167129t_unit ) ).

% boolean_algebra.compl_one
thf(fact_693_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus_uminus_assn @ top_top_assn )
    = bot_bot_assn ) ).

% boolean_algebra.compl_one
thf(fact_694_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ X ) )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_695_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ ( uminus_uminus_set_o @ X ) )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_right
thf(fact_696_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_697_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_right
thf(fact_698_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ ( uminus_uminus_assn @ X ) )
      = bot_bot_assn ) ).

% boolean_algebra.conj_cancel_right
thf(fact_699_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ X )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_700_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ X )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_left
thf(fact_701_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_702_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_left
thf(fact_703_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ X )
      = bot_bot_assn ) ).

% boolean_algebra.conj_cancel_left
thf(fact_704_inf__compl__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ ( uminus6524753893492686040at_nat @ X ) ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_right
thf(fact_705_inf__compl__bot__right,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X ) ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_right
thf(fact_706_inf__compl__bot__right,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_right
thf(fact_707_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_right
thf(fact_708_inf__compl__bot__right,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ ( uminus_uminus_assn @ X ) ) )
      = bot_bot_assn ) ).

% inf_compl_bot_right
thf(fact_709_inf__compl__bot__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ Y ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left2
thf(fact_710_inf__compl__bot__left2,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left2
thf(fact_711_inf__compl__bot__left2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left2
thf(fact_712_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left2
thf(fact_713_inf__compl__bot__left2,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ Y ) )
      = bot_bot_assn ) ).

% inf_compl_bot_left2
thf(fact_714_inf__compl__bot__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left1
thf(fact_715_inf__compl__bot__left1,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ ( inf_inf_set_o @ X @ Y ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left1
thf(fact_716_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left1
thf(fact_717_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( inf_inf_set_int @ X @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left1
thf(fact_718_inf__compl__bot__left1,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ ( inf_inf_assn @ X @ Y ) )
      = bot_bot_assn ) ).

% inf_compl_bot_left1
thf(fact_719_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( uminus935396558254630718at_nat @ X ) )
      = top_to6833984726390702231at_nat ) ).

% boolean_algebra.disj_cancel_right
thf(fact_720_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ ( uminus_uminus_set_o @ X ) )
      = top_top_set_o ) ).

% boolean_algebra.disj_cancel_right
thf(fact_721_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
      = top_top_set_nat ) ).

% boolean_algebra.disj_cancel_right
thf(fact_722_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ X @ ( uminus5944136376168626660t_unit @ X ) )
      = top_to1996260823553986621t_unit ) ).

% boolean_algebra.disj_cancel_right
thf(fact_723_boolean__algebra_Odisj__cancel__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ ( uminus_uminus_assn @ X ) )
      = top_top_assn ) ).

% boolean_algebra.disj_cancel_right
thf(fact_724_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ X )
      = top_to6833984726390702231at_nat ) ).

% boolean_algebra.disj_cancel_left
thf(fact_725_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ X )
      = top_top_set_o ) ).

% boolean_algebra.disj_cancel_left
thf(fact_726_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
      = top_top_set_nat ) ).

% boolean_algebra.disj_cancel_left
thf(fact_727_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ ( uminus5944136376168626660t_unit @ X ) @ X )
      = top_to1996260823553986621t_unit ) ).

% boolean_algebra.disj_cancel_left
thf(fact_728_boolean__algebra_Odisj__cancel__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ ( uminus_uminus_assn @ X ) @ X )
      = top_top_assn ) ).

% boolean_algebra.disj_cancel_left
thf(fact_729_sup__compl__top__left2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ Y ) )
      = top_to6833984726390702231at_nat ) ).

% sup_compl_top_left2
thf(fact_730_sup__compl__top__left2,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( sup_sup_set_o @ X @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
      = top_top_set_o ) ).

% sup_compl_top_left2
thf(fact_731_sup__compl__top__left2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
      = top_top_set_nat ) ).

% sup_compl_top_left2
thf(fact_732_sup__compl__top__left2,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ X @ ( sup_su793286257634532545t_unit @ ( uminus5944136376168626660t_unit @ X ) @ Y ) )
      = top_to1996260823553986621t_unit ) ).

% sup_compl_top_left2
thf(fact_733_sup__compl__top__left2,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ ( uminus_uminus_assn @ X ) @ Y ) )
      = top_top_assn ) ).

% sup_compl_top_left2
thf(fact_734_sup__compl__top__left1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ ( sup_su5525570899277871387at_nat @ X @ Y ) )
      = top_to6833984726390702231at_nat ) ).

% sup_compl_top_left1
thf(fact_735_sup__compl__top__left1,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ ( sup_sup_set_o @ X @ Y ) )
      = top_top_set_o ) ).

% sup_compl_top_left1
thf(fact_736_sup__compl__top__left1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( sup_sup_set_nat @ X @ Y ) )
      = top_top_set_nat ) ).

% sup_compl_top_left1
thf(fact_737_sup__compl__top__left1,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ ( uminus5944136376168626660t_unit @ X ) @ ( sup_su793286257634532545t_unit @ X @ Y ) )
      = top_to1996260823553986621t_unit ) ).

% sup_compl_top_left1
thf(fact_738_sup__compl__top__left1,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ ( uminus_uminus_assn @ X ) @ ( sup_sup_assn @ X @ Y ) )
      = top_top_assn ) ).

% sup_compl_top_left1
thf(fact_739_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_740_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_741_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_742_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_743_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_744_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_745_mod__h__bot__iff_I6_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(6)
thf(fact_746_minus__equation__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = B2 )
      = ( ( uminus_uminus_int @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_747_minus__equation__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = B2 )
      = ( ( uminus_uminus_rat @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_748_minus__equation__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = B2 )
      = ( ( uminus1351360451143612070nteger @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_749_equation__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( uminus_uminus_int @ B2 ) )
      = ( B2
        = ( uminus_uminus_int @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_750_equation__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ B2 ) )
      = ( B2
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_751_equation__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( B2
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_752_Compl__UNIV__eq,axiom,
    ( ( uminus6524753893492686040at_nat @ top_to4669805908274784177at_nat )
    = bot_bo2099793752762293965at_nat ) ).

% Compl_UNIV_eq
thf(fact_753_Compl__UNIV__eq,axiom,
    ( ( uminus1532241313380277803et_int @ top_top_set_int )
    = bot_bot_set_int ) ).

% Compl_UNIV_eq
thf(fact_754_Compl__UNIV__eq,axiom,
    ( ( uminus_uminus_set_o @ top_top_set_o )
    = bot_bot_set_o ) ).

% Compl_UNIV_eq
thf(fact_755_Compl__UNIV__eq,axiom,
    ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
    = bot_bot_set_nat ) ).

% Compl_UNIV_eq
thf(fact_756_Compl__UNIV__eq,axiom,
    ( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
    = bot_bo3957492148770167129t_unit ) ).

% Compl_UNIV_eq
thf(fact_757_Compl__empty__eq,axiom,
    ( ( uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat )
    = top_to4669805908274784177at_nat ) ).

% Compl_empty_eq
thf(fact_758_Compl__empty__eq,axiom,
    ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
    = top_top_set_int ) ).

% Compl_empty_eq
thf(fact_759_Compl__empty__eq,axiom,
    ( ( uminus_uminus_set_o @ bot_bot_set_o )
    = top_top_set_o ) ).

% Compl_empty_eq
thf(fact_760_Compl__empty__eq,axiom,
    ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
    = top_top_set_nat ) ).

% Compl_empty_eq
thf(fact_761_Compl__empty__eq,axiom,
    ( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
    = top_to1996260823553986621t_unit ) ).

% Compl_empty_eq
thf(fact_762_mod__and__dist,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        & ( rep_assn @ Q @ H2 ) ) ) ).

% mod_and_dist
thf(fact_763_ent__conjE2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( entails @ B @ C )
     => ( entails @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% ent_conjE2
thf(fact_764_ent__conjE1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ( entails @ A @ C )
     => ( entails @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% ent_conjE1
thf(fact_765_ent__conjI,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( entails @ A @ B )
     => ( ( entails @ A @ C )
       => ( entails @ A @ ( inf_inf_assn @ B @ C ) ) ) ) ).

% ent_conjI
thf(fact_766_inf__cancel__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ A2 ) @ ( inf_in2572325071724192079at_nat @ X @ B2 ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left2
thf(fact_767_inf__cancel__left2,axiom,
    ! [X: set_o,A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ A2 ) @ ( inf_inf_set_o @ X @ B2 ) )
      = bot_bot_set_o ) ).

% inf_cancel_left2
thf(fact_768_inf__cancel__left2,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( inf_inf_set_nat @ X @ B2 ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left2
thf(fact_769_inf__cancel__left2,axiom,
    ! [X: set_int,A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ A2 ) @ ( inf_inf_set_int @ X @ B2 ) )
      = bot_bot_set_int ) ).

% inf_cancel_left2
thf(fact_770_inf__cancel__left2,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ A2 ) @ ( inf_inf_assn @ X @ B2 ) )
      = bot_bot_assn ) ).

% inf_cancel_left2
thf(fact_771_inf__cancel__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ A2 ) @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ B2 ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left1
thf(fact_772_inf__cancel__left1,axiom,
    ! [X: set_o,A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ A2 ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ B2 ) )
      = bot_bot_set_o ) ).

% inf_cancel_left1
thf(fact_773_inf__cancel__left1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B2 ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left1
thf(fact_774_inf__cancel__left1,axiom,
    ! [X: set_int,A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ X @ A2 ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ B2 ) )
      = bot_bot_set_int ) ).

% inf_cancel_left1
thf(fact_775_inf__cancel__left1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ A2 ) @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ B2 ) )
      = bot_bot_assn ) ).

% inf_cancel_left1
thf(fact_776_sup__cancel__left2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ A2 ) @ ( sup_su5525570899277871387at_nat @ X @ B2 ) )
      = top_to6833984726390702231at_nat ) ).

% sup_cancel_left2
thf(fact_777_sup__cancel__left2,axiom,
    ! [X: set_o,A2: set_o,B2: set_o] :
      ( ( sup_sup_set_o @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ A2 ) @ ( sup_sup_set_o @ X @ B2 ) )
      = top_top_set_o ) ).

% sup_cancel_left2
thf(fact_778_sup__cancel__left2,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( sup_sup_set_nat @ X @ B2 ) )
      = top_top_set_nat ) ).

% sup_cancel_left2
thf(fact_779_sup__cancel__left2,axiom,
    ! [X: set_Product_unit,A2: set_Product_unit,B2: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ ( sup_su793286257634532545t_unit @ ( uminus5944136376168626660t_unit @ X ) @ A2 ) @ ( sup_su793286257634532545t_unit @ X @ B2 ) )
      = top_to1996260823553986621t_unit ) ).

% sup_cancel_left2
thf(fact_780_sup__cancel__left2,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ ( uminus_uminus_assn @ X ) @ A2 ) @ ( sup_sup_assn @ X @ B2 ) )
      = top_top_assn ) ).

% sup_cancel_left2
thf(fact_781_sup__cancel__left1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ X @ A2 ) @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ X ) @ B2 ) )
      = top_to6833984726390702231at_nat ) ).

% sup_cancel_left1
thf(fact_782_sup__cancel__left1,axiom,
    ! [X: set_o,A2: set_o,B2: set_o] :
      ( ( sup_sup_set_o @ ( sup_sup_set_o @ X @ A2 ) @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X ) @ B2 ) )
      = top_top_set_o ) ).

% sup_cancel_left1
thf(fact_783_sup__cancel__left1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ A2 ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B2 ) )
      = top_top_set_nat ) ).

% sup_cancel_left1
thf(fact_784_sup__cancel__left1,axiom,
    ! [X: set_Product_unit,A2: set_Product_unit,B2: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ ( sup_su793286257634532545t_unit @ X @ A2 ) @ ( sup_su793286257634532545t_unit @ ( uminus5944136376168626660t_unit @ X ) @ B2 ) )
      = top_to1996260823553986621t_unit ) ).

% sup_cancel_left1
thf(fact_785_sup__cancel__left1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ A2 ) @ ( sup_sup_assn @ ( uminus_uminus_assn @ X ) @ B2 ) )
      = top_top_assn ) ).

% sup_cancel_left1
thf(fact_786_pairself_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,A5: int,B5: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ A5 @ B5 ) ) ) ).

% pairself.cases
thf(fact_787_bex2I,axiom,
    ! [A2: nat,B2: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ S )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A5: nat,B5: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B5 ) @ S )
            & ( P @ A5 @ B5 ) ) ) ) ).

% bex2I
thf(fact_788_bex2I,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,S: set_Pr1281608226676607948nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A2 @ B2 ) @ S )
     => ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A5 @ B5 ) @ S )
            & ( P @ A5 @ B5 ) ) ) ) ).

% bex2I
thf(fact_789_bex2I,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ S )
     => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A5: produc3658429121746597890et_nat > $o,B5: produc3658429121746597890et_nat] :
            ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A5 @ B5 ) @ S )
            & ( P @ A5 @ B5 ) ) ) ) ).

% bex2I
thf(fact_790_bex2I,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ S )
     => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A5: produc3658429121746597890et_nat > $o,B5: produc3925858234332021118et_nat] :
            ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A5 @ B5 ) @ S )
            & ( P @ A5 @ B5 ) ) ) ) ).

% bex2I
thf(fact_791_bex2I,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,S: set_Pr9222295170931077689nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A2 @ B2 ) @ S )
     => ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A5 @ B5 ) @ S )
            & ( P @ A5 @ B5 ) ) ) ) ).

% bex2I
thf(fact_792_bex2I,axiom,
    ! [A2: int > option6357759511663192854e_term,B2: product_prod_int_int,S: set_Pr1872883991513573699nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A2 @ B2 ) @ S )
     => ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A5 @ B5 ) @ S )
            & ( P @ A5 @ B5 ) ) ) ) ).

% bex2I
thf(fact_793_set__notEmptyE,axiom,
    ! [S: set_Pr4532377907799695533_nat_o] :
      ( ( S != bot_bo7824918357723582233_nat_o )
     => ~ ! [X3: produc3658429121746597890et_nat > $o] :
            ~ ( member6576561426505652726_nat_o @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_794_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_795_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X3: $o] :
            ~ ( member_o @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_796_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_797_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_798_memb__imp__not__empty,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ S )
     => ( S != bot_bo7824918357723582233_nat_o ) ) ).

% memb_imp_not_empty
thf(fact_799_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_800_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_801_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_802_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_803_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_804_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_805_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_806_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_807_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_808_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_809_mult__minus__right,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( times_times_int @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_810_mult__minus__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_811_mult__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_812_minus__mult__minus,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) )
      = ( times_times_int @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_813_minus__mult__minus,axiom,
    ! [A2: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B2 ) )
      = ( times_times_rat @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_814_minus__mult__minus,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( times_3573771949741848930nteger @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_815_mult__minus__left,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( uminus_uminus_int @ ( times_times_int @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_816_mult__minus__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_817_mult__minus__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_818_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_819_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_820_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_821_bot__empty__eq,axiom,
    ( bot_bo7963750851167320836at_o_o
    = ( ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ).

% bot_empty_eq
thf(fact_822_bot__empty__eq,axiom,
    ( bot_bo482883023278783056_nat_o
    = ( ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq
thf(fact_823_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_824_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_825_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X2: int] : ( member_int @ X2 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_826_Collect__empty__eq__bot,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ( collec939566748876313656_nat_o @ P )
        = bot_bo7824918357723582233_nat_o )
      = ( P = bot_bo7963750851167320836at_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_827_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( P = bot_bo482883023278783056_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_828_Collect__empty__eq__bot,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( P = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_829_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_830_Collect__empty__eq__bot,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( P = bot_bot_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_831_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_832_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_833_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_834_minus__mult__commute,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( times_times_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_835_minus__mult__commute,axiom,
    ! [A2: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_836_minus__mult__commute,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_837_square__eq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ A2 )
        = ( times_times_int @ B2 @ B2 ) )
      = ( ( A2 = B2 )
        | ( A2
          = ( uminus_uminus_int @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_838_square__eq__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ A2 )
        = ( times_times_rat @ B2 @ B2 ) )
      = ( ( A2 = B2 )
        | ( A2
          = ( uminus_uminus_rat @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_839_square__eq__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ A2 )
        = ( times_3573771949741848930nteger @ B2 @ B2 ) )
      = ( ( A2 = B2 )
        | ( A2
          = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_840_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_841_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_842_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_843_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea9150928655215452349at_nat @ inf_in7913087082777306421at_nat @ sup_su5525570899277871387at_nat @ uminus935396558254630718at_nat @ bot_bo228742789529271731at_nat @ top_to6833984726390702231at_nat ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_844_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea949959532490082519at_nat @ inf_in2572325071724192079at_nat @ sup_su6327502436637775413at_nat @ uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat @ top_to4669805908274784177at_nat ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_845_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea5824373010784320748et_int @ inf_inf_set_int @ sup_sup_set_int @ uminus1532241313380277803et_int @ bot_bot_set_int @ top_top_set_int ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_846_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea379910186789422830_set_o @ inf_inf_set_o @ sup_sup_set_o @ uminus_uminus_set_o @ bot_bot_set_o @ top_top_set_o ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_847_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea778851993438741648et_nat @ inf_inf_set_nat @ sup_sup_set_nat @ uminus5710092332889474511et_nat @ bot_bot_set_nat @ top_top_set_nat ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_848_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea3191171876098928227t_unit @ inf_in4660618365625256667t_unit @ sup_su793286257634532545t_unit @ uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit @ top_to1996260823553986621t_unit ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_849_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea5875918072483028530a_assn @ inf_inf_assn @ sup_sup_assn @ uminus_uminus_assn @ bot_bot_assn @ top_top_assn ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_850_pairself_Opelims,axiom,
    ! [X: int > option6357759511663192854e_term,Xa: product_prod_int_int,Y: produc6576344331059438605e_term] :
      ( ( ( pairse3534876335208825186e_term @ X @ Xa )
        = Y )
     => ( ( accp_P8928870874622223812nt_int @ pairse6848479906795794847e_term @ ( produc4305682042979456191nt_int @ X @ Xa ) )
       => ~ ! [A5: int,B5: int] :
              ( ( Xa
                = ( product_Pair_int_int @ A5 @ B5 ) )
             => ( ( Y
                  = ( produc5950683997804057413e_term @ ( X @ A5 ) @ ( X @ B5 ) ) )
               => ~ ( accp_P8928870874622223812nt_int @ pairse6848479906795794847e_term @ ( produc4305682042979456191nt_int @ X @ ( product_Pair_int_int @ A5 @ B5 ) ) ) ) ) ) ) ).

% pairself.pelims
thf(fact_851_mod__pure,axiom,
    ! [B2: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( pure_assn @ B2 ) @ H2 )
      = ( ( ( produc8586169260539613262et_nat @ H2 )
          = bot_bot_set_nat )
        & B2 ) ) ).

% mod_pure
thf(fact_852_type__definition__assn,axiom,
    type_d3909072315231072503_nat_o @ rep_assn @ abs_assn @ ( collec939566748876313656_nat_o @ proper ) ).

% type_definition_assn
thf(fact_853_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 ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( Y
                  = ( ( As3 = bot_bot_set_nat )
                    & X ) )
               => ~ ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ) ) ) ).

% pure_assn_raw.pelims(1)
thf(fact_854_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 ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) )
               => ~ ( ( As3 = bot_bot_set_nat )
                    & X ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
thf(fact_855_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 ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( Xa
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) )
               => ( ( As3 = bot_bot_set_nat )
                  & X ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
thf(fact_856_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% dbl_dec_simps(3)
thf(fact_857_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_858_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_859_mod__emp,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ one_one_assn @ H2 )
      = ( ( produc8586169260539613262et_nat @ H2 )
        = bot_bot_set_nat ) ) ).

% mod_emp
thf(fact_860_sndE,axiom,
    ! [X: produc1908205239877642774nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,P: produc8923325533196201883nteger > $o] :
      ( ( X
        = ( produc8603105652947943368nteger @ A2 @ B2 ) )
     => ( ( P @ ( produc7856867400915047194nteger @ X ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_861_sndE,axiom,
    ! [X: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,P: produc3658429121746597890et_nat > $o] :
      ( ( X
        = ( produc5001842942810119800et_nat @ A2 @ B2 ) )
     => ( ( P @ ( produc4011572625026189258et_nat @ X ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_862_sndE,axiom,
    ! [X: produc2732055786443039994et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,P: produc3925858234332021118et_nat > $o] :
      ( ( X
        = ( produc2245416461498447860et_nat @ A2 @ B2 ) )
     => ( ( P @ ( produc5374455773327741254et_nat @ X ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_863_sndE,axiom,
    ! [X: produc2285326912895808259nt_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( X
        = ( produc5700946648718959541nt_int @ A2 @ B2 ) )
     => ( ( P @ ( produc7328097813583171335nt_int @ X ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_864_sndE,axiom,
    ! [X: produc7773217078559923341nt_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( X
        = ( produc4305682042979456191nt_int @ A2 @ B2 ) )
     => ( ( P @ ( produc3162348030201620241nt_int @ X ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_865_sndE,axiom,
    ! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B2: set_nat,P: set_nat > $o] :
      ( ( X
        = ( produc7507926704131184380et_nat @ A2 @ B2 ) )
     => ( ( P @ ( produc8586169260539613262et_nat @ X ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_866_sndE,axiom,
    ! [X: product_prod_int_int,A2: int,B2: int,P: int > $o] :
      ( ( X
        = ( product_Pair_int_int @ A2 @ B2 ) )
     => ( ( P @ ( product_snd_int_int @ X ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_867_sndE,axiom,
    ! [X: produc5835291356934675326atural,A2: code_natural,B2: produc7822875418678951345atural,P: produc7822875418678951345atural > $o] :
      ( ( X
        = ( produc6639722614265839536atural @ A2 @ B2 ) )
     => ( ( P @ ( produc6591912806276919810atural @ X ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_868_snd__eqD,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A2: produc8923325533196201883nteger] :
      ( ( ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_869_snd__eqD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat] :
      ( ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_870_snd__eqD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc3925858234332021118et_nat] :
      ( ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_871_snd__eqD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A2: product_prod_int_int] :
      ( ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_872_snd__eqD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,A2: product_prod_int_int] :
      ( ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_873_snd__eqD,axiom,
    ! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: set_nat] :
      ( ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_874_snd__eqD,axiom,
    ! [X: int,Y: int,A2: int] :
      ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_875_snd__eqD,axiom,
    ! [X: code_natural,Y: produc7822875418678951345atural,A2: produc7822875418678951345atural] :
      ( ( ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_876_snd__conv,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger] :
      ( ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_877_snd__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat] :
      ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_878_snd__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat] :
      ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_879_snd__conv,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int] :
      ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_880_snd__conv,axiom,
    ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int] :
      ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_881_snd__conv,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X22: set_nat] :
      ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_882_snd__conv,axiom,
    ! [X1: int,X22: int] :
      ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_883_snd__conv,axiom,
    ! [X1: code_natural,X22: produc7822875418678951345atural] :
      ( ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_884_sndI,axiom,
    ! [X: produc1908205239877642774nteger,Y: produc6241069584506657477e_term > option6357759511663192854e_term,Z: produc8923325533196201883nteger] :
      ( ( X
        = ( produc8603105652947943368nteger @ Y @ Z ) )
     => ( ( produc7856867400915047194nteger @ X )
        = Z ) ) ).

% sndI
thf(fact_885_sndI,axiom,
    ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
      ( ( X
        = ( produc5001842942810119800et_nat @ Y @ Z ) )
     => ( ( produc4011572625026189258et_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_886_sndI,axiom,
    ! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
      ( ( X
        = ( produc2245416461498447860et_nat @ Y @ Z ) )
     => ( ( produc5374455773327741254et_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_887_sndI,axiom,
    ! [X: produc2285326912895808259nt_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc5700946648718959541nt_int @ Y @ Z ) )
     => ( ( produc7328097813583171335nt_int @ X )
        = Z ) ) ).

% sndI
thf(fact_888_sndI,axiom,
    ! [X: produc7773217078559923341nt_int,Y: int > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc4305682042979456191nt_int @ Y @ Z ) )
     => ( ( produc3162348030201620241nt_int @ X )
        = Z ) ) ).

% sndI
thf(fact_889_sndI,axiom,
    ! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
      ( ( X
        = ( produc7507926704131184380et_nat @ Y @ Z ) )
     => ( ( produc8586169260539613262et_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_890_sndI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X
        = ( product_Pair_int_int @ Y @ Z ) )
     => ( ( product_snd_int_int @ X )
        = Z ) ) ).

% sndI
thf(fact_891_sndI,axiom,
    ! [X: produc5835291356934675326atural,Y: code_natural,Z: produc7822875418678951345atural] :
      ( ( X
        = ( produc6639722614265839536atural @ Y @ Z ) )
     => ( ( produc6591912806276919810atural @ X )
        = Z ) ) ).

% sndI
thf(fact_892_eq__snd__iff,axiom,
    ! [B2: produc8923325533196201883nteger,P5: produc1908205239877642774nteger] :
      ( ( B2
        = ( produc7856867400915047194nteger @ P5 ) )
      = ( ? [A3: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ( P5
            = ( produc8603105652947943368nteger @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_893_eq__snd__iff,axiom,
    ! [B2: produc3658429121746597890et_nat,P5: produc3925858234332021118et_nat] :
      ( ( B2
        = ( produc4011572625026189258et_nat @ P5 ) )
      = ( ? [A3: produc3658429121746597890et_nat > $o] :
            ( P5
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_894_eq__snd__iff,axiom,
    ! [B2: produc3925858234332021118et_nat,P5: produc2732055786443039994et_nat] :
      ( ( B2
        = ( produc5374455773327741254et_nat @ P5 ) )
      = ( ? [A3: produc3658429121746597890et_nat > $o] :
            ( P5
            = ( produc2245416461498447860et_nat @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_895_eq__snd__iff,axiom,
    ! [B2: product_prod_int_int,P5: produc2285326912895808259nt_int] :
      ( ( B2
        = ( produc7328097813583171335nt_int @ P5 ) )
      = ( ? [A3: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ( P5
            = ( produc5700946648718959541nt_int @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_896_eq__snd__iff,axiom,
    ! [B2: product_prod_int_int,P5: produc7773217078559923341nt_int] :
      ( ( B2
        = ( produc3162348030201620241nt_int @ P5 ) )
      = ( ? [A3: int > option6357759511663192854e_term] :
            ( P5
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_897_eq__snd__iff,axiom,
    ! [B2: set_nat,P5: produc3658429121746597890et_nat] :
      ( ( B2
        = ( produc8586169260539613262et_nat @ P5 ) )
      = ( ? [A3: heap_e7401611519738050253t_unit] :
            ( P5
            = ( produc7507926704131184380et_nat @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_898_eq__snd__iff,axiom,
    ! [B2: int,P5: product_prod_int_int] :
      ( ( B2
        = ( product_snd_int_int @ P5 ) )
      = ( ? [A3: int] :
            ( P5
            = ( product_Pair_int_int @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_899_eq__snd__iff,axiom,
    ! [B2: produc7822875418678951345atural,P5: produc5835291356934675326atural] :
      ( ( B2
        = ( produc6591912806276919810atural @ P5 ) )
      = ( ? [A3: code_natural] :
            ( P5
            = ( produc6639722614265839536atural @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_900_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_901_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_902_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_903_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% dbl_inc_simps(2)
thf(fact_904_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_905_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_906_disjoint__eq__subset__Compl,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A @ ( uminus6524753893492686040at_nat @ B ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_907_disjoint__eq__subset__Compl,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ B ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_908_disjoint__eq__subset__Compl,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ B ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_909_disjoint__eq__subset__Compl,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ B ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_910_semilattice__neutr__def,axiom,
    ( semila9081495762789891438tr_nat
    = ( ^ [F3: nat > nat > nat,Z3: nat] :
          ( ( semilattice_nat @ F3 )
          & ( comm_monoid_nat @ F3 @ Z3 ) ) ) ) ).

% semilattice_neutr_def
thf(fact_911_semilattice__neutr_Ointro,axiom,
    ! [F: nat > nat > nat,Z: nat] :
      ( ( semilattice_nat @ F )
     => ( ( comm_monoid_nat @ F @ Z )
       => ( semila9081495762789891438tr_nat @ F @ Z ) ) ) ).

% semilattice_neutr.intro
thf(fact_912_minus__assn__def,axiom,
    ( minus_minus_assn
    = ( ^ [A3: assn,B3: assn] : ( inf_inf_assn @ A3 @ ( uminus_uminus_assn @ B3 ) ) ) ) ).

% minus_assn_def
thf(fact_913_order__refl,axiom,
    ! [X: code_integer] : ( ord_le3102999989581377725nteger @ X @ X ) ).

% order_refl
thf(fact_914_order__refl,axiom,
    ! [X: assn] : ( ord_less_eq_assn @ X @ X ) ).

% order_refl
thf(fact_915_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_916_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_917_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_918_dual__order_Orefl,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_919_dual__order_Orefl,axiom,
    ! [A2: assn] : ( ord_less_eq_assn @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_920_dual__order_Orefl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_921_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_922_dual__order_Orefl,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_923_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_924_mult__zero__left,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A2 )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_left
thf(fact_925_mult__zero__left,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A2 )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_926_mult__zero__left,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_927_mult__zero__left,axiom,
    ! [A2: int] :
      ( ( times_times_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_928_mult__zero__right,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_right
thf(fact_929_mult__zero__right,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_930_mult__zero__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_931_mult__zero__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_932_mult__eq__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
      = ( ( A2 = zero_z3403309356797280102nteger )
        | ( B2 = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
thf(fact_933_mult__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_934_mult__eq__0__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_935_mult__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_936_mult__cancel__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C2 @ A2 )
        = ( times_3573771949741848930nteger @ C2 @ B2 ) )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_937_mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ( times_times_rat @ C2 @ A2 )
        = ( times_times_rat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_938_mult__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ( times_times_nat @ C2 @ A2 )
        = ( times_times_nat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_939_mult__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ( times_times_int @ C2 @ A2 )
        = ( times_times_int @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_940_mult__cancel__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ C2 )
        = ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_941_mult__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ C2 )
        = ( times_times_rat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_942_mult__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ C2 )
        = ( times_times_nat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_943_mult__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ C2 )
        = ( times_times_int @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_944_diff__self,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ A2 )
      = zero_z3403309356797280102nteger ) ).

% diff_self
thf(fact_945_diff__self,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% diff_self
thf(fact_946_diff__self,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% diff_self
thf(fact_947_diff__0__right,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% diff_0_right
thf(fact_948_diff__0__right,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_0_right
thf(fact_949_diff__0__right,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_0_right
thf(fact_950_zero__diff,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ zero_z1048942125864253310at_nat @ A2 )
      = zero_z1048942125864253310at_nat ) ).

% zero_diff
thf(fact_951_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_952_diff__zero,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% diff_zero
thf(fact_953_diff__zero,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A2 @ zero_z1048942125864253310at_nat )
      = A2 ) ).

% diff_zero
thf(fact_954_diff__zero,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_zero
thf(fact_955_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_956_diff__zero,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_zero
thf(fact_957_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ A2 )
      = zero_z3403309356797280102nteger ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_958_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A2 @ A2 )
      = zero_z1048942125864253310at_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_959_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_960_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_961_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_962_neg__le__iff__le,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% neg_le_iff_le
thf(fact_963_neg__le__iff__le,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% neg_le_iff_le
thf(fact_964_neg__le__iff__le,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% neg_le_iff_le
thf(fact_965_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_966_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_967_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_968_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_969_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_970_neg__0__equal__iff__equal,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A2 ) )
      = ( zero_zero_int = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_971_neg__0__equal__iff__equal,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A2 ) )
      = ( zero_zero_rat = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_972_neg__0__equal__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A2 ) )
      = ( zero_z3403309356797280102nteger = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_973_neg__equal__0__iff__equal,axiom,
    ! [A2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_974_neg__equal__0__iff__equal,axiom,
    ! [A2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_975_neg__equal__0__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_976_equal__neg__zero,axiom,
    ! [A2: int] :
      ( ( A2
        = ( uminus_uminus_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_977_equal__neg__zero,axiom,
    ! [A2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_978_equal__neg__zero,axiom,
    ! [A2: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_979_neg__equal__zero,axiom,
    ! [A2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = A2 )
      = ( A2 = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_980_neg__equal__zero,axiom,
    ! [A2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = A2 )
      = ( A2 = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_981_neg__equal__zero,axiom,
    ! [A2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = A2 )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_982_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( ( ord_less_eq_assn @ X @ Y )
        & ( ord_less_eq_assn @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_983_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( ( ord_less_eq_set_nat @ X @ Y )
        & ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_984_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_985_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( ord_less_eq_int @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_986_inf_Obounded__iff,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ ( inf_inf_assn @ B2 @ C2 ) )
      = ( ( ord_less_eq_assn @ A2 @ B2 )
        & ( ord_less_eq_assn @ A2 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_987_inf_Obounded__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
      = ( ( ord_less_eq_set_nat @ A2 @ B2 )
        & ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_988_inf_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
      = ( ( ord_less_eq_nat @ A2 @ B2 )
        & ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_989_inf_Obounded__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B2 @ C2 ) )
      = ( ( ord_less_eq_int @ A2 @ B2 )
        & ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_990_le__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ Z )
      = ( ( ord_le1268244103169919719at_nat @ X @ Z )
        & ( ord_le1268244103169919719at_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_991_le__sup__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( ( ord_less_eq_assn @ X @ Z )
        & ( ord_less_eq_assn @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_992_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_nat @ X @ Z )
        & ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_993_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_994_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        & ( ord_less_eq_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_995_sup_Obounded__iff,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
        & ( ord_le1268244103169919719at_nat @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_996_sup_Obounded__iff,axiom,
    ! [B2: assn,C2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_assn @ B2 @ A2 )
        & ( ord_less_eq_assn @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_997_sup_Obounded__iff,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_set_nat @ B2 @ A2 )
        & ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_998_sup_Obounded__iff,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_999_sup_Obounded__iff,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_int @ B2 @ A2 )
        & ( ord_less_eq_int @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1000_minus__diff__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( minus_minus_int @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_1001_minus__diff__eq,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( minus_minus_rat @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_1002_minus__diff__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( minus_8373710615458151222nteger @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_1003_empty__subsetI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% empty_subsetI
thf(fact_1004_empty__subsetI,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% empty_subsetI
thf(fact_1005_empty__subsetI,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% empty_subsetI
thf(fact_1006_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_1007_subset__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
thf(fact_1008_subset__empty,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1009_subset__empty,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_1010_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_1011_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1012_diff__ge__0__iff__ge,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1013_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_eq_int @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1014_neg__0__le__iff__le,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_1015_neg__0__le__iff__le,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_1016_neg__0__le__iff__le,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_1017_neg__le__0__iff__le,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_1018_neg__le__0__iff__le,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_1019_neg__le__0__iff__le,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_1020_less__eq__neg__nonpos,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_1021_less__eq__neg__nonpos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_1022_less__eq__neg__nonpos,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_1023_neg__less__eq__nonneg,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_1024_neg__less__eq__nonneg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_1025_neg__less__eq__nonneg,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_1026_mult__cancel__right2,axiom,
    ! [A2: code_integer,C2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ C2 )
        = C2 )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( A2 = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
thf(fact_1027_mult__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( ( ( times_times_rat @ A2 @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_1028_mult__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( ( ( times_times_int @ A2 @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_1029_mult__cancel__right1,axiom,
    ! [C2: code_integer,B2: code_integer] :
      ( ( C2
        = ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( B2 = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
thf(fact_1030_mult__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2
        = ( times_times_rat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_1031_mult__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2
        = ( times_times_int @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_1032_mult__cancel__left2,axiom,
    ! [C2: code_integer,A2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C2 @ A2 )
        = C2 )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( A2 = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
thf(fact_1033_mult__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( ( ( times_times_rat @ C2 @ A2 )
        = C2 )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1034_mult__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( ( ( times_times_int @ C2 @ A2 )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_1035_mult__cancel__left1,axiom,
    ! [C2: code_integer,B2: code_integer] :
      ( ( C2
        = ( times_3573771949741848930nteger @ C2 @ B2 ) )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( B2 = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
thf(fact_1036_mult__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2
        = ( times_times_rat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_1037_mult__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2
        = ( times_times_int @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_1038_diff__numeral__special_I9_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(9)
thf(fact_1039_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1040_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1041_diff__0,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ zero_zero_int @ A2 )
      = ( uminus_uminus_int @ A2 ) ) ).

% diff_0
thf(fact_1042_diff__0,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A2 )
      = ( uminus_uminus_rat @ A2 ) ) ).

% diff_0
thf(fact_1043_diff__0,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A2 )
      = ( uminus1351360451143612070nteger @ A2 ) ) ).

% diff_0
thf(fact_1044_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% diff_numeral_special(12)
thf(fact_1045_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% diff_numeral_special(12)
thf(fact_1046_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_1047_diff__eq__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C2 @ D2 ) )
     => ( ( A2 = B2 )
        = ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_1048_diff__eq__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( A2 = B2 )
        = ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_1049_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: code_integer,Z4: code_integer] : Y4 = Z4 )
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( minus_8373710615458151222nteger @ A3 @ B3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1050_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: rat,Z4: rat] : Y4 = Z4 )
    = ( ^ [A3: rat,B3: rat] :
          ( ( minus_minus_rat @ A3 @ B3 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1051_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: int,Z4: int] : Y4 = Z4 )
    = ( ^ [A3: int,B3: int] :
          ( ( minus_minus_int @ A3 @ B3 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1052_diff__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ D2 @ C2 )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C2 ) @ ( minus_minus_rat @ B2 @ D2 ) ) ) ) ).

% diff_mono
thf(fact_1053_diff__mono,axiom,
    ! [A2: code_integer,B2: code_integer,D2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ D2 @ C2 )
       => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ C2 ) @ ( minus_8373710615458151222nteger @ B2 @ D2 ) ) ) ) ).

% diff_mono
thf(fact_1054_diff__mono,axiom,
    ! [A2: int,B2: int,D2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ D2 @ C2 )
       => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).

% diff_mono
thf(fact_1055_diff__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C2 @ A2 ) @ ( minus_minus_rat @ C2 @ B2 ) ) ) ).

% diff_left_mono
thf(fact_1056_diff__left__mono,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ C2 @ A2 ) @ ( minus_8373710615458151222nteger @ C2 @ B2 ) ) ) ).

% diff_left_mono
thf(fact_1057_diff__left__mono,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A2 ) @ ( minus_minus_int @ C2 @ B2 ) ) ) ).

% diff_left_mono
thf(fact_1058_diff__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C2 ) @ ( minus_minus_rat @ B2 @ C2 ) ) ) ).

% diff_right_mono
thf(fact_1059_diff__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ C2 ) @ ( minus_8373710615458151222nteger @ B2 @ C2 ) ) ) ).

% diff_right_mono
thf(fact_1060_diff__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B2 @ C2 ) ) ) ).

% diff_right_mono
thf(fact_1061_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1062_le__iff__diff__le__0,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ zero_z3403309356797280102nteger ) ) ) ).

% le_iff_diff_le_0
thf(fact_1063_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1064_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_1065_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C2 @ D2 ) )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
        = ( ord_less_eq_rat @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1066_diff__eq__diff__less__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A2 @ B2 )
        = ( minus_8373710615458151222nteger @ C2 @ D2 ) )
     => ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
        = ( ord_le3102999989581377725nteger @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1067_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( ord_less_eq_int @ A2 @ B2 )
        = ( ord_less_eq_int @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1068_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C2 ) @ B2 )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1069_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1070_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C2 ) @ B2 )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1071_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
      = ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
thf(fact_1072_zero__reorient,axiom,
    ! [X: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat = X )
      = ( X = zero_z1048942125864253310at_nat ) ) ).

% zero_reorient
thf(fact_1073_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_1074_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1075_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1076_nle__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 ) )
      = ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_1077_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_1078_nle__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
      = ( ( ord_less_eq_int @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_1079_le__cases3,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ X @ Y )
       => ~ ( ord_le3102999989581377725nteger @ Y @ Z ) )
     => ( ( ( ord_le3102999989581377725nteger @ Y @ X )
         => ~ ( ord_le3102999989581377725nteger @ X @ Z ) )
       => ( ( ( ord_le3102999989581377725nteger @ X @ Z )
           => ~ ( ord_le3102999989581377725nteger @ Z @ Y ) )
         => ( ( ( ord_le3102999989581377725nteger @ Z @ Y )
             => ~ ( ord_le3102999989581377725nteger @ Y @ X ) )
           => ( ( ( ord_le3102999989581377725nteger @ Y @ Z )
               => ~ ( ord_le3102999989581377725nteger @ Z @ X ) )
             => ~ ( ( ord_le3102999989581377725nteger @ Z @ X )
                 => ~ ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1080_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1081_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1082_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: code_integer,Z4: code_integer] : Y4 = Z4 )
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( ( ord_le3102999989581377725nteger @ X2 @ Y2 )
          & ( ord_le3102999989581377725nteger @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1083_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: assn,Z4: assn] : Y4 = Z4 )
    = ( ^ [X2: assn,Y2: assn] :
          ( ( ord_less_eq_assn @ X2 @ Y2 )
          & ( ord_less_eq_assn @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1084_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z4: set_nat] : Y4 = Z4 )
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1085_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z4: nat] : Y4 = Z4 )
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1086_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z4: int] : Y4 = Z4 )
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1087_ord__eq__le__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( A2 = B2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ord_le3102999989581377725nteger @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1088_ord__eq__le__trans,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ord_less_eq_assn @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1089_ord__eq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1090_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1091_ord__eq__le__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1092_ord__le__eq__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le3102999989581377725nteger @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1093_ord__le__eq__trans,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_assn @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1094_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1095_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1096_ord__le__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1097_order__antisym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_le3102999989581377725nteger @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1098_order__antisym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1099_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1100_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_1101_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_1102_order_Otrans,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ord_le3102999989581377725nteger @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_1103_order_Otrans,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ord_less_eq_assn @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_1104_order_Otrans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_1105_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_1106_order_Otrans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_1107_order__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_le3102999989581377725nteger @ Y @ Z )
       => ( ord_le3102999989581377725nteger @ X @ Z ) ) ) ).

% order_trans
thf(fact_1108_order__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ Z )
       => ( ord_less_eq_assn @ X @ Z ) ) ) ).

% order_trans
thf(fact_1109_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1110_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1111_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_1112_linorder__wlog,axiom,
    ! [P: code_integer > code_integer > $o,A2: code_integer,B2: code_integer] :
      ( ! [A5: code_integer,B5: code_integer] :
          ( ( ord_le3102999989581377725nteger @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: code_integer,B5: code_integer] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1113_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1114_linorder__wlog,axiom,
    ! [P: int > int > $o,A2: int,B2: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int,B5: int] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1115_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: code_integer,Z4: code_integer] : Y4 = Z4 )
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
          & ( ord_le3102999989581377725nteger @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1116_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: assn,Z4: assn] : Y4 = Z4 )
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ B3 @ A3 )
          & ( ord_less_eq_assn @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1117_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z4: set_nat] : Y4 = Z4 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1118_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z4: nat] : Y4 = Z4 )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1119_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: int,Z4: int] : Y4 = Z4 )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1120_dual__order_Oantisym,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1121_dual__order_Oantisym,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1122_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1123_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1124_dual__order_Oantisym,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_1125_dual__order_Otrans,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ B2 )
       => ( ord_le3102999989581377725nteger @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1126_dual__order_Otrans,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ C2 @ B2 )
       => ( ord_less_eq_assn @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1127_dual__order_Otrans,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C2 @ B2 )
       => ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1128_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1129_dual__order_Otrans,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ B2 )
       => ( ord_less_eq_int @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1130_antisym,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1131_antisym,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1132_antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1133_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1134_antisym,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1135_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: code_integer,Z4: code_integer] : Y4 = Z4 )
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
          & ( ord_le3102999989581377725nteger @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1136_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: assn,Z4: assn] : Y4 = Z4 )
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ A3 @ B3 )
          & ( ord_less_eq_assn @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1137_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z4: set_nat] : Y4 = Z4 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1138_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z4: nat] : Y4 = Z4 )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1139_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z4: int] : Y4 = Z4 )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1140_order__subst1,axiom,
    ! [A2: code_integer,F: code_integer > code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1141_order__subst1,axiom,
    ! [A2: code_integer,F: assn > code_integer,B2: assn,C2: assn] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1142_order__subst1,axiom,
    ! [A2: code_integer,F: nat > code_integer,B2: nat,C2: nat] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1143_order__subst1,axiom,
    ! [A2: code_integer,F: int > code_integer,B2: int,C2: int] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1144_order__subst1,axiom,
    ! [A2: assn,F: code_integer > assn,B2: code_integer,C2: code_integer] :
      ( ( ord_less_eq_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1145_order__subst1,axiom,
    ! [A2: assn,F: assn > assn,B2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1146_order__subst1,axiom,
    ! [A2: assn,F: nat > assn,B2: nat,C2: nat] :
      ( ( ord_less_eq_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1147_order__subst1,axiom,
    ! [A2: assn,F: int > assn,B2: int,C2: int] :
      ( ( ord_less_eq_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1148_order__subst1,axiom,
    ! [A2: nat,F: code_integer > nat,B2: code_integer,C2: code_integer] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1149_order__subst1,axiom,
    ! [A2: nat,F: assn > nat,B2: assn,C2: assn] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1150_order__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1151_order__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > assn,C2: assn] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1152_order__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > nat,C2: nat] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1153_order__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > int,C2: int] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1154_order__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > code_integer,C2: code_integer] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1155_order__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1156_order__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1157_order__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > int,C2: int] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1158_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > code_integer,C2: code_integer] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1159_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1160_order__eq__refl,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X = Y )
     => ( ord_le3102999989581377725nteger @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1161_order__eq__refl,axiom,
    ! [X: assn,Y: assn] :
      ( ( X = Y )
     => ( ord_less_eq_assn @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1162_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1163_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1164_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1165_linorder__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
      | ( ord_le3102999989581377725nteger @ Y @ X ) ) ).

% linorder_linear
thf(fact_1166_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1167_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_1168_ord__eq__le__subst,axiom,
    ! [A2: code_integer,F: code_integer > code_integer,B2: code_integer,C2: code_integer] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1169_ord__eq__le__subst,axiom,
    ! [A2: assn,F: code_integer > assn,B2: code_integer,C2: code_integer] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1170_ord__eq__le__subst,axiom,
    ! [A2: nat,F: code_integer > nat,B2: code_integer,C2: code_integer] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1171_ord__eq__le__subst,axiom,
    ! [A2: int,F: code_integer > int,B2: code_integer,C2: code_integer] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1172_ord__eq__le__subst,axiom,
    ! [A2: code_integer,F: assn > code_integer,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1173_ord__eq__le__subst,axiom,
    ! [A2: assn,F: assn > assn,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1174_ord__eq__le__subst,axiom,
    ! [A2: nat,F: assn > nat,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1175_ord__eq__le__subst,axiom,
    ! [A2: int,F: assn > int,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1176_ord__eq__le__subst,axiom,
    ! [A2: code_integer,F: nat > code_integer,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1177_ord__eq__le__subst,axiom,
    ! [A2: assn,F: nat > assn,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1178_ord__le__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1179_ord__le__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > assn,C2: assn] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1180_ord__le__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > nat,C2: nat] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1181_ord__le__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > int,C2: int] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1182_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > code_integer,C2: code_integer] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1183_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1184_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1185_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > int,C2: int] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1186_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > code_integer,C2: code_integer] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le3102999989581377725nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1187_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1188_linorder__le__cases,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ord_le3102999989581377725nteger @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_1189_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_1190_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_1191_order__antisym__conv,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1192_order__antisym__conv,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_less_eq_assn @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1193_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_mult__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1202_mult__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1203_mult__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1204_mult__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1205_mult__mono_H,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1206_mult__mono_H,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1207_mult__mono_H,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1208_mult__mono_H,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1209_zero__le__square,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_1210_zero__le__square,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_1211_zero__le__square,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_1212_split__mult__pos__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_1213_split__mult__pos__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) )
        | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_1214_split__mult__pos__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_1215_mult__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1216_mult__left__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1217_mult__left__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1218_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1219_mult__nonpos__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1220_mult__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1221_mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_1222_mult__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_1223_mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_1224_mult__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_1225_mult__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1226_mult__right__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1227_mult__right__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1228_mult__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_1229_mult__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_1230_mult__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_1231_mult__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_1232_mult__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_1233_mult__le__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_1234_mult__le__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_1235_split__mult__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_1236_split__mult__neg__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ).

% split_mult_neg_le
thf(fact_1237_split__mult__neg__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
          & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_1238_split__mult__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_1239_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1240_mult__nonneg__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1241_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1242_mult__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1243_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1244_mult__nonneg__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1245_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1246_mult__nonneg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1247_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1248_mult__nonpos__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1249_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1250_mult__nonpos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1251_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1252_mult__nonneg__nonpos2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1253_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1254_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1255_zero__le__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1256_zero__le__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) )
        | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1257_zero__le__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1258_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1259_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1260_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1261_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1262_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1263_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_1264_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1265_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_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_1273_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_1274_mult__left__le,axiom,
    ! [C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_1275_mult__left__le,axiom,
    ! [C2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_1276_mult__left__le,axiom,
    ! [C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_1277_mult__left__le,axiom,
    ! [C2: int,A2: int] :
      ( ( ord_less_eq_int @ C2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_1278_mult__le__one,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_eq_rat @ B2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_1279_mult__le__one,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ one_one_Code_integer ) ) ) ) ).

% mult_le_one
thf(fact_1280_mult__le__one,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_1281_mult__le__one,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_1282_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_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_1290_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_1291_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_1292_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_1293_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_1294_left__diff__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1295_left__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1296_left__diff__distrib,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( minus_minus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1297_right__diff__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1298_right__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1299_right__diff__distrib,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1300_left__diff__distrib_H,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B2 @ C2 ) @ A2 )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_1301_left__diff__distrib_H,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C2 ) @ A2 )
      = ( minus_minus_rat @ ( times_times_rat @ B2 @ A2 ) @ ( times_times_rat @ C2 @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_1302_left__diff__distrib_H,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C2 ) @ A2 )
      = ( minus_minus_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C2 @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_1303_left__diff__distrib_H,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( times_times_int @ ( minus_minus_int @ B2 @ C2 ) @ A2 )
      = ( minus_minus_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C2 @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_1304_right__diff__distrib_H,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1305_right__diff__distrib_H,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1306_right__diff__distrib_H,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ A2 @ ( minus_minus_nat @ B2 @ C2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1307_right__diff__distrib_H,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1308_minus__diff__commute,axiom,
    ! [B2: int,A2: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A2 )
      = ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_1309_minus__diff__commute,axiom,
    ! [B2: rat,A2: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B2 ) @ A2 )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_1310_minus__diff__commute,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A2 )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_1311_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_1312_le__numeral__extra_I4_J,axiom,
    ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).

% le_numeral_extra(4)
thf(fact_1313_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_1314_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_1315_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
     => ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1316_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
     => ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_1317_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_1318_bot_Oextremum__uniqueI,axiom,
    ! [A2: assn] :
      ( ( ord_less_eq_assn @ A2 @ bot_bot_assn )
     => ( A2 = bot_bot_assn ) ) ).

% bot.extremum_uniqueI
thf(fact_1319_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1320_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1321_bot_Oextremum__unique,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
thf(fact_1322_bot_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_1323_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_1324_bot_Oextremum__unique,axiom,
    ! [A2: assn] :
      ( ( ord_less_eq_assn @ A2 @ bot_bot_assn )
      = ( A2 = bot_bot_assn ) ) ).

% bot.extremum_unique
thf(fact_1325_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_1326_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1327_bot_Oextremum,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).

% bot.extremum
thf(fact_1328_bot_Oextremum,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% bot.extremum
thf(fact_1329_bot_Oextremum,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% bot.extremum
thf(fact_1330_bot_Oextremum,axiom,
    ! [A2: assn] : ( ord_less_eq_assn @ bot_bot_assn @ A2 ) ).

% bot.extremum
thf(fact_1331_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_1332_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_1333_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1334_subrelI,axiom,
    ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
      ( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
          ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R2 )
         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le653643898420964396nteger @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1335_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_1336_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_1337_subrelI,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R2 )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le8725513860283290265nt_int @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1338_subrelI,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R2 )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S2 ) )
     => ( ord_le135402666524580259nt_int @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1339_le__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ ( uminus_uminus_rat @ A2 ) ) ) ).

% le_minus_iff
thf(fact_1340_le__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( ord_le3102999989581377725nteger @ B2 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% le_minus_iff
thf(fact_1341_le__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).

% le_minus_iff
thf(fact_1342_minus__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_1343_minus__le__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_1344_minus__le__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_1345_le__imp__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_1346_le__imp__neg__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_1347_le__imp__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_1348_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_1349_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).

% compl_mono
thf(fact_1350_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_1351_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
     => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% compl_le_swap1
thf(fact_1352_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_1353_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_1354_top_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
     => ( A2 = top_top_set_o ) ) ).

% top.extremum_uniqueI
thf(fact_1355_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Product_unit] :
      ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
     => ( A2 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_uniqueI
thf(fact_1356_top_Oextremum__uniqueI,axiom,
    ! [A2: assn] :
      ( ( ord_less_eq_assn @ top_top_assn @ A2 )
     => ( A2 = top_top_assn ) ) ).

% top.extremum_uniqueI
thf(fact_1357_top_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
     => ( A2 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_1358_top_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
      = ( A2 = top_top_set_o ) ) ).

% top.extremum_unique
thf(fact_1359_top_Oextremum__unique,axiom,
    ! [A2: set_Product_unit] :
      ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
      = ( A2 = top_to1996260823553986621t_unit ) ) ).

% top.extremum_unique
thf(fact_1360_top_Oextremum__unique,axiom,
    ! [A2: assn] :
      ( ( ord_less_eq_assn @ top_top_assn @ A2 )
      = ( A2 = top_top_assn ) ) ).

% top.extremum_unique
thf(fact_1361_top_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
      = ( A2 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_1362_top__greatest,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).

% top_greatest
thf(fact_1363_top__greatest,axiom,
    ! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).

% top_greatest
thf(fact_1364_top__greatest,axiom,
    ! [A2: assn] : ( ord_less_eq_assn @ A2 @ top_top_assn ) ).

% top_greatest
thf(fact_1365_top__greatest,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% top_greatest
thf(fact_1366_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_1367_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_1368_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_1369_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_1370_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_1371_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_1372_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_1373_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_1374_inf__le1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1375_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_1376_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1377_inf__le1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1378_inf__le2,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1379_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_1380_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1381_inf__le2,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1382_le__infE,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_assn @ X @ A2 )
         => ~ ( ord_less_eq_assn @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1383_le__infE,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_set_nat @ X @ A2 )
         => ~ ( ord_less_eq_set_nat @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1384_le__infE,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_nat @ X @ A2 )
         => ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1385_le__infE,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_int @ X @ A2 )
         => ~ ( ord_less_eq_int @ X @ B2 ) ) ) ).

% le_infE
thf(fact_1386_le__infI,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ X @ A2 )
     => ( ( ord_less_eq_assn @ X @ B2 )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1387_le__infI,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ X @ B2 )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1388_le__infI,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ( ord_less_eq_nat @ X @ B2 )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1389_le__infI,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ( ord_less_eq_int @ X @ B2 )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_1390_inf__mono,axiom,
    ! [A2: assn,C2: assn,B2: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A2 @ C2 )
     => ( ( ord_less_eq_assn @ B2 @ D2 )
       => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ ( inf_inf_assn @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1391_inf__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C2 )
     => ( ( ord_less_eq_set_nat @ B2 @ D2 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1392_inf__mono,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1393_inf__mono,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ C2 )
     => ( ( ord_less_eq_int @ B2 @ D2 )
       => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ ( inf_inf_int @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1394_le__infI1,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1395_le__infI1,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1396_le__infI1,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1397_le__infI1,axiom,
    ! [A2: int,X: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_1398_le__infI2,axiom,
    ! [B2: assn,X: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1399_le__infI2,axiom,
    ! [B2: set_nat,X: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1400_le__infI2,axiom,
    ! [B2: nat,X: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1401_le__infI2,axiom,
    ! [B2: int,X: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_1402_inf_OorderE,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( A2
        = ( inf_inf_assn @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1403_inf_OorderE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( A2
        = ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1404_inf_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( A2
        = ( inf_inf_nat @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1405_inf_OorderE,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( A2
        = ( inf_inf_int @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_1406_inf_OorderI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2
        = ( inf_inf_assn @ A2 @ B2 ) )
     => ( ord_less_eq_assn @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1407_inf_OorderI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2
        = ( inf_inf_set_nat @ A2 @ B2 ) )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1408_inf_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( inf_inf_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1409_inf_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( inf_inf_int @ A2 @ B2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_1410_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_1411_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_1412_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_1413_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_1414_le__iff__inf,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y2: assn] :
          ( ( inf_inf_assn @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1415_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( inf_inf_set_nat @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1416_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( inf_inf_nat @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1417_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y2: int] :
          ( ( inf_inf_int @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1418_inf_Oabsorb1,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( inf_inf_assn @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1419_inf_Oabsorb1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1420_inf_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1421_inf_Oabsorb1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( inf_inf_int @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_1422_inf_Oabsorb2,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( inf_inf_assn @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1423_inf_Oabsorb2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1424_inf_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1425_inf_Oabsorb2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( inf_inf_int @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_1426_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( inf_inf_assn @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1427_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_1428_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1429_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( inf_inf_int @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1430_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( inf_inf_assn @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1431_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_1432_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1433_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( inf_inf_int @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1434_inf_OboundedE,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ ( inf_inf_assn @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_assn @ A2 @ B2 )
         => ~ ( ord_less_eq_assn @ A2 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1435_inf_OboundedE,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1436_inf_OboundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1437_inf_OboundedE,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_int @ A2 @ B2 )
         => ~ ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_1438_inf_OboundedI,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ A2 @ C2 )
       => ( ord_less_eq_assn @ A2 @ ( inf_inf_assn @ B2 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1439_inf_OboundedI,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ C2 )
       => ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1440_inf_OboundedI,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ C2 )
       => ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1441_inf_OboundedI,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ A2 @ C2 )
       => ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B2 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_1442_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Z )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1443_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Z )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1444_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1445_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Z )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1446_inf_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( A3
          = ( inf_inf_assn @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1447_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( A3
          = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1448_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( A3
          = ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1449_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( A3
          = ( inf_inf_int @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_1450_inf_Ocobounded1,axiom,
    ! [A2: assn,B2: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1451_inf_Ocobounded1,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1452_inf_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1453_inf_Ocobounded1,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_1454_inf_Ocobounded2,axiom,
    ! [A2: assn,B2: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1455_inf_Ocobounded2,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1456_inf_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1457_inf_Ocobounded2,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_1458_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( inf_inf_assn @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1459_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( inf_inf_set_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1460_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( inf_inf_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1461_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( inf_inf_int @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_1462_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( inf_inf_assn @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1463_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( inf_inf_set_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1464_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( inf_inf_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1465_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( inf_inf_int @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_1466_inf_OcoboundedI1,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ C2 )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1467_inf_OcoboundedI1,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1468_inf_OcoboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1469_inf_OcoboundedI1,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ C2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_1470_inf_OcoboundedI2,axiom,
    ! [B2: assn,C2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ C2 )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1471_inf_OcoboundedI2,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1472_inf_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1473_inf_OcoboundedI2,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ C2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_1474_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_1475_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1476_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_1477_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_1478_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_1479_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_1480_inf__sup__ord_I3_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1481_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_1482_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_1483_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_1484_le__supE,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le1268244103169919719at_nat @ A2 @ X )
         => ~ ( ord_le1268244103169919719at_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_1485_le__supE,axiom,
    ! [A2: assn,B2: assn,X: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_assn @ A2 @ X )
         => ~ ( ord_less_eq_assn @ B2 @ X ) ) ) ).

% le_supE
thf(fact_1486_le__supE,axiom,
    ! [A2: set_nat,B2: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ X )
         => ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_1487_le__supE,axiom,
    ! [A2: nat,B2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_nat @ A2 @ X )
         => ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_1488_le__supE,axiom,
    ! [A2: int,B2: int,X: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_int @ A2 @ X )
         => ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).

% le_supE
thf(fact_1489_le__supI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ X )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ X )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_1490_le__supI,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ X )
     => ( ( ord_less_eq_assn @ B2 @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_1491_le__supI,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ X )
     => ( ( ord_less_eq_set_nat @ B2 @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_1492_le__supI,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ( ord_less_eq_nat @ B2 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_1493_le__supI,axiom,
    ! [A2: int,X: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ X )
     => ( ( ord_less_eq_int @ B2 @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_1494_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1495_sup__ge1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge1
thf(fact_1496_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_1497_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1498_sup__ge1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_1499_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1500_sup__ge2,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge2
thf(fact_1501_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_1502_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1503_sup__ge2,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_1504_le__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X @ A2 )
     => ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_1505_le__supI1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ X @ A2 )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_1506_le__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A2 )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_1507_le__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_1508_le__supI1,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_1509_le__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X @ B2 )
     => ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_1510_le__supI2,axiom,
    ! [X: assn,B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ X @ B2 )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_1511_le__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ B2 )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_1512_le__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ X @ B2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_1513_le__supI2,axiom,
    ! [X: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ X @ B2 )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_1514_sup_Omono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ D2 @ B2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C2 @ D2 ) @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1515_sup_Omono,axiom,
    ! [C2: assn,A2: assn,D2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ C2 @ A2 )
     => ( ( ord_less_eq_assn @ D2 @ B2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ C2 @ D2 ) @ ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1516_sup_Omono,axiom,
    ! [C2: set_nat,A2: set_nat,D2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A2 )
     => ( ( ord_less_eq_set_nat @ D2 @ B2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D2 ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1517_sup_Omono,axiom,
    ! [C2: nat,A2: nat,D2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ A2 )
     => ( ( ord_less_eq_nat @ D2 @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1518_sup_Omono,axiom,
    ! [C2: int,A2: int,D2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ A2 )
     => ( ( ord_less_eq_int @ D2 @ B2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ C2 @ D2 ) @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1519_sup__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ D2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) @ ( sup_su5525570899277871387at_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1520_sup__mono,axiom,
    ! [A2: assn,C2: assn,B2: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A2 @ C2 )
     => ( ( ord_less_eq_assn @ B2 @ D2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ ( sup_sup_assn @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1521_sup__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C2 )
     => ( ( ord_less_eq_set_nat @ B2 @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1522_sup__mono,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1523_sup__mono,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ C2 )
     => ( ( ord_less_eq_int @ B2 @ D2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ ( sup_sup_int @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1524_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ Y @ X )
     => ( ( ord_le1268244103169919719at_nat @ Z @ X )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1525_sup__least,axiom,
    ! [Y: assn,X: assn,Z: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_less_eq_assn @ Z @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1526_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ Z @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1527_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1528_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ Z @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1529_le__iff__sup,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1530_le__iff__sup,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y2: assn] :
          ( ( sup_sup_assn @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1531_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( sup_sup_set_nat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1532_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( sup_sup_nat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1533_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y2: int] :
          ( ( sup_sup_int @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1534_sup_OorderE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
     => ( A2
        = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1535_sup_OorderE,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( A2
        = ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1536_sup_OorderE,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1537_sup_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1538_sup_OorderE,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( A2
        = ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1539_sup_OorderI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( A2
        = ( sup_su5525570899277871387at_nat @ A2 @ B2 ) )
     => ( ord_le1268244103169919719at_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_1540_sup_OorderI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2
        = ( sup_sup_assn @ A2 @ B2 ) )
     => ( ord_less_eq_assn @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_1541_sup_OorderI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2
        = ( sup_sup_set_nat @ A2 @ B2 ) )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_1542_sup_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( sup_sup_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_1543_sup_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( sup_sup_int @ A2 @ B2 ) )
     => ( ord_less_eq_int @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_1544_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_1545_sup__unique,axiom,
    ! [F: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ X3 @ ( F @ X3 @ Y3 ) )
     => ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ Y3 @ ( F @ X3 @ Y3 ) )
       => ( ! [X3: assn,Y3: assn,Z2: assn] :
              ( ( ord_less_eq_assn @ Y3 @ X3 )
             => ( ( ord_less_eq_assn @ Z2 @ X3 )
               => ( ord_less_eq_assn @ ( F @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_assn @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1546_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_1547_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_1548_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_1549_sup_Oabsorb1,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1550_sup_Oabsorb1,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1551_sup_Oabsorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1552_sup_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1553_sup_Oabsorb1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1554_sup_Oabsorb2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1555_sup_Oabsorb2,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1556_sup_Oabsorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1557_sup_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1558_sup_Oabsorb2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1559_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_1560_sup__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( sup_sup_assn @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1561_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_1562_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1563_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( sup_sup_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1564_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_1565_sup__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( sup_sup_assn @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1566_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_1567_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1568_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( sup_sup_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1569_sup_OboundedE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
         => ~ ( ord_le1268244103169919719at_nat @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_1570_sup_OboundedE,axiom,
    ! [B2: assn,C2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_assn @ B2 @ A2 )
         => ~ ( ord_less_eq_assn @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_1571_sup_OboundedE,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_1572_sup_OboundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_1573_sup_OboundedE,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_int @ B2 @ A2 )
         => ~ ( ord_less_eq_int @ C2 @ A2 ) ) ) ).

% sup.boundedE
thf(fact_1574_sup_OboundedI,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B2 @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ C2 @ A2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_1575_sup_OboundedI,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ C2 @ A2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_1576_sup_OboundedI,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C2 @ A2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_1577_sup_OboundedI,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ A2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_1578_sup_OboundedI,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ A2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C2 ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_1579_sup_Oorder__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
          ( A3
          = ( sup_su5525570899277871387at_nat @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_1580_sup_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [B3: assn,A3: assn] :
          ( A3
          = ( sup_sup_assn @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_1581_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( A3
          = ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_1582_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( A3
          = ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_1583_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( A3
          = ( sup_sup_int @ A3 @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_1584_sup_Ocobounded1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_1585_sup_Ocobounded1,axiom,
    ! [A2: assn,B2: assn] : ( ord_less_eq_assn @ A2 @ ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_1586_sup_Ocobounded1,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_1587_sup_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_1588_sup_Ocobounded1,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_1589_sup_Ocobounded2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_1590_sup_Ocobounded2,axiom,
    ! [B2: assn,A2: assn] : ( ord_less_eq_assn @ B2 @ ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_1591_sup_Ocobounded2,axiom,
    ! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_1592_sup_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_1593_sup_Ocobounded2,axiom,
    ! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_1594_sup_Oabsorb__iff1,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_1595_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( sup_sup_assn @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_1596_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( sup_sup_set_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_1597_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( sup_sup_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_1598_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( sup_sup_int @ A3 @ B3 )
          = A3 ) ) ) ).

% sup.absorb_iff1
thf(fact_1599_sup_Oabsorb__iff2,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_1600_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( sup_sup_assn @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_1601_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( sup_sup_set_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_1602_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( sup_sup_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_1603_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( sup_sup_int @ A3 @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_1604_sup_OcoboundedI1,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ A2 )
     => ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_1605_sup_OcoboundedI1,axiom,
    ! [C2: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ C2 @ A2 )
     => ( ord_less_eq_assn @ C2 @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_1606_sup_OcoboundedI1,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A2 )
     => ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_1607_sup_OcoboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ A2 )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_1608_sup_OcoboundedI1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ A2 )
     => ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_1609_sup_OcoboundedI2,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ B2 )
     => ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_1610_sup_OcoboundedI2,axiom,
    ! [C2: assn,B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ C2 @ B2 )
     => ( ord_less_eq_assn @ C2 @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_1611_sup_OcoboundedI2,axiom,
    ! [C2: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ B2 )
     => ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_1612_sup_OcoboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C2 @ B2 )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_1613_sup_OcoboundedI2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ C2 @ B2 )
     => ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_1614_mult__not__zero,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ B2 )
       != zero_z3403309356797280102nteger )
     => ( ( A2 != zero_z3403309356797280102nteger )
        & ( B2 != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
thf(fact_1615_mult__not__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
       != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_1616_mult__not__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
       != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_1617_mult__not__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
       != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_1618_divisors__zero,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
     => ( ( A2 = zero_z3403309356797280102nteger )
        | ( B2 = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
thf(fact_1619_divisors__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
        = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_1620_divisors__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
        = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_1621_divisors__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
        = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_1622_no__zero__divisors,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( B2 != zero_z3403309356797280102nteger )
       => ( ( times_3573771949741848930nteger @ A2 @ B2 )
         != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
thf(fact_1623_no__zero__divisors,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( times_times_rat @ A2 @ B2 )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_1624_no__zero__divisors,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( ( times_times_nat @ A2 @ B2 )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_1625_no__zero__divisors,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( ( times_times_int @ A2 @ B2 )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_1626_mult__left__cancel,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( C2 != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ C2 @ A2 )
          = ( times_3573771949741848930nteger @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1627_mult__left__cancel,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ C2 @ A2 )
          = ( times_times_rat @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1628_mult__left__cancel,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ C2 @ A2 )
          = ( times_times_nat @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1629_mult__left__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( ( times_times_int @ C2 @ A2 )
          = ( times_times_int @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1630_mult__right__cancel,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( C2 != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ A2 @ C2 )
          = ( times_3573771949741848930nteger @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1631_mult__right__cancel,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C2 )
          = ( times_times_rat @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1632_mult__right__cancel,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ A2 @ C2 )
          = ( times_times_nat @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1633_mult__right__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( ( times_times_int @ A2 @ C2 )
          = ( times_times_int @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1634_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
thf(fact_1635_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1636_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1637_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1638_mult_Oabel__semigroup__axioms,axiom,
    abel_s8772020223019846471nteger @ times_3573771949741848930nteger ).

% mult.abel_semigroup_axioms
thf(fact_1639_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn @ times_times_assn ).

% mult.abel_semigroup_axioms
thf(fact_1640_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_rat @ times_times_rat ).

% mult.abel_semigroup_axioms
thf(fact_1641_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_nat @ times_times_nat ).

% mult.abel_semigroup_axioms
thf(fact_1642_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_int @ times_times_int ).

% mult.abel_semigroup_axioms
thf(fact_1643_inf_Oabel__semigroup__axioms,axiom,
    abel_s3907380067910933104et_nat @ inf_inf_set_nat ).

% inf.abel_semigroup_axioms
thf(fact_1644_inf_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn @ inf_inf_assn ).

% inf.abel_semigroup_axioms
thf(fact_1645_sup_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn @ sup_sup_assn ).

% sup.abel_semigroup_axioms
thf(fact_1646_sup_Oabel__semigroup__axioms,axiom,
    abel_s3907380067910933104et_nat @ sup_sup_set_nat ).

% sup.abel_semigroup_axioms
thf(fact_1647_sup_Oabel__semigroup__axioms,axiom,
    abel_s6408242399240741597at_nat @ sup_su5525570899277871387at_nat ).

% sup.abel_semigroup_axioms
thf(fact_1648_relH__subset,axiom,
    ! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( relH @ Bs @ H2 @ H5 )
     => ( ( ord_less_eq_set_nat @ As @ Bs )
       => ( relH @ As @ H2 @ H5 ) ) ) ).

% relH_subset
thf(fact_1649_inf_Osemilattice__axioms,axiom,
    semilattice_set_nat @ inf_inf_set_nat ).

% inf.semilattice_axioms
thf(fact_1650_inf_Osemilattice__axioms,axiom,
    semilattice_assn @ inf_inf_assn ).

% inf.semilattice_axioms
thf(fact_1651_sup_Osemilattice__axioms,axiom,
    semilattice_assn @ sup_sup_assn ).

% sup.semilattice_axioms
thf(fact_1652_sup_Osemilattice__axioms,axiom,
    semilattice_set_nat @ sup_sup_set_nat ).

% sup.semilattice_axioms
thf(fact_1653_sup_Osemilattice__axioms,axiom,
    semila366160313143161959at_nat @ sup_su5525570899277871387at_nat ).

% sup.semilattice_axioms
thf(fact_1654_semilattice__neutr_Oaxioms_I1_J,axiom,
    ! [F: nat > nat > nat,Z: nat] :
      ( ( semila9081495762789891438tr_nat @ F @ Z )
     => ( semilattice_nat @ F ) ) ).

% semilattice_neutr.axioms(1)
thf(fact_1655_diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ) ).

% diff_eq
thf(fact_1656_diff__eq,axiom,
    ( minus_minus_assn
    = ( ^ [X2: assn,Y2: assn] : ( inf_inf_assn @ X2 @ ( uminus_uminus_assn @ Y2 ) ) ) ) ).

% diff_eq
thf(fact_1657_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_1658_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_1659_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_1660_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_1661_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_1662_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_1663_distrib__sup__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X @ ( inf_in7913087082777306421at_nat @ Y @ Z ) ) @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) @ ( sup_su5525570899277871387at_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1664_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) @ ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1665_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1666_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1667_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1668_distrib__inf__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ ( inf_in7913087082777306421at_nat @ X @ Z ) ) @ ( inf_in7913087082777306421at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1669_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) @ ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1670_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1671_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1672_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1673_disjoint__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B2 @ B4 )
       => ( ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
            = bot_bo2099793752762293965at_nat )
         => ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
            = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
thf(fact_1674_disjoint__mono,axiom,
    ! [A2: set_o,A4: set_o,B2: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ A4 )
     => ( ( ord_less_eq_set_o @ B2 @ B4 )
       => ( ( ( inf_inf_set_o @ A4 @ B4 )
            = bot_bot_set_o )
         => ( ( inf_inf_set_o @ A2 @ B2 )
            = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
thf(fact_1675_disjoint__mono,axiom,
    ! [A2: set_int,A4: set_int,B2: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ A4 )
     => ( ( ord_less_eq_set_int @ B2 @ B4 )
       => ( ( ( inf_inf_set_int @ A4 @ B4 )
            = bot_bot_set_int )
         => ( ( inf_inf_set_int @ A2 @ B2 )
            = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
thf(fact_1676_disjoint__mono,axiom,
    ! [A2: set_nat,A4: set_nat,B2: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ A4 )
     => ( ( ord_less_eq_set_nat @ B2 @ B4 )
       => ( ( ( inf_inf_set_nat @ A4 @ B4 )
            = bot_bot_set_nat )
         => ( ( inf_inf_set_nat @ A2 @ B2 )
            = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
thf(fact_1677_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_1678_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_1679_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_1680_subset__Compl__self__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( uminus6524753893492686040at_nat @ A ) )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
thf(fact_1681_subset__Compl__self__eq,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_1682_subset__Compl__self__eq,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_1683_subset__Compl__self__eq,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_1684_in__range__subset,axiom,
    ! [As: set_nat,As2: set_nat,H2: heap_e7401611519738050253t_unit] :
      ( ( ord_less_eq_set_nat @ As @ As2 )
     => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).

% in_range_subset
thf(fact_1685_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_1686_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_1687_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_1688_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_1689_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_1690_sup__shunt,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ X @ Y )
        = top_to6833984726390702231at_nat )
      = ( ord_le1268244103169919719at_nat @ ( uminus935396558254630718at_nat @ X ) @ Y ) ) ).

% sup_shunt
thf(fact_1691_sup__shunt,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( sup_sup_set_o @ X @ Y )
        = top_top_set_o )
      = ( ord_less_eq_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) ) ).

% sup_shunt
thf(fact_1692_sup__shunt,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( sup_su793286257634532545t_unit @ X @ Y )
        = top_to1996260823553986621t_unit )
      = ( ord_le3507040750410214029t_unit @ ( uminus5944136376168626660t_unit @ X ) @ Y ) ) ).

% sup_shunt
thf(fact_1693_sup__shunt,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( sup_sup_assn @ X @ Y )
        = top_top_assn )
      = ( ord_less_eq_assn @ ( uminus_uminus_assn @ X ) @ Y ) ) ).

% sup_shunt
thf(fact_1694_sup__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( sup_sup_set_nat @ X @ Y )
        = top_top_set_nat )
      = ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).

% sup_shunt
thf(fact_1695_sup__neg__inf,axiom,
    ! [P5: set_Pr4329608150637261639at_nat,Q5: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ P5 @ ( sup_su5525570899277871387at_nat @ Q5 @ R2 ) )
      = ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ P5 @ ( uminus935396558254630718at_nat @ Q5 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_1696_sup__neg__inf,axiom,
    ! [P5: assn,Q5: assn,R2: assn] :
      ( ( ord_less_eq_assn @ P5 @ ( sup_sup_assn @ Q5 @ R2 ) )
      = ( ord_less_eq_assn @ ( inf_inf_assn @ P5 @ ( uminus_uminus_assn @ Q5 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_1697_sup__neg__inf,axiom,
    ! [P5: set_nat,Q5: set_nat,R2: set_nat] :
      ( ( ord_less_eq_set_nat @ P5 @ ( sup_sup_set_nat @ Q5 @ R2 ) )
      = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P5 @ ( uminus5710092332889474511et_nat @ Q5 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_1698_shunt2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X @ ( uminus935396558254630718at_nat @ Y ) ) @ Z )
      = ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_1699_shunt2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( inf_inf_assn @ X @ ( uminus_uminus_assn @ Y ) ) @ Z )
      = ( ord_less_eq_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% shunt2
thf(fact_1700_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) @ Z )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_1701_shunt1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X @ Y ) @ Z )
      = ( ord_le1268244103169919719at_nat @ X @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_1702_shunt1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( ord_less_eq_assn @ X @ ( sup_sup_assn @ ( uminus_uminus_assn @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_1703_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_1704_subset__emptyI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ~ ( member6576561426505652726_nat_o @ X3 @ A )
     => ( ord_le2965882846123202637_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).

% subset_emptyI
thf(fact_1705_subset__emptyI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ X3 @ A )
     => ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% subset_emptyI
thf(fact_1706_subset__emptyI,axiom,
    ! [A: set_o] :
      ( ! [X3: $o] :
          ~ ( member_o @ X3 @ A )
     => ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_1707_subset__emptyI,axiom,
    ! [A: set_int] :
      ( ! [X3: int] :
          ~ ( member_int @ X3 @ A )
     => ( ord_less_eq_set_int @ A @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_1708_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A )
     => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1709_inf__period_I2_J,axiom,
    ! [P: code_integer > $o,D3: code_integer,Q: code_integer > $o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( ( P @ X3 )
          = ( P @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
       => ! [X4: code_integer,K3: code_integer] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) )
              | ( Q @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_1710_inf__period_I2_J,axiom,
    ! [P: rat > $o,D3: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
       => ! [X4: rat,K3: rat] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) )
              | ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_1711_inf__period_I2_J,axiom,
    ! [P: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
       => ! [X4: int,K3: int] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) )
              | ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_1712_inf__period_I1_J,axiom,
    ! [P: code_integer > $o,D3: code_integer,Q: code_integer > $o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( ( P @ X3 )
          = ( P @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) ) )
       => ! [X4: code_integer,K3: code_integer] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) )
              & ( Q @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_1713_inf__period_I1_J,axiom,
    ! [P: rat > $o,D3: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D3 ) ) ) )
       => ! [X4: rat,K3: rat] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) )
              & ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_1714_inf__period_I1_J,axiom,
    ! [P: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
       => ! [X4: int,K3: int] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) )
              & ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_1715_mod__star__trueE_H,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 )
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ( ( ( produc1824681642469235216et_nat @ H4 )
              = ( produc1824681642469235216et_nat @ H2 ) )
           => ( ( ord_less_eq_set_nat @ ( produc8586169260539613262et_nat @ H4 ) @ ( produc8586169260539613262et_nat @ H2 ) )
             => ~ ( rep_assn @ P @ H4 ) ) ) ) ).

% mod_star_trueE'
thf(fact_1716_divides__aux__eq,axiom,
    ! [Q5: code_integer,R2: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q5 @ R2 ) )
      = ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_1717_divides__aux__eq,axiom,
    ! [Q5: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q5 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_1718_divides__aux__eq,axiom,
    ! [Q5: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q5 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_1719_mult__le__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1720_mult__le__cancel__left1,axiom,
    ! [C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B2 ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B2 @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1721_mult__le__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1722_mult__le__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1723_mult__le__cancel__left2,axiom,
    ! [C2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1724_mult__le__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1725_Diff__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ bot_bo2099793752762293965at_nat )
      = A ) ).

% Diff_empty
thf(fact_1726_Diff__empty,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ bot_bot_set_o )
      = A ) ).

% Diff_empty
thf(fact_1727_Diff__empty,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ bot_bot_set_int )
      = A ) ).

% Diff_empty
thf(fact_1728_Diff__empty,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Diff_empty
thf(fact_1729_empty__Diff,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A )
      = bot_bo2099793752762293965at_nat ) ).

% empty_Diff
thf(fact_1730_empty__Diff,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_1731_empty__Diff,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_1732_empty__Diff,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_1733_Diff__cancel,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ A )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_cancel
thf(fact_1734_Diff__cancel,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ A )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_1735_Diff__cancel,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ A )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_1736_Diff__cancel,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ A )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_1737_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1738_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_1739_neg__less__iff__less,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% neg_less_iff_less
thf(fact_1740_neg__less__iff__less,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% neg_less_iff_less
thf(fact_1741_neg__less__iff__less,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% neg_less_iff_less
thf(fact_1742_Diff__eq__empty__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1743_Diff__eq__empty__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( minus_minus_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1744_Diff__eq__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( minus_minus_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1745_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1746_Diff__UNIV,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ top_to4669805908274784177at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_UNIV
thf(fact_1747_Diff__UNIV,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ top_top_set_int )
      = bot_bot_set_int ) ).

% Diff_UNIV
thf(fact_1748_Diff__UNIV,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ top_top_set_o )
      = bot_bot_set_o ) ).

% Diff_UNIV
thf(fact_1749_Diff__UNIV,axiom,
    ! [A: set_Product_unit] :
      ( ( minus_6452836326544984404t_unit @ A @ top_to1996260823553986621t_unit )
      = bot_bo3957492148770167129t_unit ) ).

% Diff_UNIV
thf(fact_1750_Diff__UNIV,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ top_top_set_nat )
      = bot_bot_set_nat ) ).

% Diff_UNIV
thf(fact_1751_Diff__disjoint,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( minus_1356011639430497352at_nat @ B @ A ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_1752_Diff__disjoint,axiom,
    ! [A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B @ A ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_1753_Diff__disjoint,axiom,
    ! [A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ A @ ( minus_minus_set_int @ B @ A ) )
      = bot_bot_set_int ) ).

% Diff_disjoint
thf(fact_1754_Diff__disjoint,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_1755_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_rat @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1756_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_int @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1757_diff__gt__0__iff__gt,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1758_neg__less__0__iff__less,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_1759_neg__less__0__iff__less,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_1760_neg__less__0__iff__less,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_1761_neg__0__less__iff__less,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_1762_neg__0__less__iff__less,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_1763_neg__0__less__iff__less,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_1764_neg__less__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% neg_less_pos
thf(fact_1765_neg__less__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_pos
thf(fact_1766_neg__less__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_pos
thf(fact_1767_less__neg__neg,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_1768_less__neg__neg,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_1769_less__neg__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_1770_prod_Ocollapse,axiom,
    ! [Prod: produc1908205239877642774nteger] :
      ( ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Prod ) @ ( produc7856867400915047194nteger @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1771_prod_Ocollapse,axiom,
    ! [Prod: produc3925858234332021118et_nat] :
      ( ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1772_prod_Ocollapse,axiom,
    ! [Prod: produc2732055786443039994et_nat] :
      ( ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1773_prod_Ocollapse,axiom,
    ! [Prod: produc2285326912895808259nt_int] :
      ( ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1774_prod_Ocollapse,axiom,
    ! [Prod: produc7773217078559923341nt_int] :
      ( ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1775_prod_Ocollapse,axiom,
    ! [Prod: produc3658429121746597890et_nat] :
      ( ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1776_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] :
      ( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1777_prod_Ocollapse,axiom,
    ! [Prod: produc5835291356934675326atural] :
      ( ( produc6639722614265839536atural @ ( produc497848011034438852atural @ Prod ) @ ( produc6591912806276919810atural @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_1778_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_1779_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_1780_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_1781_order__less__imp__not__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_1782_order__less__imp__not__less,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ~ ( ord_le5570908160329646204atural @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_1783_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1784_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1785_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1786_order__less__imp__not__eq2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1787_order__less__imp__not__eq2,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1788_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1789_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1790_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1791_order__less__imp__not__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1792_order__less__imp__not__eq,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1793_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_1794_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_1795_linorder__less__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
      | ( X = Y )
      | ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_1796_linorder__less__linear,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
      | ( X = Y )
      | ( ord_le5570908160329646204atural @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_1797_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,P: $o] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1798_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1799_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1800_order__less__imp__triv,axiom,
    ! [X: code_integer,Y: code_integer,P: $o] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( ord_le6747313008572928689nteger @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1801_order__less__imp__triv,axiom,
    ! [X: code_natural,Y: code_natural,P: $o] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ( ( ord_le5570908160329646204atural @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1802_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_1803_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_1804_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_1805_order__less__not__sym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_1806_order__less__not__sym,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ~ ( ord_le5570908160329646204atural @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_1807_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_assn @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1808_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1809_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > int,C2: int] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1810_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > code_integer,C2: code_integer] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1811_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > code_natural,C2: code_natural] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1812_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_assn @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1813_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1814_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1815_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > code_integer,C2: code_integer] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1816_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > code_natural,C2: code_natural] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1817_order__less__subst1,axiom,
    ! [A2: assn,F: assn > assn,B2: assn,C2: assn] :
      ( ( ord_less_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1818_order__less__subst1,axiom,
    ! [A2: assn,F: nat > assn,B2: nat,C2: nat] :
      ( ( ord_less_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1819_order__less__subst1,axiom,
    ! [A2: assn,F: int > assn,B2: int,C2: int] :
      ( ( ord_less_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1820_order__less__subst1,axiom,
    ! [A2: assn,F: code_integer > assn,B2: code_integer,C2: code_integer] :
      ( ( ord_less_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1821_order__less__subst1,axiom,
    ! [A2: assn,F: code_natural > assn,B2: code_natural,C2: code_natural] :
      ( ( ord_less_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ! [X3: code_natural,Y3: code_natural] :
              ( ( ord_le5570908160329646204atural @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1822_order__less__subst1,axiom,
    ! [A2: nat,F: assn > nat,B2: assn,C2: assn] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1823_order__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1824_order__less__subst1,axiom,
    ! [A2: nat,F: int > nat,B2: int,C2: int] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1825_order__less__subst1,axiom,
    ! [A2: nat,F: code_integer > nat,B2: code_integer,C2: code_integer] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1826_order__less__subst1,axiom,
    ! [A2: nat,F: code_natural > nat,B2: code_natural,C2: code_natural] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ! [X3: code_natural,Y3: code_natural] :
              ( ( ord_le5570908160329646204atural @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1827_order__less__irrefl,axiom,
    ! [X: assn] :
      ~ ( ord_less_assn @ X @ X ) ).

% order_less_irrefl
thf(fact_1828_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_1829_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_1830_order__less__irrefl,axiom,
    ! [X: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ X @ X ) ).

% order_less_irrefl
thf(fact_1831_order__less__irrefl,axiom,
    ! [X: code_natural] :
      ~ ( ord_le5570908160329646204atural @ X @ X ) ).

% order_less_irrefl
thf(fact_1832_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1833_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1834_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > int,C2: int] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1835_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > code_integer,C2: code_integer] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1836_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F: assn > code_natural,C2: code_natural] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1837_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1838_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1839_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1840_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > code_integer,C2: code_integer] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1841_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > code_natural,C2: code_natural] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1842_ord__eq__less__subst,axiom,
    ! [A2: assn,F: assn > assn,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1843_ord__eq__less__subst,axiom,
    ! [A2: nat,F: assn > nat,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1844_ord__eq__less__subst,axiom,
    ! [A2: int,F: assn > int,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1845_ord__eq__less__subst,axiom,
    ! [A2: code_integer,F: assn > code_integer,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1846_ord__eq__less__subst,axiom,
    ! [A2: code_natural,F: assn > code_natural,B2: assn,C2: assn] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1847_ord__eq__less__subst,axiom,
    ! [A2: assn,F: nat > assn,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1848_ord__eq__less__subst,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1849_ord__eq__less__subst,axiom,
    ! [A2: int,F: nat > int,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1850_ord__eq__less__subst,axiom,
    ! [A2: code_integer,F: nat > code_integer,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1851_ord__eq__less__subst,axiom,
    ! [A2: code_natural,F: nat > code_natural,B2: nat,C2: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1852_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_1853_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_1854_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_1855_order__less__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( ord_le6747313008572928689nteger @ Y @ Z )
       => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_1856_order__less__trans,axiom,
    ! [X: code_natural,Y: code_natural,Z: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ( ( ord_le5570908160329646204atural @ Y @ Z )
       => ( ord_le5570908160329646204atural @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_1857_order__less__asym_H,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ~ ( ord_less_assn @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1858_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1859_order__less__asym_H,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ~ ( ord_less_int @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1860_order__less__asym_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ~ ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1861_order__less__asym_H,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ~ ( ord_le5570908160329646204atural @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1862_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_1863_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_1864_linorder__neq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
      = ( ( ord_le6747313008572928689nteger @ X @ Y )
        | ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1865_linorder__neq__iff,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( X != Y )
      = ( ( ord_le5570908160329646204atural @ X @ Y )
        | ( ord_le5570908160329646204atural @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_1866_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_asym
thf(fact_1867_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_1868_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_1869_order__less__asym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% order_less_asym
thf(fact_1870_order__less__asym,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ~ ( ord_le5570908160329646204atural @ Y @ X ) ) ).

% order_less_asym
thf(fact_1871_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1872_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1873_linorder__neqE,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
     => ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
       => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1874_linorder__neqE,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( X != Y )
     => ( ~ ( ord_le5570908160329646204atural @ X @ Y )
       => ( ord_le5570908160329646204atural @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1875_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1876_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1877_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1878_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1879_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( ord_le5570908160329646204atural @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1880_order_Ostrict__implies__not__eq,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1881_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1882_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1883_order_Ostrict__implies__not__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1884_order_Ostrict__implies__not__eq,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1885_dual__order_Ostrict__trans,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( ord_less_assn @ C2 @ B2 )
       => ( ord_less_assn @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1886_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1887_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C2 @ B2 )
       => ( ord_less_int @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1888_dual__order_Ostrict__trans,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ B2 )
       => ( ord_le6747313008572928689nteger @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1889_dual__order_Ostrict__trans,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ B2 @ A2 )
     => ( ( ord_le5570908160329646204atural @ C2 @ B2 )
       => ( ord_le5570908160329646204atural @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1890_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_1891_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_1892_not__less__iff__gr__or__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1893_not__less__iff__gr__or__eq,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ~ ( ord_le5570908160329646204atural @ X @ Y ) )
      = ( ( ord_le5570908160329646204atural @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1894_order_Ostrict__trans,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ord_less_assn @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1895_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1896_order_Ostrict__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1897_order_Ostrict__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ord_le6747313008572928689nteger @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1898_order_Ostrict__trans,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ord_le5570908160329646204atural @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1899_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1900_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A2: int,B2: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int] : ( P @ A5 @ A5 )
       => ( ! [A5: int,B5: int] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1901_linorder__less__wlog,axiom,
    ! [P: code_integer > code_integer > $o,A2: code_integer,B2: code_integer] :
      ( ! [A5: code_integer,B5: code_integer] :
          ( ( ord_le6747313008572928689nteger @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: code_integer] : ( P @ A5 @ A5 )
       => ( ! [A5: code_integer,B5: code_integer] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1902_linorder__less__wlog,axiom,
    ! [P: code_natural > code_natural > $o,A2: code_natural,B2: code_natural] :
      ( ! [A5: code_natural,B5: code_natural] :
          ( ( ord_le5570908160329646204atural @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: code_natural] : ( P @ A5 @ A5 )
       => ( ! [A5: code_natural,B5: code_natural] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1903_exists__least__iff,axiom,
    ( ( ^ [P6: nat > $o] :
        ? [X5: nat] : ( P6 @ X5 ) )
    = ( ^ [P3: nat > $o] :
        ? [N2: nat] :
          ( ( P3 @ N2 )
          & ! [M: nat] :
              ( ( ord_less_nat @ M @ N2 )
             => ~ ( P3 @ M ) ) ) ) ) ).

% exists_least_iff
thf(fact_1904_dual__order_Oirrefl,axiom,
    ! [A2: assn] :
      ~ ( ord_less_assn @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1905_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1906_dual__order_Oirrefl,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1907_dual__order_Oirrefl,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1908_dual__order_Oirrefl,axiom,
    ! [A2: code_natural] :
      ~ ( ord_le5570908160329646204atural @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1909_dual__order_Oasym,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ~ ( ord_less_assn @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1910_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1911_dual__order_Oasym,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ~ ( ord_less_int @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1912_dual__order_Oasym,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ~ ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1913_dual__order_Oasym,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( ord_le5570908160329646204atural @ B2 @ A2 )
     => ~ ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1914_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1915_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1916_linorder__cases,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( X != Y )
       => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1917_linorder__cases,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ~ ( ord_le5570908160329646204atural @ X @ Y )
     => ( ( X != Y )
       => ( ord_le5570908160329646204atural @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1918_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1919_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1920_antisym__conv3,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ Y @ X )
     => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1921_antisym__conv3,axiom,
    ! [Y: code_natural,X: code_natural] :
      ( ~ ( ord_le5570908160329646204atural @ Y @ X )
     => ( ( ~ ( ord_le5570908160329646204atural @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1922_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P @ Y5 ) )
         => ( P @ X3 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_1923_ord__less__eq__trans,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_assn @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1924_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1925_ord__less__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1926_ord__less__eq__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le6747313008572928689nteger @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1927_ord__less__eq__trans,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le5570908160329646204atural @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1928_ord__eq__less__trans,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( A2 = B2 )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ord_less_assn @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1929_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1930_ord__eq__less__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1931_ord__eq__less__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( A2 = B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ord_le6747313008572928689nteger @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1932_ord__eq__less__trans,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( A2 = B2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ord_le5570908160329646204atural @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1933_order_Oasym,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ~ ( ord_less_assn @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1934_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1935_order_Oasym,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ~ ( ord_less_int @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1936_order_Oasym,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ~ ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1937_order_Oasym,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ~ ( ord_le5570908160329646204atural @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1938_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1939_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1940_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1941_less__imp__neq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1942_less__imp__neq,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1943_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_1944_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_1945_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_1946_disjoint__alt__simp3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A @ B ) @ A )
      = ( ( inf_in2572325071724192079at_nat @ A @ B )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
thf(fact_1947_disjoint__alt__simp3,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ord_less_set_o @ ( minus_minus_set_o @ A @ B ) @ A )
      = ( ( inf_inf_set_o @ A @ B )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
thf(fact_1948_disjoint__alt__simp3,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ ( minus_minus_set_int @ A @ B ) @ A )
      = ( ( inf_inf_set_int @ A @ B )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
thf(fact_1949_disjoint__alt__simp3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A )
      = ( ( inf_inf_set_nat @ A @ B )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
thf(fact_1950_fstE,axiom,
    ! [X: produc1908205239877642774nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o] :
      ( ( X
        = ( produc8603105652947943368nteger @ A2 @ B2 ) )
     => ( ( P @ ( produc7822682618958472924nteger @ X ) )
       => ( P @ A2 ) ) ) ).

% fstE
thf(fact_1951_fstE,axiom,
    ! [X: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( X
        = ( produc5001842942810119800et_nat @ A2 @ B2 ) )
     => ( ( P @ ( produc995936583742144908et_nat @ X ) )
       => ( P @ A2 ) ) ) ).

% fstE
thf(fact_1952_fstE,axiom,
    ! [X: produc2732055786443039994et_nat,A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( X
        = ( produc2245416461498447860et_nat @ A2 @ B2 ) )
     => ( ( P @ ( produc180342877477747464et_nat @ X ) )
       => ( P @ A2 ) ) ) ).

% fstE
thf(fact_1953_fstE,axiom,
    ! [X: produc2285326912895808259nt_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o] :
      ( ( X
        = ( produc5700946648718959541nt_int @ A2 @ B2 ) )
     => ( ( P @ ( produc6816164490631068361nt_int @ X ) )
       => ( P @ A2 ) ) ) ).

% fstE
thf(fact_1954_fstE,axiom,
    ! [X: produc7773217078559923341nt_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int,P: ( int > option6357759511663192854e_term ) > $o] :
      ( ( X
        = ( produc4305682042979456191nt_int @ A2 @ B2 ) )
     => ( ( P @ ( produc6230002227079971283nt_int @ X ) )
       => ( P @ A2 ) ) ) ).

% fstE
thf(fact_1955_fstE,axiom,
    ! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B2: set_nat,P: heap_e7401611519738050253t_unit > $o] :
      ( ( X
        = ( produc7507926704131184380et_nat @ A2 @ B2 ) )
     => ( ( P @ ( produc1824681642469235216et_nat @ X ) )
       => ( P @ A2 ) ) ) ).

% fstE
thf(fact_1956_fstE,axiom,
    ! [X: product_prod_int_int,A2: int,B2: int,P: int > $o] :
      ( ( X
        = ( product_Pair_int_int @ A2 @ B2 ) )
     => ( ( P @ ( product_fst_int_int @ X ) )
       => ( P @ A2 ) ) ) ).

% fstE
thf(fact_1957_fst__eqD,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_1958_fst__eqD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o] :
      ( ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_1959_fst__eqD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o] :
      ( ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_1960_fst__eqD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_1961_fst__eqD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,A2: int > option6357759511663192854e_term] :
      ( ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_1962_fst__eqD,axiom,
    ! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: heap_e7401611519738050253t_unit] :
      ( ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_1963_fst__eqD,axiom,
    ! [X: int,Y: int,A2: int] :
      ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_1964_fst__conv,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger] :
      ( ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1965_fst__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat] :
      ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1966_fst__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat] :
      ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1967_fst__conv,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int] :
      ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1968_fst__conv,axiom,
    ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int] :
      ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1969_fst__conv,axiom,
    ! [X1: heap_e7401611519738050253t_unit,X22: set_nat] :
      ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1970_fst__conv,axiom,
    ! [X1: int,X22: int] :
      ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_1971_fstI,axiom,
    ! [X: produc1908205239877642774nteger,Y: produc6241069584506657477e_term > option6357759511663192854e_term,Z: produc8923325533196201883nteger] :
      ( ( X
        = ( produc8603105652947943368nteger @ Y @ Z ) )
     => ( ( produc7822682618958472924nteger @ X )
        = Y ) ) ).

% fstI
thf(fact_1972_fstI,axiom,
    ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
      ( ( X
        = ( produc5001842942810119800et_nat @ Y @ Z ) )
     => ( ( produc995936583742144908et_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_1973_fstI,axiom,
    ! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
      ( ( X
        = ( produc2245416461498447860et_nat @ Y @ Z ) )
     => ( ( produc180342877477747464et_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_1974_fstI,axiom,
    ! [X: produc2285326912895808259nt_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc5700946648718959541nt_int @ Y @ Z ) )
     => ( ( produc6816164490631068361nt_int @ X )
        = Y ) ) ).

% fstI
thf(fact_1975_fstI,axiom,
    ! [X: produc7773217078559923341nt_int,Y: int > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc4305682042979456191nt_int @ Y @ Z ) )
     => ( ( produc6230002227079971283nt_int @ X )
        = Y ) ) ).

% fstI
thf(fact_1976_fstI,axiom,
    ! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
      ( ( X
        = ( produc7507926704131184380et_nat @ Y @ Z ) )
     => ( ( produc1824681642469235216et_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_1977_fstI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X
        = ( product_Pair_int_int @ Y @ Z ) )
     => ( ( product_fst_int_int @ X )
        = Y ) ) ).

% fstI
thf(fact_1978_eq__fst__iff,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,P5: produc1908205239877642774nteger] :
      ( ( A2
        = ( produc7822682618958472924nteger @ P5 ) )
      = ( ? [B3: produc8923325533196201883nteger] :
            ( P5
            = ( produc8603105652947943368nteger @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1979_eq__fst__iff,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,P5: produc3925858234332021118et_nat] :
      ( ( A2
        = ( produc995936583742144908et_nat @ P5 ) )
      = ( ? [B3: produc3658429121746597890et_nat] :
            ( P5
            = ( produc5001842942810119800et_nat @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1980_eq__fst__iff,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,P5: produc2732055786443039994et_nat] :
      ( ( A2
        = ( produc180342877477747464et_nat @ P5 ) )
      = ( ? [B3: produc3925858234332021118et_nat] :
            ( P5
            = ( produc2245416461498447860et_nat @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1981_eq__fst__iff,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,P5: produc2285326912895808259nt_int] :
      ( ( A2
        = ( produc6816164490631068361nt_int @ P5 ) )
      = ( ? [B3: product_prod_int_int] :
            ( P5
            = ( produc5700946648718959541nt_int @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1982_eq__fst__iff,axiom,
    ! [A2: int > option6357759511663192854e_term,P5: produc7773217078559923341nt_int] :
      ( ( A2
        = ( produc6230002227079971283nt_int @ P5 ) )
      = ( ? [B3: product_prod_int_int] :
            ( P5
            = ( produc4305682042979456191nt_int @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1983_eq__fst__iff,axiom,
    ! [A2: heap_e7401611519738050253t_unit,P5: produc3658429121746597890et_nat] :
      ( ( A2
        = ( produc1824681642469235216et_nat @ P5 ) )
      = ( ? [B3: set_nat] :
            ( P5
            = ( produc7507926704131184380et_nat @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1984_eq__fst__iff,axiom,
    ! [A2: int,P5: product_prod_int_int] :
      ( ( A2
        = ( product_fst_int_int @ P5 ) )
      = ( ? [B3: int] :
            ( P5
            = ( product_Pair_int_int @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_1985_order__le__imp__less__or__eq,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le1926595141338095240atural @ X @ Y )
     => ( ( ord_le5570908160329646204atural @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1986_order__le__imp__less__or__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_le6747313008572928689nteger @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1987_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_1988_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1989_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_1990_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_1991_linorder__le__less__linear,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le1926595141338095240atural @ X @ Y )
      | ( ord_le5570908160329646204atural @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1992_linorder__le__less__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
      | ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1993_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_1994_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_1995_order__less__le__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > code_natural,C2: code_natural] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_le1926595141338095240atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1996_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > code_natural,C2: code_natural] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_le1926595141338095240atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1997_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F: int > code_natural,C2: code_natural] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_le1926595141338095240atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1998_order__less__le__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > code_natural,C2: code_natural] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le1926595141338095240atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1999_order__less__le__subst2,axiom,
    ! [A2: code_natural,B2: code_natural,F: code_natural > code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le1926595141338095240atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_natural,Y3: code_natural] :
              ( ( ord_le5570908160329646204atural @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2000_order__less__le__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > code_integer,C2: code_integer] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2001_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > code_integer,C2: code_integer] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2002_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F: int > code_integer,C2: code_integer] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2003_order__less__le__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2004_order__less__le__subst2,axiom,
    ! [A2: code_natural,B2: code_natural,F: code_natural > code_integer,C2: code_integer] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_natural,Y3: code_natural] :
              ( ( ord_le5570908160329646204atural @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2005_order__less__le__subst1,axiom,
    ! [A2: code_natural,F: code_integer > code_natural,B2: code_integer,C2: code_integer] :
      ( ( ord_le5570908160329646204atural @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_le1926595141338095240atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2006_order__less__le__subst1,axiom,
    ! [A2: code_integer,F: code_integer > code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2007_order__less__le__subst1,axiom,
    ! [A2: assn,F: code_integer > assn,B2: code_integer,C2: code_integer] :
      ( ( ord_less_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2008_order__less__le__subst1,axiom,
    ! [A2: nat,F: code_integer > nat,B2: code_integer,C2: code_integer] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2009_order__less__le__subst1,axiom,
    ! [A2: int,F: code_integer > int,B2: code_integer,C2: code_integer] :
      ( ( ord_less_int @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2010_order__less__le__subst1,axiom,
    ! [A2: code_natural,F: assn > code_natural,B2: assn,C2: assn] :
      ( ( ord_le5570908160329646204atural @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_le1926595141338095240atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2011_order__less__le__subst1,axiom,
    ! [A2: code_integer,F: assn > code_integer,B2: assn,C2: assn] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2012_order__less__le__subst1,axiom,
    ! [A2: assn,F: assn > assn,B2: assn,C2: assn] :
      ( ( ord_less_assn @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2013_order__less__le__subst1,axiom,
    ! [A2: nat,F: assn > nat,B2: assn,C2: assn] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2014_order__less__le__subst1,axiom,
    ! [A2: int,F: assn > int,B2: assn,C2: assn] :
      ( ( ord_less_int @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2015_order__le__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > code_natural,C2: code_natural] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_le1926595141338095240atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2016_order__le__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2017_order__le__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > assn,C2: assn] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_less_assn @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2018_order__le__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > nat,C2: nat] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2019_order__le__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F: code_integer > int,C2: int] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le3102999989581377725nteger @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2020_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > code_natural,C2: code_natural] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_le1926595141338095240atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2021_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > code_integer,C2: code_integer] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2022_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_assn @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_assn @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2023_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2024_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F: assn > int,C2: int] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2025_order__le__less__subst1,axiom,
    ! [A2: code_natural,F: assn > code_natural,B2: assn,C2: assn] :
      ( ( ord_le1926595141338095240atural @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2026_order__le__less__subst1,axiom,
    ! [A2: code_natural,F: nat > code_natural,B2: nat,C2: nat] :
      ( ( ord_le1926595141338095240atural @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2027_order__le__less__subst1,axiom,
    ! [A2: code_natural,F: int > code_natural,B2: int,C2: int] :
      ( ( ord_le1926595141338095240atural @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2028_order__le__less__subst1,axiom,
    ! [A2: code_natural,F: code_integer > code_natural,B2: code_integer,C2: code_integer] :
      ( ( ord_le1926595141338095240atural @ A2 @ ( F @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2029_order__le__less__subst1,axiom,
    ! [A2: code_natural,F: code_natural > code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A2 @ ( F @ B2 ) )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ! [X3: code_natural,Y3: code_natural] :
              ( ( ord_le5570908160329646204atural @ X3 @ Y3 )
             => ( ord_le5570908160329646204atural @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le5570908160329646204atural @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2030_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: assn > code_integer,B2: assn,C2: assn] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ! [X3: assn,Y3: assn] :
              ( ( ord_less_assn @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2031_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: nat > code_integer,B2: nat,C2: nat] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2032_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: int > code_integer,B2: int,C2: int] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2033_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: code_integer > code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ! [X3: code_integer,Y3: code_integer] :
              ( ( ord_le6747313008572928689nteger @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2034_order__le__less__subst1,axiom,
    ! [A2: code_integer,F: code_natural > code_integer,B2: code_natural,C2: code_natural] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B2 ) )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ! [X3: code_natural,Y3: code_natural] :
              ( ( ord_le5570908160329646204atural @ X3 @ Y3 )
             => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2035_order__less__le__trans,axiom,
    ! [X: code_natural,Y: code_natural,Z: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ( ( ord_le1926595141338095240atural @ Y @ Z )
       => ( ord_le5570908160329646204atural @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2036_order__less__le__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( ord_le3102999989581377725nteger @ Y @ Z )
       => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2037_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2038_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2039_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2040_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2041_order__le__less__trans,axiom,
    ! [X: code_natural,Y: code_natural,Z: code_natural] :
      ( ( ord_le1926595141338095240atural @ X @ Y )
     => ( ( ord_le5570908160329646204atural @ Y @ Z )
       => ( ord_le5570908160329646204atural @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2042_order__le__less__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_le6747313008572928689nteger @ Y @ Z )
       => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2043_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2044_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2045_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2046_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2047_order__neq__le__trans,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( A2 != B2 )
     => ( ( ord_le1926595141338095240atural @ A2 @ B2 )
       => ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_2048_order__neq__le__trans,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != B2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_2049_order__neq__le__trans,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_assn @ A2 @ B2 )
       => ( ord_less_assn @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_2050_order__neq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_2051_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_2052_order__neq__le__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_2053_order__le__neq__trans,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_2054_order__le__neq__trans,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_2055_order__le__neq__trans,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_assn @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_2056_order__le__neq__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_2057_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_2058_order__le__neq__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_2059_order__less__imp__le,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ X @ Y )
     => ( ord_le1926595141338095240atural @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_2060_order__less__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ord_le3102999989581377725nteger @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_2061_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_2062_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_2063_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_2064_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_2065_linorder__not__less,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ~ ( ord_le5570908160329646204atural @ X @ Y ) )
      = ( ord_le1926595141338095240atural @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2066_linorder__not__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
      = ( ord_le3102999989581377725nteger @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2067_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2068_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2069_linorder__not__le,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ~ ( ord_le1926595141338095240atural @ X @ Y ) )
      = ( ord_le5570908160329646204atural @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2070_linorder__not__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ~ ( ord_le3102999989581377725nteger @ X @ Y ) )
      = ( ord_le6747313008572928689nteger @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2071_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2072_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2073_order__less__le,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [X2: code_natural,Y2: code_natural] :
          ( ( ord_le1926595141338095240atural @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2074_order__less__le,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( ( ord_le3102999989581377725nteger @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2075_order__less__le,axiom,
    ( ord_less_assn
    = ( ^ [X2: assn,Y2: assn] :
          ( ( ord_less_eq_assn @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2076_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2077_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2078_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2079_order__le__less,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [X2: code_natural,Y2: code_natural] :
          ( ( ord_le5570908160329646204atural @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2080_order__le__less,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2081_order__le__less,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y2: assn] :
          ( ( ord_less_assn @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2082_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_set_nat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2083_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_nat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2084_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_int @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2085_dual__order_Ostrict__implies__order,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( ord_le5570908160329646204atural @ B2 @ A2 )
     => ( ord_le1926595141338095240atural @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_2086_dual__order_Ostrict__implies__order,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_2087_dual__order_Ostrict__implies__order,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ord_less_eq_assn @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_2088_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_2089_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_2090_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ord_less_eq_int @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_2091_order_Ostrict__implies__order,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ord_le1926595141338095240atural @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_2092_order_Ostrict__implies__order,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_2093_order_Ostrict__implies__order,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ord_less_eq_assn @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_2094_order_Ostrict__implies__order,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_2095_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_2096_order_Ostrict__implies__order,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_2097_dual__order_Ostrict__iff__not,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [B3: code_natural,A3: code_natural] :
          ( ( ord_le1926595141338095240atural @ B3 @ A3 )
          & ~ ( ord_le1926595141338095240atural @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2098_dual__order_Ostrict__iff__not,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B3: code_integer,A3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
          & ~ ( ord_le3102999989581377725nteger @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2099_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( ord_less_eq_assn @ B3 @ A3 )
          & ~ ( ord_less_eq_assn @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2100_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2101_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2102_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2103_dual__order_Ostrict__trans2,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ B2 @ A2 )
     => ( ( ord_le1926595141338095240atural @ C2 @ B2 )
       => ( ord_le5570908160329646204atural @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2104_dual__order_Ostrict__trans2,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ B2 )
       => ( ord_le6747313008572928689nteger @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2105_dual__order_Ostrict__trans2,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ C2 @ B2 )
       => ( ord_less_assn @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2106_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C2 @ B2 )
       => ( ord_less_set_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2107_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2108_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ B2 )
       => ( ord_less_int @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2109_dual__order_Ostrict__trans1,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( ord_le1926595141338095240atural @ B2 @ A2 )
     => ( ( ord_le5570908160329646204atural @ C2 @ B2 )
       => ( ord_le5570908160329646204atural @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2110_dual__order_Ostrict__trans1,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ B2 )
       => ( ord_le6747313008572928689nteger @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2111_dual__order_Ostrict__trans1,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_assn @ C2 @ B2 )
       => ( ord_less_assn @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2112_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_set_nat @ C2 @ B2 )
       => ( ord_less_set_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2113_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2114_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_int @ C2 @ B2 )
       => ( ord_less_int @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2115_dual__order_Ostrict__iff__order,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [B3: code_natural,A3: code_natural] :
          ( ( ord_le1926595141338095240atural @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2116_dual__order_Ostrict__iff__order,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B3: code_integer,A3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2117_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( ord_less_eq_assn @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2118_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2119_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2120_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2121_dual__order_Oorder__iff__strict,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [B3: code_natural,A3: code_natural] :
          ( ( ord_le5570908160329646204atural @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2122_dual__order_Oorder__iff__strict,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B3: code_integer,A3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2123_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( ord_less_assn @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2124_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_set_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2125_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2126_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_int @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2127_order_Ostrict__iff__not,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [A3: code_natural,B3: code_natural] :
          ( ( ord_le1926595141338095240atural @ A3 @ B3 )
          & ~ ( ord_le1926595141338095240atural @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2128_order_Ostrict__iff__not,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
          & ~ ( ord_le3102999989581377725nteger @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2129_order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ A3 @ B3 )
          & ~ ( ord_less_eq_assn @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2130_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2131_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2132_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2133_order_Ostrict__trans2,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le1926595141338095240atural @ B2 @ C2 )
       => ( ord_le5570908160329646204atural @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2134_order_Ostrict__trans2,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ord_le6747313008572928689nteger @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2135_order_Ostrict__trans2,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( ord_less_eq_assn @ B2 @ C2 )
       => ( ord_less_assn @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2136_order_Ostrict__trans2,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_set_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2137_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2138_order_Ostrict__trans2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2139_order_Ostrict__trans1,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ord_le5570908160329646204atural @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2140_order_Ostrict__trans1,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ord_le6747313008572928689nteger @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2141_order_Ostrict__trans1,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( ord_less_assn @ B2 @ C2 )
       => ( ord_less_assn @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2142_order_Ostrict__trans1,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C2 )
       => ( ord_less_set_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2143_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2144_order_Ostrict__trans1,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2145_order_Ostrict__iff__order,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [A3: code_natural,B3: code_natural] :
          ( ( ord_le1926595141338095240atural @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2146_order_Ostrict__iff__order,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2147_order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2148_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2149_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2150_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2151_order_Oorder__iff__strict,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [A3: code_natural,B3: code_natural] :
          ( ( ord_le5570908160329646204atural @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2152_order_Oorder__iff__strict,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2153_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_assn @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2154_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_set_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2155_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2156_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2157_not__le__imp__less,axiom,
    ! [Y: code_natural,X: code_natural] :
      ( ~ ( ord_le1926595141338095240atural @ Y @ X )
     => ( ord_le5570908160329646204atural @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2158_not__le__imp__less,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ~ ( ord_le3102999989581377725nteger @ Y @ X )
     => ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2159_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_2160_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_2161_less__le__not__le,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [X2: code_natural,Y2: code_natural] :
          ( ( ord_le1926595141338095240atural @ X2 @ Y2 )
          & ~ ( ord_le1926595141338095240atural @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2162_less__le__not__le,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( ( ord_le3102999989581377725nteger @ X2 @ Y2 )
          & ~ ( ord_le3102999989581377725nteger @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2163_less__le__not__le,axiom,
    ( ord_less_assn
    = ( ^ [X2: assn,Y2: assn] :
          ( ( ord_less_eq_assn @ X2 @ Y2 )
          & ~ ( ord_less_eq_assn @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2164_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ~ ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2165_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2166_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2167_antisym__conv2,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le1926595141338095240atural @ X @ Y )
     => ( ( ~ ( ord_le5570908160329646204atural @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2168_antisym__conv2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2169_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ~ ( ord_less_assn @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2170_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2171_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2172_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2173_antisym__conv1,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ~ ( ord_le5570908160329646204atural @ X @ Y )
     => ( ( ord_le1926595141338095240atural @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2174_antisym__conv1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ( ord_le3102999989581377725nteger @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2175_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2176_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2177_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2178_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2179_nless__le,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ~ ( ord_le5570908160329646204atural @ A2 @ B2 ) )
      = ( ~ ( ord_le1926595141338095240atural @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2180_nless__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2181_nless__le,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ~ ( ord_less_assn @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_assn @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2182_nless__le,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2183_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2184_nless__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ~ ( ord_less_int @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2185_leI,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ~ ( ord_le5570908160329646204atural @ X @ Y )
     => ( ord_le1926595141338095240atural @ Y @ X ) ) ).

% leI
thf(fact_2186_leI,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
     => ( ord_le3102999989581377725nteger @ Y @ X ) ) ).

% leI
thf(fact_2187_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_2188_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_2189_leD,axiom,
    ! [Y: code_natural,X: code_natural] :
      ( ( ord_le1926595141338095240atural @ Y @ X )
     => ~ ( ord_le5570908160329646204atural @ X @ Y ) ) ).

% leD
thf(fact_2190_leD,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y @ X )
     => ~ ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% leD
thf(fact_2191_leD,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ~ ( ord_less_assn @ X @ Y ) ) ).

% leD
thf(fact_2192_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_2193_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_2194_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_2195_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_2196_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_2197_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_2198_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_2199_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_2200_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_2201_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_2202_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_2203_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ one_one_Code_natural ) ).

% less_numeral_extra(4)
thf(fact_2204_diff__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C2 ) @ ( minus_minus_rat @ B2 @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_2205_diff__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B2 @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_2206_diff__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C2 ) @ ( minus_8373710615458151222nteger @ B2 @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_2207_diff__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ord_less_rat @ ( minus_minus_rat @ C2 @ A2 ) @ ( minus_minus_rat @ C2 @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_2208_diff__strict__left__mono,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ord_less_int @ ( minus_minus_int @ C2 @ A2 ) @ ( minus_minus_int @ C2 @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_2209_diff__strict__left__mono,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ C2 @ A2 ) @ ( minus_8373710615458151222nteger @ C2 @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_2210_diff__eq__diff__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C2 @ D2 ) )
     => ( ( ord_less_rat @ A2 @ B2 )
        = ( ord_less_rat @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_2211_diff__eq__diff__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( ord_less_int @ A2 @ B2 )
        = ( ord_less_int @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_2212_diff__eq__diff__less,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A2 @ B2 )
        = ( minus_8373710615458151222nteger @ C2 @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
        = ( ord_le6747313008572928689nteger @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_2213_diff__strict__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ D2 @ C2 )
       => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C2 ) @ ( minus_minus_rat @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_2214_diff__strict__mono,axiom,
    ! [A2: int,B2: int,D2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ D2 @ C2 )
       => ( ord_less_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_2215_diff__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,D2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ D2 @ C2 )
       => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C2 ) @ ( minus_8373710615458151222nteger @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_2216_bot_Oextremum__strict,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_2217_bot_Oextremum__strict,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_2218_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_2219_bot_Oextremum__strict,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_2220_bot_Oextremum__strict,axiom,
    ! [A2: assn] :
      ~ ( ord_less_assn @ A2 @ bot_bot_assn ) ).

% bot.extremum_strict
thf(fact_2221_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_2222_bot_Onot__eq__extremum,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
      = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_2223_bot_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_2224_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_2225_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_2226_bot_Onot__eq__extremum,axiom,
    ! [A2: assn] :
      ( ( A2 != bot_bot_assn )
      = ( ord_less_assn @ bot_bot_assn @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_2227_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_2228_not__psubset__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_2229_not__psubset__empty,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_2230_not__psubset__empty,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_2231_not__psubset__empty,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_2232_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_2233_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_2234_minus__less__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_2235_minus__less__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_2236_minus__less__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_2237_less__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).

% less_minus_iff
thf(fact_2238_less__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
      = ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ A2 ) ) ) ).

% less_minus_iff
thf(fact_2239_less__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( ord_le6747313008572928689nteger @ B2 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% less_minus_iff
thf(fact_2240_top_Oextremum__strict,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ top_top_set_o @ A2 ) ).

% top.extremum_strict
thf(fact_2241_top_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).

% top.extremum_strict
thf(fact_2242_top_Oextremum__strict,axiom,
    ! [A2: set_Product_unit] :
      ~ ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ A2 ) ).

% top.extremum_strict
thf(fact_2243_top_Oextremum__strict,axiom,
    ! [A2: assn] :
      ~ ( ord_less_assn @ top_top_assn @ A2 ) ).

% top.extremum_strict
thf(fact_2244_top_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != top_top_set_o )
      = ( ord_less_set_o @ A2 @ top_top_set_o ) ) ).

% top.not_eq_extremum
thf(fact_2245_top_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != top_top_set_nat )
      = ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).

% top.not_eq_extremum
thf(fact_2246_top_Onot__eq__extremum,axiom,
    ! [A2: set_Product_unit] :
      ( ( A2 != top_to1996260823553986621t_unit )
      = ( ord_le8056459307392131481t_unit @ A2 @ top_to1996260823553986621t_unit ) ) ).

% top.not_eq_extremum
thf(fact_2247_top_Onot__eq__extremum,axiom,
    ! [A2: assn] :
      ( ( A2 != top_top_assn )
      = ( ord_less_assn @ A2 @ top_top_assn ) ) ).

% top.not_eq_extremum
thf(fact_2248_inf_Ostrict__coboundedI2,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ C2 )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_2249_inf_Ostrict__coboundedI2,axiom,
    ! [B2: assn,C2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ C2 )
     => ( ord_less_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_2250_inf_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ C2 )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_2251_inf_Ostrict__coboundedI2,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_int @ B2 @ C2 )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B2 ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_2252_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ C2 )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_2253_inf_Ostrict__coboundedI1,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ C2 )
     => ( ord_less_assn @ ( inf_inf_assn @ A2 @ B2 ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_2254_inf_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ C2 )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_2255_inf_Ostrict__coboundedI1,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ A2 @ C2 )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B2 ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_2256_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( A3
            = ( inf_inf_set_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2257_inf_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( A3
            = ( inf_inf_assn @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2258_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( A3
            = ( inf_inf_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2259_inf_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( A3
            = ( inf_inf_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_2260_inf_Ostrict__boundedE,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
     => ~ ( ( ord_less_set_nat @ A2 @ B2 )
         => ~ ( ord_less_set_nat @ A2 @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_2261_inf_Ostrict__boundedE,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( ord_less_assn @ A2 @ ( inf_inf_assn @ B2 @ C2 ) )
     => ~ ( ( ord_less_assn @ A2 @ B2 )
         => ~ ( ord_less_assn @ A2 @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_2262_inf_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
     => ~ ( ( ord_less_nat @ A2 @ B2 )
         => ~ ( ord_less_nat @ A2 @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_2263_inf_Ostrict__boundedE,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ ( inf_inf_int @ B2 @ C2 ) )
     => ~ ( ( ord_less_int @ A2 @ B2 )
         => ~ ( ord_less_int @ A2 @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_2264_inf_Oabsorb4,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_2265_inf_Oabsorb4,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( inf_inf_assn @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_2266_inf_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_2267_inf_Oabsorb4,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( inf_inf_int @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb4
thf(fact_2268_inf_Oabsorb3,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_2269_inf_Oabsorb3,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( inf_inf_assn @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_2270_inf_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( inf_inf_nat @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_2271_inf_Oabsorb3,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( inf_inf_int @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_2272_less__infI2,axiom,
    ! [B2: set_nat,X: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_2273_less__infI2,axiom,
    ! [B2: assn,X: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_2274_less__infI2,axiom,
    ! [B2: nat,X: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_2275_less__infI2,axiom,
    ! [B2: int,X: int,A2: int] :
      ( ( ord_less_int @ B2 @ X )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B2 ) @ X ) ) ).

% less_infI2
thf(fact_2276_less__infI1,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_2277_less__infI1,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_2278_less__infI1,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_2279_less__infI1,axiom,
    ! [A2: int,X: int,B2: int] :
      ( ( ord_less_int @ A2 @ X )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B2 ) @ X ) ) ).

% less_infI1
thf(fact_2280_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ C2 @ B2 )
     => ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2281_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C2 @ B2 )
     => ( ord_le2604355607129572851at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2282_sup_Ostrict__coboundedI2,axiom,
    ! [C2: assn,B2: assn,A2: assn] :
      ( ( ord_less_assn @ C2 @ B2 )
     => ( ord_less_assn @ C2 @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2283_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ C2 @ B2 )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2284_sup_Ostrict__coboundedI2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( ord_less_int @ C2 @ B2 )
     => ( ord_less_int @ C2 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_2285_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ C2 @ A2 )
     => ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2286_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C2 @ A2 )
     => ( ord_le2604355607129572851at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2287_sup_Ostrict__coboundedI1,axiom,
    ! [C2: assn,A2: assn,B2: assn] :
      ( ( ord_less_assn @ C2 @ A2 )
     => ( ord_less_assn @ C2 @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2288_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ C2 @ A2 )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2289_sup_Ostrict__coboundedI1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ C2 @ A2 )
     => ( ord_less_int @ C2 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_2290_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( A3
            = ( sup_sup_set_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2291_sup_Ostrict__order__iff,axiom,
    ( ord_le2604355607129572851at_nat
    = ( ^ [B3: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
          ( ( A3
            = ( sup_su5525570899277871387at_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2292_sup_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [B3: assn,A3: assn] :
          ( ( A3
            = ( sup_sup_assn @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2293_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( A3
            = ( sup_sup_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2294_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( A3
            = ( sup_sup_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_2295_sup_Ostrict__boundedE,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_set_nat @ B2 @ A2 )
         => ~ ( ord_less_set_nat @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_2296_sup_Ostrict__boundedE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_le2604355607129572851at_nat @ B2 @ A2 )
         => ~ ( ord_le2604355607129572851at_nat @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_2297_sup_Ostrict__boundedE,axiom,
    ! [B2: assn,C2: assn,A2: assn] :
      ( ( ord_less_assn @ ( sup_sup_assn @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_assn @ B2 @ A2 )
         => ~ ( ord_less_assn @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_2298_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_nat @ B2 @ A2 )
         => ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_2299_sup_Ostrict__boundedE,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_int @ B2 @ A2 )
         => ~ ( ord_less_int @ C2 @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_2300_sup_Oabsorb4,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_2301_sup_Oabsorb4,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A2 @ B2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_2302_sup_Oabsorb4,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_2303_sup_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_2304_sup_Oabsorb4,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_2305_sup_Oabsorb3,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_2306_sup_Oabsorb3,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ B2 @ A2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_2307_sup_Oabsorb3,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_2308_sup_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_2309_sup_Oabsorb3,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_2310_less__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ X @ B2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_2311_less__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ X @ B2 )
     => ( ord_le2604355607129572851at_nat @ X @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_2312_less__supI2,axiom,
    ! [X: assn,B2: assn,A2: assn] :
      ( ( ord_less_assn @ X @ B2 )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_2313_less__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ X @ B2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_2314_less__supI2,axiom,
    ! [X: int,B2: int,A2: int] :
      ( ( ord_less_int @ X @ B2 )
     => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_2315_less__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ X @ A2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_2316_less__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ X @ A2 )
     => ( ord_le2604355607129572851at_nat @ X @ ( sup_su5525570899277871387at_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_2317_less__supI1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_assn @ X @ A2 )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_2318_less__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ X @ A2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_2319_less__supI1,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_int @ X @ A2 )
     => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_2320_less__eq__assn__def,axiom,
    ( ord_less_eq_assn
    = ( ^ [A3: assn,B3: assn] :
          ( A3
          = ( inf_inf_assn @ A3 @ B3 ) ) ) ) ).

% less_eq_assn_def
thf(fact_2321_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1908205239877642774nteger] :
      ( Prod
      = ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Prod ) @ ( produc7856867400915047194nteger @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_2322_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3925858234332021118et_nat] :
      ( Prod
      = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_2323_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2732055786443039994et_nat] :
      ( Prod
      = ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_2324_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2285326912895808259nt_int] :
      ( Prod
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_2325_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7773217078559923341nt_int] :
      ( Prod
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_2326_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3658429121746597890et_nat] :
      ( Prod
      = ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_2327_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] :
      ( Prod
      = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_2328_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5835291356934675326atural] :
      ( Prod
      = ( produc6639722614265839536atural @ ( produc497848011034438852atural @ Prod ) @ ( produc6591912806276919810atural @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_2329_surjective__pairing,axiom,
    ! [T: produc1908205239877642774nteger] :
      ( T
      = ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ T ) @ ( produc7856867400915047194nteger @ T ) ) ) ).

% surjective_pairing
thf(fact_2330_surjective__pairing,axiom,
    ! [T: produc3925858234332021118et_nat] :
      ( T
      = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ T ) @ ( produc4011572625026189258et_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_2331_surjective__pairing,axiom,
    ! [T: produc2732055786443039994et_nat] :
      ( T
      = ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ T ) @ ( produc5374455773327741254et_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_2332_surjective__pairing,axiom,
    ! [T: produc2285326912895808259nt_int] :
      ( T
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ T ) @ ( produc7328097813583171335nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_2333_surjective__pairing,axiom,
    ! [T: produc7773217078559923341nt_int] :
      ( T
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ T ) @ ( produc3162348030201620241nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_2334_surjective__pairing,axiom,
    ! [T: produc3658429121746597890et_nat] :
      ( T
      = ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ T ) @ ( produc8586169260539613262et_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_2335_surjective__pairing,axiom,
    ! [T: product_prod_int_int] :
      ( T
      = ( product_Pair_int_int @ ( product_fst_int_int @ T ) @ ( product_snd_int_int @ T ) ) ) ).

% surjective_pairing
thf(fact_2336_surjective__pairing,axiom,
    ! [T: produc5835291356934675326atural] :
      ( T
      = ( produc6639722614265839536atural @ ( produc497848011034438852atural @ T ) @ ( produc6591912806276919810atural @ T ) ) ) ).

% surjective_pairing
thf(fact_2337_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2338_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2339_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2340_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2341_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C2 )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C2 @ A2 ) @ ( times_2397367101498566445atural @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2342_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2343_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A2 @ B2 ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2344_mult__less__cancel__right__disj,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
          & ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
        | ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2345_mult__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2346_mult__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2347_mult__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2348_mult__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2349_mult__strict__right__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C2 )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2350_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2351_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2352_mult__strict__right__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2353_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2354_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A2 @ B2 ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2355_mult__less__cancel__left__disj,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
          & ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
        | ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2356_mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2357_mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2358_mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2359_mult__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2360_mult__strict__left__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C2 )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C2 @ A2 ) @ ( times_2397367101498566445atural @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2361_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2362_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2363_mult__strict__left__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2364_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2365_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2366_mult__less__cancel__left__pos,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
        = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2367_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( ord_less_rat @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2368_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( ord_less_int @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2369_mult__less__cancel__left__neg,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
        = ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2370_zero__less__mult__pos2,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2371_zero__less__mult__pos2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2372_zero__less__mult__pos2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2373_zero__less__mult__pos2,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2374_zero__less__mult__pos2,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ B2 @ A2 ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2375_zero__less__mult__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2376_zero__less__mult__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2377_zero__less__mult__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2378_zero__less__mult__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2379_zero__less__mult__pos,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ A2 @ B2 ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2380_zero__less__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2381_zero__less__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2382_zero__less__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) )
        | ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2383_mult__pos__neg2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2384_mult__pos__neg2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2385_mult__pos__neg2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2386_mult__pos__neg2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg2
thf(fact_2387_mult__pos__neg2,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ B2 @ A2 ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_pos_neg2
thf(fact_2388_mult__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2389_mult__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2390_mult__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2391_mult__pos__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2392_mult__pos__pos,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2393_mult__pos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2394_mult__pos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2395_mult__pos__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2396_mult__pos__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg
thf(fact_2397_mult__pos__neg,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_pos_neg
thf(fact_2398_mult__neg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2399_mult__neg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2400_mult__neg__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2401_mult__neg__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_neg_pos
thf(fact_2402_mult__neg__pos,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_neg_pos
thf(fact_2403_mult__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2404_mult__less__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2405_mult__less__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
          & ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2406_not__square__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2407_not__square__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2408_not__square__less__zero,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger ) ).

% not_square_less_zero
thf(fact_2409_mult__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2410_mult__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2411_mult__neg__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2412_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2413_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2414_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2415_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_2416_less__numeral__extra_I1_J,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% less_numeral_extra(1)
thf(fact_2417_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2418_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2419_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2420_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_2421_zero__less__one,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% zero_less_one
thf(fact_2422_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2423_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2424_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2425_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_2426_not__one__less__zero,axiom,
    ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).

% not_one_less_zero
thf(fact_2427_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2428_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2429_less__iff__diff__less__0,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ zero_z3403309356797280102nteger ) ) ) ).

% less_iff_diff_less_0
thf(fact_2430_subset__minus__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( minus_1356011639430497352at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
thf(fact_2431_subset__minus__empty,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ( minus_minus_set_o @ A @ B )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_2432_subset__minus__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( minus_minus_set_int @ A @ B )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_2433_subset__minus__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( minus_minus_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_2434_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M2 )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2435_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2436_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2437_less__1__mult,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M2 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2438_less__1__mult,axiom,
    ! [M2: code_natural,N: code_natural] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ M2 )
     => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ N )
       => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2439_disjoint__alt__simp2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A @ B )
       != A )
      = ( ( inf_in2572325071724192079at_nat @ A @ B )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
thf(fact_2440_disjoint__alt__simp2,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( minus_minus_set_o @ A @ B )
       != A )
      = ( ( inf_inf_set_o @ A @ B )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
thf(fact_2441_disjoint__alt__simp2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( minus_minus_set_int @ A @ B )
       != A )
      = ( ( inf_inf_set_int @ A @ B )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
thf(fact_2442_disjoint__alt__simp2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B )
       != A )
      = ( ( inf_inf_set_nat @ A @ B )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
thf(fact_2443_disjoint__alt__simp1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A @ B )
        = A )
      = ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
thf(fact_2444_disjoint__alt__simp1,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( minus_minus_set_o @ A @ B )
        = A )
      = ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
thf(fact_2445_disjoint__alt__simp1,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( minus_minus_set_int @ A @ B )
        = A )
      = ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
thf(fact_2446_disjoint__alt__simp1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B )
        = A )
      = ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
thf(fact_2447_Int__Diff__disjoint,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ A @ B ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_2448_Int__Diff__disjoint,axiom,
    ! [A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B ) @ ( minus_minus_set_o @ A @ B ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_2449_Int__Diff__disjoint,axiom,
    ! [A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ A @ B ) @ ( minus_minus_set_int @ A @ B ) )
      = bot_bot_set_int ) ).

% Int_Diff_disjoint
thf(fact_2450_Int__Diff__disjoint,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_2451_Diff__triv,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2452_Diff__triv,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2453_Diff__triv,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int )
     => ( ( minus_minus_set_int @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2454_Diff__triv,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A @ B )
        = A ) ) ).

% Diff_triv
thf(fact_2455_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_2456_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_2457_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_2458_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_2459_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_2460_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_2461_mult__le__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2462_mult__le__cancel__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A2 @ B2 ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2463_mult__le__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A2 @ B2 ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2464_mult__le__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2465_mult__le__cancel__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A2 @ B2 ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2466_mult__le__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A2 @ B2 ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2467_mult__left__less__imp__less,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2468_mult__left__less__imp__less,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C2 @ A2 ) @ ( times_2397367101498566445atural @ C2 @ B2 ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C2 )
       => ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2469_mult__left__less__imp__less,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2470_mult__left__less__imp__less,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2471_mult__left__less__imp__less,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2472_mult__strict__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2473_mult__strict__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural,D2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ C2 @ D2 )
       => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C2 )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2474_mult__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2475_mult__strict__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2476_mult__strict__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2477_mult__less__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2478_mult__less__cancel__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2479_mult__less__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A2 @ B2 ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2480_mult__right__less__imp__less,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2481_mult__right__less__imp__less,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ C2 ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C2 )
       => ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2482_mult__right__less__imp__less,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2483_mult__right__less__imp__less,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2484_mult__right__less__imp__less,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2485_mult__strict__mono_H,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2486_mult__strict__mono_H,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural,D2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ C2 @ D2 )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C2 )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2487_mult__strict__mono_H,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2488_mult__strict__mono_H,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2489_mult__strict__mono_H,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2490_mult__less__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2491_mult__less__cancel__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A2 @ B2 ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2492_mult__less__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A2 @ B2 ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2493_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2494_mult__le__cancel__left__neg,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
        = ( ord_le3102999989581377725nteger @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2495_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2496_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2497_mult__le__cancel__left__pos,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
        = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2498_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2499_mult__left__le__imp__le,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2500_mult__left__le__imp__le,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ C2 @ A2 ) @ ( times_2397367101498566445atural @ C2 @ B2 ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C2 )
       => ( ord_le1926595141338095240atural @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2501_mult__left__le__imp__le,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2502_mult__left__le__imp__le,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2503_mult__left__le__imp__le,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2504_mult__right__le__imp__le,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2505_mult__right__le__imp__le,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ C2 ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C2 )
       => ( ord_le1926595141338095240atural @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2506_mult__right__le__imp__le,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
       => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2507_mult__right__le__imp__le,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2508_mult__right__le__imp__le,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2509_mult__le__less__imp__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2510_mult__le__less__imp__less,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural,D2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ C2 @ D2 )
       => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C2 )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2511_mult__le__less__imp__less,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2512_mult__le__less__imp__less,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2513_mult__le__less__imp__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2514_mult__less__le__imp__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2515_mult__less__le__imp__less,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural,D2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le1926595141338095240atural @ C2 @ D2 )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
         => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C2 )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2516_mult__less__le__imp__less,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2517_mult__less__le__imp__less,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2518_mult__less__le__imp__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2519_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_2520_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_2521_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_2522_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_2523_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_2524_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_2525_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2526_ssubst__Pair__rhs,axiom,
    ! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S2: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S3: produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2527_ssubst__Pair__rhs,axiom,
    ! [R2: produc3658429121746597890et_nat > $o,S2: produc3658429121746597890et_nat,R: set_Pr3286484037609594932et_nat,S3: produc3658429121746597890et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2528_ssubst__Pair__rhs,axiom,
    ! [R2: produc3658429121746597890et_nat > $o,S2: produc3925858234332021118et_nat,R: set_Pr8536935166611901872et_nat,S3: produc3925858234332021118et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2529_ssubst__Pair__rhs,axiom,
    ! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S2: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S3: product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2530_ssubst__Pair__rhs,axiom,
    ! [R2: int > option6357759511663192854e_term,S2: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S3: product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_2531_mult__less__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2532_mult__less__cancel__right2,axiom,
    ! [A2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2533_mult__less__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2534_mult__less__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2535_mult__less__cancel__right1,axiom,
    ! [C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2536_mult__less__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2537_mult__less__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2538_mult__less__cancel__left2,axiom,
    ! [C2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2539_mult__less__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2540_mult__less__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2541_mult__less__cancel__left1,axiom,
    ! [C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2542_mult__less__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2543_mult__le__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2544_mult__le__cancel__right2,axiom,
    ! [A2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2545_mult__le__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2546_mult__le__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2547_mult__le__cancel__right1,axiom,
    ! [C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B2 ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B2 @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2548_mult__le__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2549_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_2550_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2551_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z @ X ) @ ( times_3573771949741848930nteger @ Z @ Y ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2552_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2553_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2554_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2555_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2556_exI__realizer,axiom,
    ! [P: produc8923325533196201883nteger > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,Y: produc8923325533196201883nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X @ Y ) ) @ ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_2557_exI__realizer,axiom,
    ! [P: produc3658429121746597890et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_2558_exI__realizer,axiom,
    ! [P: produc3925858234332021118et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_2559_exI__realizer,axiom,
    ! [P: product_prod_int_int > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,Y: product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_2560_exI__realizer,axiom,
    ! [P: product_prod_int_int > ( int > option6357759511663192854e_term ) > $o,Y: product_prod_int_int,X: int > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_2561_exI__realizer,axiom,
    ! [P: set_nat > heap_e7401611519738050253t_unit > $o,Y: set_nat,X: heap_e7401611519738050253t_unit] :
      ( ( P @ Y @ X )
     => ( P @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) ) @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_2562_exI__realizer,axiom,
    ! [P: int > int > $o,Y: int,X: int] :
      ( ( P @ Y @ X )
     => ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_2563_exI__realizer,axiom,
    ! [P: produc7822875418678951345atural > code_natural > $o,Y: produc7822875418678951345atural,X: code_natural] :
      ( ( P @ Y @ X )
     => ( P @ ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X @ Y ) ) @ ( produc497848011034438852atural @ ( produc6639722614265839536atural @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_2564_conjI__realizer,axiom,
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,P5: produc6241069584506657477e_term > option6357759511663192854e_term,Q: produc8923325533196201883nteger > $o,Q5: produc8923325533196201883nteger] :
      ( ( P @ P5 )
     => ( ( Q @ Q5 )
       => ( ( P @ ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ P5 @ Q5 ) ) )
          & ( Q @ ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ P5 @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_2565_conjI__realizer,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P5: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,Q5: produc3658429121746597890et_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q5 )
       => ( ( P @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ P5 @ Q5 ) ) )
          & ( Q @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ P5 @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_2566_conjI__realizer,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P5: produc3658429121746597890et_nat > $o,Q: produc3925858234332021118et_nat > $o,Q5: produc3925858234332021118et_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q5 )
       => ( ( P @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ P5 @ Q5 ) ) )
          & ( Q @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ P5 @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_2567_conjI__realizer,axiom,
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,P5: produc8551481072490612790e_term > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q5: product_prod_int_int] :
      ( ( P @ P5 )
     => ( ( Q @ Q5 )
       => ( ( P @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ P5 @ Q5 ) ) )
          & ( Q @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ P5 @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_2568_conjI__realizer,axiom,
    ! [P: ( int > option6357759511663192854e_term ) > $o,P5: int > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q5: product_prod_int_int] :
      ( ( P @ P5 )
     => ( ( Q @ Q5 )
       => ( ( P @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ P5 @ Q5 ) ) )
          & ( Q @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ P5 @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_2569_conjI__realizer,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,P5: heap_e7401611519738050253t_unit,Q: set_nat > $o,Q5: set_nat] :
      ( ( P @ P5 )
     => ( ( Q @ Q5 )
       => ( ( P @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ P5 @ Q5 ) ) )
          & ( Q @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ P5 @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_2570_conjI__realizer,axiom,
    ! [P: int > $o,P5: int,Q: int > $o,Q5: int] :
      ( ( P @ P5 )
     => ( ( Q @ Q5 )
       => ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P5 @ Q5 ) ) )
          & ( Q @ ( product_snd_int_int @ ( product_Pair_int_int @ P5 @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_2571_conjI__realizer,axiom,
    ! [P: code_natural > $o,P5: code_natural,Q: produc7822875418678951345atural > $o,Q5: produc7822875418678951345atural] :
      ( ( P @ P5 )
     => ( ( Q @ Q5 )
       => ( ( P @ ( produc497848011034438852atural @ ( produc6639722614265839536atural @ P5 @ Q5 ) ) )
          & ( Q @ ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ P5 @ Q5 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_2572_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A2: produc1908205239877642774nteger] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc8603105652947943368nteger @ X @ Y ) )
       => ( P @ ( produc7822682618958472924nteger @ A2 ) @ ( produc7856867400915047194nteger @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2573_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3925858234332021118et_nat] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc5001842942810119800et_nat @ X @ Y ) )
       => ( P @ ( produc995936583742144908et_nat @ A2 ) @ ( produc4011572625026189258et_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2574_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc2732055786443039994et_nat] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc2245416461498447860et_nat @ X @ Y ) )
       => ( P @ ( produc180342877477747464et_nat @ A2 ) @ ( produc5374455773327741254et_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2575_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A2: produc2285326912895808259nt_int] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc5700946648718959541nt_int @ X @ Y ) )
       => ( P @ ( produc6816164490631068361nt_int @ A2 ) @ ( produc7328097813583171335nt_int @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2576_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,X: int > option6357759511663192854e_term,Y: product_prod_int_int,A2: produc7773217078559923341nt_int] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc4305682042979456191nt_int @ X @ Y ) )
       => ( P @ ( produc6230002227079971283nt_int @ A2 ) @ ( produc3162348030201620241nt_int @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2577_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: heap_e7401611519738050253t_unit > set_nat > $o,X: heap_e7401611519738050253t_unit,Y: set_nat,A2: produc3658429121746597890et_nat] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc7507926704131184380et_nat @ X @ Y ) )
       => ( P @ ( produc1824681642469235216et_nat @ A2 ) @ ( produc8586169260539613262et_nat @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2578_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: int > int > $o,X: int,Y: int,A2: product_prod_int_int] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( product_Pair_int_int @ X @ Y ) )
       => ( P @ ( product_fst_int_int @ A2 ) @ ( product_snd_int_int @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2579_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: code_natural > produc7822875418678951345atural > $o,X: code_natural,Y: produc7822875418678951345atural,A2: produc5835291356934675326atural] :
      ( ( P @ X @ Y )
     => ( ( A2
          = ( produc6639722614265839536atural @ X @ Y ) )
       => ( P @ ( produc497848011034438852atural @ A2 ) @ ( produc6591912806276919810atural @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2580_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2581_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2582_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
        = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2583_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila2554085542299052326_set_o @ inf_inf_set_o @ top_top_set_o @ ord_less_eq_set_o @ ord_less_set_o ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2584_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4423693302617054363t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ ord_le3507040750410214029t_unit @ ord_le8056459307392131481t_unit ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2585_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila8603258263270017530r_assn @ inf_inf_assn @ top_top_assn @ ord_less_eq_assn @ ord_less_assn ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2586_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2587_sgn__neg,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( sgn_sgn_int @ A2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% sgn_neg
thf(fact_2588_sgn__neg,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( sgn_sgn_rat @ A2 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% sgn_neg
thf(fact_2589_sgn__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( sgn_sgn_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% sgn_neg
thf(fact_2590_top_Oordering__top__axioms,axiom,
    ordering_top_set_o @ ord_less_eq_set_o @ ord_less_set_o @ top_top_set_o ).

% top.ordering_top_axioms
thf(fact_2591_top_Oordering__top__axioms,axiom,
    orderi2935705153647960990t_unit @ ord_le3507040750410214029t_unit @ ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit ).

% top.ordering_top_axioms
thf(fact_2592_top_Oordering__top__axioms,axiom,
    ordering_top_assn @ ord_less_eq_assn @ ord_less_assn @ top_top_assn ).

% top.ordering_top_axioms
thf(fact_2593_top_Oordering__top__axioms,axiom,
    ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).

% top.ordering_top_axioms
thf(fact_2594_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_2595_sgn__1,axiom,
    ( ( sgn_sgn_rat @ one_one_rat )
    = one_one_rat ) ).

% sgn_1
thf(fact_2596_sgn__1,axiom,
    ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% sgn_1
thf(fact_2597_sgn__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( sgn_sgn_rat @ A2 )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_2598_sgn__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( sgn_sgn_int @ A2 )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_2599_sgn__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( sgn_sgn_Code_integer @ A2 )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_2600_less__assn__def,axiom,
    ( ord_less_assn
    = ( ^ [A3: assn,B3: assn] :
          ( ( ord_less_eq_assn @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% less_assn_def
thf(fact_2601_sgn__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A2 ) @ ( sgn_sgn_Code_integer @ B2 ) ) ) ).

% sgn_mult
thf(fact_2602_sgn__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( sgn_sgn_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( times_times_rat @ ( sgn_sgn_rat @ A2 ) @ ( sgn_sgn_rat @ B2 ) ) ) ).

% sgn_mult
thf(fact_2603_sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A2 @ B2 ) )
      = ( times_times_int @ ( sgn_sgn_int @ A2 ) @ ( sgn_sgn_int @ B2 ) ) ) ).

% sgn_mult
thf(fact_2604_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_2605_sgn__minus__1,axiom,
    ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% sgn_minus_1
thf(fact_2606_sgn__minus__1,axiom,
    ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% sgn_minus_1
thf(fact_2607_semilattice__neutr__order_Oaxioms_I1_J,axiom,
    ! [F: nat > nat > nat,Z: nat,Less_eq: nat > nat > $o,Less: nat > nat > $o] :
      ( ( semila1623282765462674594er_nat @ F @ Z @ Less_eq @ Less )
     => ( semila9081495762789891438tr_nat @ F @ Z ) ) ).

% semilattice_neutr_order.axioms(1)
thf(fact_2608_sgn__1__pos,axiom,
    ! [A2: rat] :
      ( ( ( sgn_sgn_rat @ A2 )
        = one_one_rat )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% sgn_1_pos
thf(fact_2609_sgn__1__pos,axiom,
    ! [A2: int] :
      ( ( ( sgn_sgn_int @ A2 )
        = one_one_int )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% sgn_1_pos
thf(fact_2610_sgn__1__pos,axiom,
    ! [A2: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A2 )
        = one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% sgn_1_pos
thf(fact_2611_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X2: int] : ( if_int @ ( X2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_2612_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X2: rat] : ( if_rat @ ( X2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_2613_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X2: code_integer] : ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X2 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_2614_sgn__1__neg,axiom,
    ! [A2: int] :
      ( ( ( sgn_sgn_int @ A2 )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% sgn_1_neg
thf(fact_2615_sgn__1__neg,axiom,
    ! [A2: rat] :
      ( ( ( sgn_sgn_rat @ A2 )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% sgn_1_neg
thf(fact_2616_sgn__1__neg,axiom,
    ! [A2: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% sgn_1_neg
thf(fact_2617_convex__bound__lt,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A2 )
     => ( ( ord_less_rat @ Y @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2618_convex__bound__lt,axiom,
    ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ A2 )
     => ( ( ord_le6747313008572928689nteger @ Y @ A2 )
       => ( ( 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2619_convex__bound__lt,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A2 )
     => ( ( ord_less_int @ Y @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2620_le__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_2621_minus__divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_2622_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_2623_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_2624_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_2625_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_2626_prod_Oswap__def,axiom,
    ( produc3254817899252985374e_term
    = ( ^ [P7: produc998928222018024388e_term] : ( produc8603105652947943368nteger @ ( produc1152141638515386128e_term @ P7 ) @ ( produc1117956856558811858e_term @ P7 ) ) ) ) ).

% prod.swap_def
thf(fact_2627_prod_Oswap__def,axiom,
    ( produc1078497009909708626_nat_o
    = ( ^ [P7: produc6907119039986491712_nat_o] : ( produc5001842942810119800et_nat @ ( produc1838968430882301508_nat_o @ P7 ) @ ( produc8046704426453032966_nat_o @ P7 ) ) ) ) ).

% prod.swap_def
thf(fact_2628_prod_Oswap__def,axiom,
    ( produc2580581461503396054_nat_o
    = ( ^ [P7: produc4962441903510821572_nat_o] : ( produc2245416461498447860et_nat @ ( produc6803565743942592456_nat_o @ P7 ) @ ( produc1609452848092598666_nat_o @ P7 ) ) ) ) ).

% prod.swap_def
thf(fact_2629_prod_Oswap__def,axiom,
    ( produc1360913580328244269e_term
    = ( ^ [P7: produc1821397396195235155e_term] : ( produc5700946648718959541nt_int @ ( produc6847057335086834975e_term @ P7 ) @ ( produc6335124012134732001e_term @ P7 ) ) ) ) ).

% prod.swap_def
thf(fact_2630_prod_Oswap__def,axiom,
    ( produc2386993194027891363e_term
    = ( ^ [P7: produc3832618445565238089e_term] : ( produc4305682042979456191nt_int @ ( produc6373639247523685013e_term @ P7 ) @ ( produc217921407547260247e_term @ P7 ) ) ) ) ).

% prod.swap_def
thf(fact_2631_prod_Oswap__def,axiom,
    ( produc6300697153377358172et_nat
    = ( ^ [P7: produc3658429121746597890et_nat] : ( produc2471974422597015918t_unit @ ( produc8586169260539613262et_nat @ P7 ) @ ( produc1824681642469235216et_nat @ P7 ) ) ) ) ).

% prod.swap_def
thf(fact_2632_prod_Oswap__def,axiom,
    ( product_swap_int_int
    = ( ^ [P7: product_prod_int_int] : ( product_Pair_int_int @ ( product_snd_int_int @ P7 ) @ ( product_fst_int_int @ P7 ) ) ) ) ).

% prod.swap_def
thf(fact_2633_prod_Oswap__def,axiom,
    ( produc850777735424141840atural
    = ( ^ [P7: produc5835291356934675326atural] : ( produc1272419604612767874atural @ ( produc6591912806276919810atural @ P7 ) @ ( produc497848011034438852atural @ P7 ) ) ) ) ).

% prod.swap_def
thf(fact_2634_le__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
        = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2635_add__right__cancel,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B2 @ A2 )
        = ( plus_p5714425477246183910nteger @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2636_add__right__cancel,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B2 @ A2 )
        = ( plus_p7104986032573967614at_nat @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2637_add__right__cancel,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2638_add__right__cancel,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2639_add__right__cancel,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2640_add__left__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = ( plus_p5714425477246183910nteger @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2641_add__left__cancel,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A2 @ B2 )
        = ( plus_p7104986032573967614at_nat @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2642_add__left__cancel,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2643_add__left__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2644_add__left__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2645_add__le__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_2646_add__le__cancel__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_2647_add__le__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_2648_add__le__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_2649_add__le__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_2650_add__le__cancel__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C2 @ A2 ) @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_2651_add__le__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_2652_add__le__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_2653_add__0,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
      = A2 ) ).

% add_0
thf(fact_2654_add__0,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_2655_add__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add_0
thf(fact_2656_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_2657_add__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add_0
thf(fact_2658_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_2659_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_2660_add__cancel__right__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
thf(fact_2661_add__cancel__right__right,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A2
        = ( plus_p7104986032573967614at_nat @ A2 @ B2 ) )
      = ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_right
thf(fact_2662_add__cancel__right__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( plus_plus_rat @ A2 @ B2 ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_2663_add__cancel__right__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ A2 @ B2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_2664_add__cancel__right__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( plus_plus_int @ A2 @ B2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_2665_add__cancel__right__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ B2 @ A2 ) )
      = ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
thf(fact_2666_add__cancel__right__left,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A2
        = ( plus_p7104986032573967614at_nat @ B2 @ A2 ) )
      = ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_left
thf(fact_2667_add__cancel__right__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( plus_plus_rat @ B2 @ A2 ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_2668_add__cancel__right__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ B2 @ A2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_2669_add__cancel__right__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( plus_plus_int @ B2 @ A2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_2670_add__cancel__left__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
thf(fact_2671_add__cancel__left__right,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_right
thf(fact_2672_add__cancel__left__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_2673_add__cancel__left__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_2674_add__cancel__left__right,axiom,
    ! [A2: int,B2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_2675_add__cancel__left__left,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
thf(fact_2676_add__cancel__left__left,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_left
thf(fact_2677_add__cancel__left__left,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_2678_add__cancel__left__left,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_2679_add__cancel__left__left,axiom,
    ! [B2: int,A2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_2680_double__zero__sym,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
thf(fact_2681_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A2 @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2682_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A2 @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2683_add_Oright__neutral,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% add.right_neutral
thf(fact_2684_add_Oright__neutral,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A2 @ zero_z1048942125864253310at_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_2685_add_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.right_neutral
thf(fact_2686_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_2687_add_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.right_neutral
thf(fact_2688_add__less__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2689_add__less__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2690_add__less__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2691_add__less__cancel__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2692_add__less__cancel__right,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ C2 ) @ ( plus_p4538020629002901425atural @ B2 @ C2 ) )
      = ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2693_add__less__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2694_add__less__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2695_add__less__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2696_add__less__cancel__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C2 @ A2 ) @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2697_add__less__cancel__left,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ C2 @ A2 ) @ ( plus_p4538020629002901425atural @ C2 @ B2 ) )
      = ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2698_add__diff__cancel__right_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2699_add__diff__cancel__right_H,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2700_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2701_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2702_add__diff__cancel__right_H,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2703_add__diff__cancel__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
      = ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2704_add__diff__cancel__right,axiom,
    ! [A2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ C2 ) @ ( plus_p7104986032573967614at_nat @ B2 @ C2 ) )
      = ( minus_4286766774447292334at_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2705_add__diff__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2706_add__diff__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2707_add__diff__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2708_add__diff__cancel__left_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2709_add__diff__cancel__left_H,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2710_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2711_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2712_add__diff__cancel__left_H,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2713_add__diff__cancel__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ C2 @ A2 ) @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) )
      = ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2714_add__diff__cancel__left,axiom,
    ! [C2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ C2 @ A2 ) @ ( plus_p7104986032573967614at_nat @ C2 @ B2 ) )
      = ( minus_4286766774447292334at_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2715_add__diff__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
      = ( minus_minus_rat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2716_add__diff__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2717_add__diff__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
      = ( minus_minus_int @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2718_diff__add__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_2719_diff__add__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_2720_diff__add__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_2721_add__diff__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_2722_add__diff__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_2723_add__diff__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_2724_minus__add__distrib,axiom,
    ! [A2: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_2725_minus__add__distrib,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_2726_minus__add__distrib,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_2727_minus__add__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_2728_minus__add__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_2729_minus__add__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_2730_add__minus__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_2731_add__minus__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_2732_add__minus__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_2733_times__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 ) ) ).

% times_divide_eq_right
thf(fact_2734_divide__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( divide_divide_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_2735_divide__divide__eq__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A2 @ B2 ) @ C2 )
      = ( divide_divide_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% divide_divide_eq_left
thf(fact_2736_times__divide__eq__left,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
      = ( divide_divide_rat @ ( times_times_rat @ B2 @ A2 ) @ C2 ) ) ).

% times_divide_eq_left
thf(fact_2737_div__by__1,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ one_one_nat )
      = A2 ) ).

% div_by_1
thf(fact_2738_div__by__1,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ one_one_int )
      = A2 ) ).

% div_by_1
thf(fact_2739_div__by__1,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ one_one_rat )
      = A2 ) ).

% div_by_1
thf(fact_2740_div__by__1,axiom,
    ! [A2: code_integer] :
      ( ( divide6298287555418463151nteger @ A2 @ one_one_Code_integer )
      = A2 ) ).

% div_by_1
thf(fact_2741_div__by__1,axiom,
    ! [A2: code_natural] :
      ( ( divide5121882707175180666atural @ A2 @ one_one_Code_natural )
      = A2 ) ).

% div_by_1
thf(fact_2742_swap__simp,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( produc3254817899252985374e_term @ ( produc1898379890548282302e_term @ X @ Y ) )
      = ( produc8603105652947943368nteger @ Y @ X ) ) ).

% swap_simp
thf(fact_2743_swap__simp,axiom,
    ! [X: produc3658429121746597890et_nat,Y: produc3658429121746597890et_nat > $o] :
      ( ( produc1078497009909708626_nat_o @ ( produc2829238748666232050_nat_o @ X @ Y ) )
      = ( produc5001842942810119800et_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_2744_swap__simp,axiom,
    ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o] :
      ( ( produc2580581461503396054_nat_o @ ( produc3674526432113299062_nat_o @ X @ Y ) )
      = ( produc2245416461498447860et_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_2745_swap__simp,axiom,
    ! [X: product_prod_int_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( produc1360913580328244269e_term @ ( produc5219906170222623181e_term @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ Y @ X ) ) ).

% swap_simp
thf(fact_2746_swap__simp,axiom,
    ! [X: product_prod_int_int,Y: int > option6357759511663192854e_term] :
      ( ( produc2386993194027891363e_term @ ( produc7516973260301520963e_term @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ Y @ X ) ) ).

% swap_simp
thf(fact_2747_swap__simp,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc736171624797870632nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( produc1898379890548282302e_term @ Y @ X ) ) ).

% swap_simp
thf(fact_2748_swap__simp,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( produc3251101204053596376et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
      = ( produc2829238748666232050_nat_o @ Y @ X ) ) ).

% swap_simp
thf(fact_2749_swap__simp,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( produc1151471490888544852et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
      = ( produc3674526432113299062_nat_o @ Y @ X ) ) ).

% swap_simp
thf(fact_2750_swap__simp,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc1841954058824580629nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc5219906170222623181e_term @ Y @ X ) ) ).

% swap_simp
thf(fact_2751_swap__simp,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc8399074013560602399nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc7516973260301520963e_term @ Y @ X ) ) ).

% swap_simp
thf(fact_2752_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2753_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2754_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2755_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2756_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2757_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2758_le__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B2 @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_2759_le__add__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_2760_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_2761_le__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_2762_le__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_2763_le__add__same__cancel1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_2764_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_2765_le__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_2766_add__le__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_2767_add__le__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ B2 )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel2
thf(fact_2768_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_2769_add__le__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_2770_add__le__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_2771_add__le__same__cancel1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) @ B2 )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel1
thf(fact_2772_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_2773_add__le__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_2774_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2775_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2776_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2777_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2778_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2779_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2780_less__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B2 @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2781_less__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2782_less__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2783_less__add__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2784_less__add__same__cancel2,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ ( plus_p4538020629002901425atural @ B2 @ A2 ) )
      = ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2785_less__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2786_less__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2787_less__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2788_less__add__same__cancel1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2789_less__add__same__cancel1,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ ( plus_p4538020629002901425atural @ A2 @ B2 ) )
      = ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2790_add__less__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_2791_add__less__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_2792_add__less__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_2793_add__less__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ B2 )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel2
thf(fact_2794_add__less__same__cancel2,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) @ B2 )
      = ( ord_le5570908160329646204atural @ A2 @ zero_z2226904508553997617atural ) ) ).

% add_less_same_cancel2
thf(fact_2795_add__less__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A2 ) @ B2 )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_2796_add__less__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_2797_add__less__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_2798_add__less__same__cancel1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) @ B2 )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel1
thf(fact_2799_add__less__same__cancel1,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ B2 @ A2 ) @ B2 )
      = ( ord_le5570908160329646204atural @ A2 @ zero_z2226904508553997617atural ) ) ).

% add_less_same_cancel1
thf(fact_2800_diff__add__zero,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) )
      = zero_z1048942125864253310at_nat ) ).

% diff_add_zero
thf(fact_2801_diff__add__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_2802_ab__left__minus,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_2803_ab__left__minus,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_2804_ab__left__minus,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_2805_add_Oright__inverse,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_2806_add_Oright__inverse,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_2807_add_Oright__inverse,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_2808_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2809_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2810_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2811_nonzero__mult__div__cancel__right,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2812_nonzero__mult__div__cancel__right,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2813_nonzero__mult__div__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2814_nonzero__mult__div__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2815_nonzero__mult__div__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2816_nonzero__mult__div__cancel__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2817_nonzero__mult__div__cancel__left,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( A2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2818_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2819_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2820_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ B2 @ C2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2821_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2822_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ( C2 = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
          = zero_zero_rat ) )
      & ( ( C2 != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
          = ( divide_divide_rat @ A2 @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2823_div__self,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( divide_divide_nat @ A2 @ A2 )
        = one_one_nat ) ) ).

% div_self
thf(fact_2824_div__self,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( divide_divide_int @ A2 @ A2 )
        = one_one_int ) ) ).

% div_self
thf(fact_2825_div__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ A2 )
        = one_one_rat ) ) ).

% div_self
thf(fact_2826_div__self,axiom,
    ! [A2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ A2 @ A2 )
        = one_one_Code_integer ) ) ).

% div_self
thf(fact_2827_div__self,axiom,
    ! [A2: code_natural] :
      ( ( A2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ A2 @ A2 )
        = one_one_Code_natural ) ) ).

% div_self
thf(fact_2828_divide__eq__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ B2 )
        = one_one_rat )
      = ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_2829_one__eq__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A2 @ B2 ) )
      = ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_2830_divide__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ A2 )
        = one_one_rat ) ) ).

% divide_self
thf(fact_2831_divide__self__if,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( ( divide_divide_rat @ A2 @ A2 )
          = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( ( divide_divide_rat @ A2 @ A2 )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_2832_divide__eq__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ A2 )
        = one_one_rat )
      = ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
thf(fact_2833_eq__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B2 @ A2 ) )
      = ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
thf(fact_2834_one__divide__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_2835_zero__eq__1__divide__iff,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_2836_uminus__add__conv__diff,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( minus_minus_int @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_2837_uminus__add__conv__diff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( minus_minus_rat @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_2838_uminus__add__conv__diff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( minus_8373710615458151222nteger @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_2839_diff__minus__eq__add,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( plus_plus_int @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_2840_diff__minus__eq__add,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
      = ( plus_plus_rat @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_2841_diff__minus__eq__add,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_2842_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_2843_divide__sgn,axiom,
    ! [A2: rat,B2: rat] :
      ( ( divide_divide_rat @ A2 @ ( sgn_sgn_rat @ B2 ) )
      = ( times_times_rat @ A2 @ ( sgn_sgn_rat @ B2 ) ) ) ).

% divide_sgn
thf(fact_2844_divide__le__0__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_2845_zero__le__divide__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% zero_le_divide_1_iff
thf(fact_2846_divide__less__0__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_2847_divide__less__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
        = ( ord_less_rat @ A2 @ B2 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2848_divide__less__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
        = ( ord_less_rat @ B2 @ A2 ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2849_less__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
        = ( ord_less_rat @ B2 @ A2 ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2850_less__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
        = ( ord_less_rat @ A2 @ B2 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2851_zero__less__divide__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% zero_less_divide_1_iff
thf(fact_2852_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_2853_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_2854_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_2855_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_2856_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_2857_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_2858_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ ( times_times_rat @ A2 @ B2 ) )
        = ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2859_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A2 @ B2 ) )
        = ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2860_divide__le__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
        = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2861_divide__le__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
        = ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2862_le__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
        = ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2863_add__right__imp__eq,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B2 @ A2 )
        = ( plus_p5714425477246183910nteger @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2864_add__right__imp__eq,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B2 @ A2 )
        = ( plus_p7104986032573967614at_nat @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2865_add__right__imp__eq,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2866_add__right__imp__eq,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2867_add__right__imp__eq,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2868_add__left__imp__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = ( plus_p5714425477246183910nteger @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2869_add__left__imp__eq,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A2 @ B2 )
        = ( plus_p7104986032573967614at_nat @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2870_add__left__imp__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2871_add__left__imp__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2872_add__left__imp__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2873_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) )
      = ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2874_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ B2 @ ( plus_p7104986032573967614at_nat @ A2 @ C2 ) )
      = ( plus_p7104986032573967614at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2875_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2876_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2877_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2878_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_p5714425477246183910nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2879_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_p7104986032573967614at_nat
    = ( ^ [A3: multis2468970476368604999at_nat,B3: multis2468970476368604999at_nat] : ( plus_p7104986032573967614at_nat @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2880_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2881_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2882_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2883_add_Oright__cancel,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B2 @ A2 )
        = ( plus_p5714425477246183910nteger @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add.right_cancel
thf(fact_2884_add_Oright__cancel,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add.right_cancel
thf(fact_2885_add_Oright__cancel,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add.right_cancel
thf(fact_2886_add_Oleft__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = ( plus_p5714425477246183910nteger @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add.left_cancel
thf(fact_2887_add_Oleft__cancel,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add.left_cancel
thf(fact_2888_add_Oleft__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add.left_cancel
thf(fact_2889_add_Oassoc,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2890_add_Oassoc,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) @ C2 )
      = ( plus_p7104986032573967614at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2891_add_Oassoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2892_add_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2893_add_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2894_add_Oright__commute,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ B2 ) ) ).

% add.right_commute
thf(fact_2895_add_Oright__commute,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) @ C2 )
      = ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ C2 ) @ B2 ) ) ).

% add.right_commute
thf(fact_2896_add_Oright__commute,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ).

% add.right_commute
thf(fact_2897_add_Oright__commute,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B2 ) ) ).

% add.right_commute
thf(fact_2898_add_Oright__commute,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ).

% add.right_commute
thf(fact_2899_add_Oright__assoc,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) ) ) ).

% add.right_assoc
thf(fact_2900_add_Oright__assoc,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) @ C2 )
      = ( plus_p7104986032573967614at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ B2 @ C2 ) ) ) ).

% add.right_assoc
thf(fact_2901_add_Oright__assoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% add.right_assoc
thf(fact_2902_add_Oright__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% add.right_assoc
thf(fact_2903_add_Oright__assoc,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% add.right_assoc
thf(fact_2904_group__cancel_Oadd2,axiom,
    ! [B: code_integer,K: code_integer,B2: code_integer,A2: code_integer] :
      ( ( B
        = ( plus_p5714425477246183910nteger @ K @ B2 ) )
     => ( ( plus_p5714425477246183910nteger @ A2 @ B )
        = ( plus_p5714425477246183910nteger @ K @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2905_group__cancel_Oadd2,axiom,
    ! [B: multis2468970476368604999at_nat,K: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( B
        = ( plus_p7104986032573967614at_nat @ K @ B2 ) )
     => ( ( plus_p7104986032573967614at_nat @ A2 @ B )
        = ( plus_p7104986032573967614at_nat @ K @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2906_group__cancel_Oadd2,axiom,
    ! [B: rat,K: rat,B2: rat,A2: rat] :
      ( ( B
        = ( plus_plus_rat @ K @ B2 ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2907_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,B2: nat,A2: nat] :
      ( ( B
        = ( plus_plus_nat @ K @ B2 ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2908_group__cancel_Oadd2,axiom,
    ! [B: int,K: int,B2: int,A2: int] :
      ( ( B
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2909_group__cancel_Oadd1,axiom,
    ! [A: code_integer,K: code_integer,A2: code_integer,B2: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ K @ A2 ) )
     => ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = ( plus_p5714425477246183910nteger @ K @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2910_group__cancel_Oadd1,axiom,
    ! [A: multis2468970476368604999at_nat,K: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A
        = ( plus_p7104986032573967614at_nat @ K @ A2 ) )
     => ( ( plus_p7104986032573967614at_nat @ A @ B2 )
        = ( plus_p7104986032573967614at_nat @ K @ ( plus_p7104986032573967614at_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2911_group__cancel_Oadd1,axiom,
    ! [A: rat,K: rat,A2: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2912_group__cancel_Oadd1,axiom,
    ! [A: nat,K: nat,A2: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ K @ A2 ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2913_group__cancel_Oadd1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2914_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_p5714425477246183910nteger @ I @ K )
        = ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2915_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2916_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2917_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2918_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2919_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2920_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2921_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2922_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2923_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B2 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B2 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2924_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B2 )
          = ( divide_divide_rat @ ( plus_plus_rat @ A2 @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2925_gt__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B2 ) ) ).

% gt_half_sum
thf(fact_2926_less__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ A2 @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_2927_add__le__imp__le__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2928_add__le__imp__le__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
     => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2929_add__le__imp__le__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2930_add__le__imp__le__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2931_add__le__imp__le__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2932_add__le__imp__le__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C2 @ A2 ) @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) )
     => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2933_add__le__imp__le__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2934_add__le__imp__le__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2935_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C4: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_2936_add__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% add_right_mono
thf(fact_2937_add__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) ) ) ).

% add_right_mono
thf(fact_2938_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% add_right_mono
thf(fact_2939_add__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% add_right_mono
thf(fact_2940_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ~ ! [C3: nat] :
            ( B2
           != ( plus_plus_nat @ A2 @ C3 ) ) ) ).

% less_eqE
thf(fact_2941_add__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% add_left_mono
thf(fact_2942_add__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C2 @ A2 ) @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) ) ) ).

% add_left_mono
thf(fact_2943_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).

% add_left_mono
thf(fact_2944_add__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) ) ) ).

% add_left_mono
thf(fact_2945_add__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_2946_add__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ D2 )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_2947_add__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_2948_add__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_2949_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2950_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I @ J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2951_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2952_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2953_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2954_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I = J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2955_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2956_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2957_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2958_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I @ J )
        & ( K = L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2959_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2960_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2961_add_Ogroup__left__neutral,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2962_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2963_add_Ogroup__left__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2964_add_Ocomm__neutral,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% add.comm_neutral
thf(fact_2965_add_Ocomm__neutral,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A2 @ zero_z1048942125864253310at_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_2966_add_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.comm_neutral
thf(fact_2967_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_2968_add_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.comm_neutral
thf(fact_2969_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2970_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2971_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2972_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2973_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2974_divide__divide__eq__left_H,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A2 @ B2 ) @ C2 )
      = ( divide_divide_rat @ A2 @ ( times_times_rat @ C2 @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_2975_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_2976_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_2977_add__less__imp__less__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
     => ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2978_add__less__imp__less__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2979_add__less__imp__less__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
     => ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2980_add__less__imp__less__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
     => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2981_add__less__imp__less__right,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ C2 ) @ ( plus_p4538020629002901425atural @ B2 @ C2 ) )
     => ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2982_add__less__imp__less__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
     => ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2983_add__less__imp__less__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2984_add__less__imp__less__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
     => ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2985_add__less__imp__less__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C2 @ A2 ) @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) )
     => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2986_add__less__imp__less__left,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ C2 @ A2 ) @ ( plus_p4538020629002901425atural @ C2 @ B2 ) )
     => ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2987_add__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2988_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2989_add__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2990_add__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2991_add__strict__right__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ C2 ) @ ( plus_p4538020629002901425atural @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2992_add__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2993_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2994_add__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2995_add__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C2 @ A2 ) @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2996_add__strict__left__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ C2 @ A2 ) @ ( plus_p4538020629002901425atural @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2997_add__strict__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2998_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2999_add__strict__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3000_add__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3001_add__strict__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural,D2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ C2 @ D2 )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ C2 ) @ ( plus_p4538020629002901425atural @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3002_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3003_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3004_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3005_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( K = L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3006_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
      ( ( ( ord_le5570908160329646204atural @ I @ J )
        & ( K = L ) )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ I @ K ) @ ( plus_p4538020629002901425atural @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3007_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3008_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3009_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3010_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I = J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3011_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
      ( ( ( I = J )
        & ( ord_le5570908160329646204atural @ K @ L ) )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ I @ K ) @ ( plus_p4538020629002901425atural @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3012_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3013_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3014_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3015_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3016_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
      ( ( ( ord_le5570908160329646204atural @ I @ J )
        & ( ord_le5570908160329646204atural @ K @ L ) )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ I @ K ) @ ( plus_p4538020629002901425atural @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3017_combine__common__factor,axiom,
    ! [A2: code_integer,E: code_integer,B2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ C2 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_3018_combine__common__factor,axiom,
    ! [A2: rat,E: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_3019_combine__common__factor,axiom,
    ! [A2: nat,E: nat,B2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_3020_combine__common__factor,axiom,
    ! [A2: int,E: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_3021_distrib__right,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% distrib_right
thf(fact_3022_distrib__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% distrib_right
thf(fact_3023_distrib__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% distrib_right
thf(fact_3024_distrib__right,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).

% distrib_right
thf(fact_3025_distrib__left,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ).

% distrib_left
thf(fact_3026_distrib__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ).

% distrib_left
thf(fact_3027_distrib__left,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).

% distrib_left
thf(fact_3028_distrib__left,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).

% distrib_left
thf(fact_3029_comm__semiring__class_Odistrib,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3030_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3031_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3032_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3033_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3034_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3035_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3036_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3037_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3038_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3039_diff__diff__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
      = ( minus_8373710615458151222nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_3040_diff__diff__eq,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C2: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( minus_4286766774447292334at_nat @ A2 @ B2 ) @ C2 )
      = ( minus_4286766774447292334at_nat @ A2 @ ( plus_p7104986032573967614at_nat @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_3041_diff__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_3042_diff__diff__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_3043_diff__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_3044_add__implies__diff,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3045_add__implies__diff,axiom,
    ! [C2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_4286766774447292334at_nat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3046_add__implies__diff,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ( plus_plus_rat @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3047_add__implies__diff,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_minus_nat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3048_add__implies__diff,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( ( plus_plus_int @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_minus_int @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_3049_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
      = ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3050_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3051_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3052_diff__add__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq
thf(fact_3053_diff__add__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq
thf(fact_3054_diff__add__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq
thf(fact_3055_diff__diff__eq2,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_3056_diff__diff__eq2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_3057_diff__diff__eq2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_3058_add__diff__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 ) ) ).

% add_diff_eq
thf(fact_3059_add__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 ) ) ).

% add_diff_eq
thf(fact_3060_add__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% add_diff_eq
thf(fact_3061_eq__diff__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( A2
        = ( minus_8373710615458151222nteger @ C2 @ B2 ) )
      = ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_3062_eq__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( A2
        = ( minus_minus_rat @ C2 @ B2 ) )
      = ( ( plus_plus_rat @ A2 @ B2 )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_3063_eq__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( A2
        = ( minus_minus_int @ C2 @ B2 ) )
      = ( ( plus_plus_int @ A2 @ B2 )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_3064_diff__eq__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A2 @ B2 )
        = C2 )
      = ( A2
        = ( plus_p5714425477246183910nteger @ C2 @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_3065_diff__eq__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = C2 )
      = ( A2
        = ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_3066_diff__eq__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = C2 )
      = ( A2
        = ( plus_plus_int @ C2 @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_3067_group__cancel_Osub1,axiom,
    ! [A: code_integer,K: code_integer,A2: code_integer,B2: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ K @ A2 ) )
     => ( ( minus_8373710615458151222nteger @ A @ B2 )
        = ( plus_p5714425477246183910nteger @ K @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3068_group__cancel_Osub1,axiom,
    ! [A: rat,K: rat,A2: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( minus_minus_rat @ A @ B2 )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3069_group__cancel_Osub1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3070_add_Oinverse__distrib__swap,axiom,
    ! [A2: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3071_add_Oinverse__distrib__swap,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3072_add_Oinverse__distrib__swap,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3073_group__cancel_Oneg1,axiom,
    ! [A: int,K: int,A2: int] :
      ( ( A
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( uminus_uminus_int @ A )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_3074_group__cancel_Oneg1,axiom,
    ! [A: rat,K: rat,A2: rat] :
      ( ( A
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( uminus_uminus_rat @ A )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_3075_group__cancel_Oneg1,axiom,
    ! [A: code_integer,K: code_integer,A2: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ K @ A2 ) )
     => ( ( uminus1351360451143612070nteger @ A )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_3076_add_Oabel__semigroup__axioms,axiom,
    abel_s8772020223019846471nteger @ plus_p5714425477246183910nteger ).

% add.abel_semigroup_axioms
thf(fact_3077_add_Oabel__semigroup__axioms,axiom,
    abel_s8335990944309670109at_nat @ plus_p7104986032573967614at_nat ).

% add.abel_semigroup_axioms
thf(fact_3078_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_rat @ plus_plus_rat ).

% add.abel_semigroup_axioms
thf(fact_3079_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_nat @ plus_plus_nat ).

% add.abel_semigroup_axioms
thf(fact_3080_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_int @ plus_plus_int ).

% add.abel_semigroup_axioms
thf(fact_3081_add_Osemigroup__axioms,axiom,
    semigr7036097719740861967nteger @ plus_p5714425477246183910nteger ).

% add.semigroup_axioms
thf(fact_3082_add_Osemigroup__axioms,axiom,
    semigr4652511051630528533at_nat @ plus_p7104986032573967614at_nat ).

% add.semigroup_axioms
thf(fact_3083_add_Osemigroup__axioms,axiom,
    semigroup_rat @ plus_plus_rat ).

% add.semigroup_axioms
thf(fact_3084_add_Osemigroup__axioms,axiom,
    semigroup_nat @ plus_plus_nat ).

% add.semigroup_axioms
thf(fact_3085_add_Osemigroup__axioms,axiom,
    semigroup_int @ plus_plus_int ).

% add.semigroup_axioms
thf(fact_3086_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_3087_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B2 )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_3088_add_Oac__operator__axioms,axiom,
    syntax414191934074104506nteger @ plus_p5714425477246183910nteger ).

% add.ac_operator_axioms
thf(fact_3089_add_Oac__operator__axioms,axiom,
    syntax4222486904863270186at_nat @ plus_p7104986032573967614at_nat ).

% add.ac_operator_axioms
thf(fact_3090_add_Oac__operator__axioms,axiom,
    syntax8165409007564224565or_rat @ plus_plus_rat ).

% add.ac_operator_axioms
thf(fact_3091_add_Oac__operator__axioms,axiom,
    syntax8800539067650720301or_nat @ plus_plus_nat ).

% add.ac_operator_axioms
thf(fact_3092_add_Oac__operator__axioms,axiom,
    syntax8798048597141670025or_int @ plus_plus_int ).

% add.ac_operator_axioms
thf(fact_3093_add_Osafe__commute,axiom,
    ! [X: code_integer,Y: code_integer,A2: code_integer,B2: code_integer] :
      ( ( syntax380677951125133566nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ A2 )
     => ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = ( plus_p5714425477246183910nteger @ B2 @ A2 ) ) ) ).

% add.safe_commute
thf(fact_3094_add_Osafe__commute,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( syntax5084084251851430274at_nat @ ( plus_p7104986032573967614at_nat @ X @ Y ) @ A2 )
     => ( ( plus_p7104986032573967614at_nat @ A2 @ B2 )
        = ( plus_p7104986032573967614at_nat @ B2 @ A2 ) ) ) ).

% add.safe_commute
thf(fact_3095_add_Osafe__commute,axiom,
    ! [X: rat,Y: rat,A2: rat,B2: rat] :
      ( ( syntax3730441303064801268at_rat @ ( plus_plus_rat @ X @ Y ) @ A2 )
     => ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ B2 @ A2 ) ) ) ).

% add.safe_commute
thf(fact_3096_add_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A2: nat,B2: nat] :
      ( ( syntax4682126007086162916at_nat @ ( plus_plus_nat @ X @ Y ) @ A2 )
     => ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% add.safe_commute
thf(fact_3097_add_Osafe__commute,axiom,
    ! [X: int,Y: int,A2: int,B2: int] :
      ( ( syntax5678989248478167196nt_int @ ( plus_plus_int @ X @ Y ) @ A2 )
     => ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ B2 @ A2 ) ) ) ).

% add.safe_commute
thf(fact_3098_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_3099_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A2
          = ( divide_divide_rat @ B2 @ C2 ) )
        = ( ( times_times_rat @ A2 @ C2 )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3100_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B2 @ C2 )
          = A2 )
        = ( B2
          = ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3101_eq__divide__imp,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C2 )
          = B2 )
       => ( A2
          = ( divide_divide_rat @ B2 @ C2 ) ) ) ) ).

% eq_divide_imp
thf(fact_3102_divide__eq__imp,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B2
          = ( times_times_rat @ A2 @ C2 ) )
       => ( ( divide_divide_rat @ B2 @ C2 )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_3103_eq__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2
        = ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_3104_divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ C2 )
        = A2 )
      = ( ( ( C2 != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A2 @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_3105_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3106_right__inverse__eq,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A2 @ B2 )
          = one_one_rat )
        = ( A2 = B2 ) ) ) ).

% right_inverse_eq
thf(fact_3107_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_3108_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_3109_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_3110_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_3111_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_3112_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_3113_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_3114_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_3115_add__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3116_add__nonpos__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_nonpos
thf(fact_3117_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3118_add__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_3119_add__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3120_add__nonneg__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3121_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3122_add__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3123_add__increasing2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ B2 @ A2 )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_3124_add__increasing2,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
       => ( ord_le3102999989581377725nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_3125_add__increasing2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_3126_add__increasing2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_3127_add__decreasing2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_3128_add__decreasing2,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_3129_add__decreasing2,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_3130_add__decreasing2,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_3131_add__increasing,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% add_increasing
thf(fact_3132_add__increasing,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ord_le3102999989581377725nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ) ) ).

% add_increasing
thf(fact_3133_add__increasing,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% add_increasing
thf(fact_3134_add__increasing,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% add_increasing
thf(fact_3135_add__decreasing,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C2 @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_3136_add__decreasing,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ C2 @ B2 )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_3137_add__decreasing,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_3138_add__decreasing,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ C2 @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_3139_add__less__le__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3140_add__less__le__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural,D2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_le1926595141338095240atural @ C2 @ D2 )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ C2 ) @ ( plus_p4538020629002901425atural @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3141_add__less__le__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ C2 @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3142_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3143_add__less__le__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3144_add__le__less__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3145_add__le__less__mono,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural,D2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A2 @ B2 )
     => ( ( ord_le5570908160329646204atural @ C2 @ D2 )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ C2 ) @ ( plus_p4538020629002901425atural @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3146_add__le__less__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) @ ( plus_p5714425477246183910nteger @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3147_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3148_add__le__less__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3149_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3150_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
      ( ( ( ord_le5570908160329646204atural @ I @ J )
        & ( ord_le1926595141338095240atural @ K @ L ) )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ I @ K ) @ ( plus_p4538020629002901425atural @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3151_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3152_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3153_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3154_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3155_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: code_natural,J: code_natural,K: code_natural,L: code_natural] :
      ( ( ( ord_le1926595141338095240atural @ I @ J )
        & ( ord_le5570908160329646204atural @ K @ L ) )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ I @ K ) @ ( plus_p4538020629002901425atural @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3156_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I @ J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3157_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3158_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3159_pos__add__strict,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_3160_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_3161_pos__add__strict,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_3162_pos__add__strict,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ord_le6747313008572928689nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_3163_pos__add__strict,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ord_le5570908160329646204atural @ B2 @ ( plus_p4538020629002901425atural @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_3164_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ! [C3: nat] :
            ( ( B2
              = ( plus_plus_nat @ A2 @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_3165_add__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_3166_add__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_3167_add__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_3168_add__pos__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_3169_add__pos__pos,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_3170_add__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_3171_add__neg__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_3172_add__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_3173_add__neg__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_neg
thf(fact_3174_add__neg__neg,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ B2 @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) @ zero_z2226904508553997617atural ) ) ) ).

% add_neg_neg
thf(fact_3175_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A2 )
            = C2 )
          = ( B2
            = ( plus_plus_nat @ C2 @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3176_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_3177_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3178_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3179_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3180_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B2 ) @ A2 )
        = ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3181_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B2 ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3182_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3183_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% le_add_diff
thf(fact_3184_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_3185_le__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C2 @ B2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 ) ) ).

% le_diff_eq
thf(fact_3186_le__diff__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( minus_8373710615458151222nteger @ C2 @ B2 ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 ) ) ).

% le_diff_eq
thf(fact_3187_le__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C2 @ B2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% le_diff_eq
thf(fact_3188_diff__le__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% diff_le_eq
thf(fact_3189_diff__le__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
      = ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) ) ) ).

% diff_le_eq
thf(fact_3190_diff__le__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C2 @ B2 ) ) ) ).

% diff_le_eq
thf(fact_3191_less__add__one,axiom,
    ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).

% less_add_one
thf(fact_3192_less__add__one,axiom,
    ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).

% less_add_one
thf(fact_3193_less__add__one,axiom,
    ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).

% less_add_one
thf(fact_3194_less__add__one,axiom,
    ! [A2: code_integer] : ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_3195_less__add__one,axiom,
    ! [A2: code_natural] : ( ord_le5570908160329646204atural @ A2 @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) ) ).

% less_add_one
thf(fact_3196_add__mono1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_3197_add__mono1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_3198_add__mono1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).

% add_mono1
thf(fact_3199_add__mono1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B2 @ one_one_Code_integer ) ) ) ).

% add_mono1
thf(fact_3200_add__mono1,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ B2 @ one_one_Code_natural ) ) ) ).

% add_mono1
thf(fact_3201_less__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C2 @ B2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 ) ) ).

% less_diff_eq
thf(fact_3202_less__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( minus_minus_int @ C2 @ B2 ) )
      = ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% less_diff_eq
thf(fact_3203_less__diff__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( minus_8373710615458151222nteger @ C2 @ B2 ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 ) ) ).

% less_diff_eq
thf(fact_3204_diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% diff_less_eq
thf(fact_3205_diff__less__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( ord_less_int @ A2 @ ( plus_plus_int @ C2 @ B2 ) ) ) ).

% diff_less_eq
thf(fact_3206_diff__less__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
      = ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ C2 @ B2 ) ) ) ).

% diff_less_eq
thf(fact_3207_neg__eq__iff__add__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = B2 )
      = ( ( plus_plus_int @ A2 @ B2 )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3208_neg__eq__iff__add__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = B2 )
      = ( ( plus_plus_rat @ A2 @ B2 )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3209_neg__eq__iff__add__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = B2 )
      = ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3210_eq__neg__iff__add__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( uminus_uminus_int @ B2 ) )
      = ( ( plus_plus_int @ A2 @ B2 )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3211_eq__neg__iff__add__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ B2 ) )
      = ( ( plus_plus_rat @ A2 @ B2 )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3212_eq__neg__iff__add__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3213_add_Oinverse__unique,axiom,
    ! [A2: int,B2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_3214_add_Oinverse__unique,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_3215_add_Oinverse__unique,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_3216_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3217_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3218_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3219_add__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = zero_zero_int )
      = ( B2
        = ( uminus_uminus_int @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_3220_add__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = zero_zero_rat )
      = ( B2
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_3221_add__eq__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
      = ( B2
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_3222_square__diff__square__factored,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( minus_8373710615458151222nteger @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3223_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_3224_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_3225_eq__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C2 )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( C2
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3226_eq__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( C2
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3227_eq__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( C2
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3228_eq__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C2 )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ E ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3229_eq__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3230_eq__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3231_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3232_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3233_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3234_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3235_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3236_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3237_group__cancel_Osub2,axiom,
    ! [B: int,K: int,B2: int,A2: int] :
      ( ( B
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3238_group__cancel_Osub2,axiom,
    ! [B: rat,K: rat,B2: rat,A2: rat] :
      ( ( B
        = ( plus_plus_rat @ K @ B2 ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3239_group__cancel_Osub2,axiom,
    ! [B: code_integer,K: code_integer,B2: code_integer,A2: code_integer] :
      ( ( B
        = ( plus_p5714425477246183910nteger @ K @ B2 ) )
     => ( ( minus_8373710615458151222nteger @ A2 @ B )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3240_add_Ocomm__monoid__axioms,axiom,
    comm_m1654649897431166824nteger @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).

% add.comm_monoid_axioms
thf(fact_3241_add_Ocomm__monoid__axioms,axiom,
    comm_m1862543879040115196at_nat @ plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ).

% add.comm_monoid_axioms
thf(fact_3242_add_Ocomm__monoid__axioms,axiom,
    comm_monoid_rat @ plus_plus_rat @ zero_zero_rat ).

% add.comm_monoid_axioms
thf(fact_3243_add_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ plus_plus_nat @ zero_zero_nat ).

% add.comm_monoid_axioms
thf(fact_3244_add_Ocomm__monoid__axioms,axiom,
    comm_monoid_int @ plus_plus_int @ zero_zero_int ).

% add.comm_monoid_axioms
thf(fact_3245_add_Omonoid__axioms,axiom,
    monoid_Code_integer @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).

% add.monoid_axioms
thf(fact_3246_add_Omonoid__axioms,axiom,
    monoid66256789251010760at_nat @ plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ).

% add.monoid_axioms
thf(fact_3247_add_Omonoid__axioms,axiom,
    monoid_rat @ plus_plus_rat @ zero_zero_rat ).

% add.monoid_axioms
thf(fact_3248_add_Omonoid__axioms,axiom,
    monoid_nat @ plus_plus_nat @ zero_zero_nat ).

% add.monoid_axioms
thf(fact_3249_add_Omonoid__axioms,axiom,
    monoid_int @ plus_plus_int @ zero_zero_int ).

% add.monoid_axioms
thf(fact_3250_sum__list_Omonoid__list__axioms,axiom,
    groups5100778238730182072nteger @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).

% sum_list.monoid_list_axioms
thf(fact_3251_sum__list_Omonoid__list__axioms,axiom,
    groups5598625667918239660at_nat @ plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ).

% sum_list.monoid_list_axioms
thf(fact_3252_sum__list_Omonoid__list__axioms,axiom,
    groups7826448149055759155st_rat @ plus_plus_rat @ zero_zero_rat ).

% sum_list.monoid_list_axioms
thf(fact_3253_sum__list_Omonoid__list__axioms,axiom,
    groups8461578209142254891st_nat @ plus_plus_nat @ zero_zero_nat ).

% sum_list.monoid_list_axioms
thf(fact_3254_sum__list_Omonoid__list__axioms,axiom,
    groups8459087738633204615st_int @ plus_plus_int @ zero_zero_int ).

% sum_list.monoid_list_axioms
thf(fact_3255_sum__list_Ocomm__monoid__list__axioms,axiom,
    groups2567640417189887108nteger @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).

% sum_list.comm_monoid_list_axioms
thf(fact_3256_sum__list_Ocomm__monoid__list__axioms,axiom,
    groups7159712745983395808at_nat @ plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ).

% sum_list.comm_monoid_list_axioms
thf(fact_3257_sum__list_Ocomm__monoid__list__axioms,axiom,
    groups593308882113688063st_rat @ plus_plus_rat @ zero_zero_rat ).

% sum_list.comm_monoid_list_axioms
thf(fact_3258_sum__list_Ocomm__monoid__list__axioms,axiom,
    groups1228438942200183799st_nat @ plus_plus_nat @ zero_zero_nat ).

% sum_list.comm_monoid_list_axioms
thf(fact_3259_sum__list_Ocomm__monoid__list__axioms,axiom,
    groups1225948471691133523st_int @ plus_plus_int @ zero_zero_int ).

% sum_list.comm_monoid_list_axioms
thf(fact_3260_sum_Ocomm__monoid__list__set__axioms,axiom,
    groups7165323350619936122nteger @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).

% sum.comm_monoid_list_set_axioms
thf(fact_3261_sum_Ocomm__monoid__list__set__axioms,axiom,
    groups4109576622710581866at_nat @ plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ).

% sum.comm_monoid_list_set_axioms
thf(fact_3262_sum_Ocomm__monoid__list__set__axioms,axiom,
    groups2612948464029442805et_rat @ plus_plus_rat @ zero_zero_rat ).

% sum.comm_monoid_list_set_axioms
thf(fact_3263_sum_Ocomm__monoid__list__set__axioms,axiom,
    groups3248078524115938541et_nat @ plus_plus_nat @ zero_zero_nat ).

% sum.comm_monoid_list_set_axioms
thf(fact_3264_sum_Ocomm__monoid__list__set__axioms,axiom,
    groups3245588053606888265et_int @ plus_plus_int @ zero_zero_int ).

% sum.comm_monoid_list_set_axioms
thf(fact_3265_dbl__inc__def,axiom,
    ( neg_nu5831290666863070958nteger
    = ( ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).

% dbl_inc_def
thf(fact_3266_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_3267_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_3268_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3269_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3270_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3271_divide__strict__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C2 @ A2 ) @ ( divide_divide_rat @ C2 @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3272_divide__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C2 @ A2 ) @ ( divide_divide_rat @ C2 @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3273_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3274_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3275_pos__less__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_3276_pos__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3277_neg__less__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3278_neg__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_3279_less__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3280_divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3281_divide__less__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B2 @ A2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_3282_less__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% less_divide_eq_1
thf(fact_3283_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3284_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3285_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3286_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B2 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B2 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3287_add__strict__increasing2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_3288_add__strict__increasing2,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ C2 )
       => ( ord_le5570908160329646204atural @ B2 @ ( plus_p4538020629002901425atural @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_3289_add__strict__increasing2,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
       => ( ord_le6747313008572928689nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_3290_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_3291_add__strict__increasing2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_3292_add__strict__increasing,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_3293_add__strict__increasing,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le1926595141338095240atural @ B2 @ C2 )
       => ( ord_le5570908160329646204atural @ B2 @ ( plus_p4538020629002901425atural @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_3294_add__strict__increasing,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
       => ( ord_le6747313008572928689nteger @ B2 @ ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_3295_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_3296_add__strict__increasing,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_3297_add__pos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3298_add__pos__nonneg,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B2 )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3299_add__pos__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3300_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3301_add__pos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3302_add__nonpos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_3303_add__nonpos__neg,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A2 @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ B2 @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) @ zero_z2226904508553997617atural ) ) ) ).

% add_nonpos_neg
thf(fact_3304_add__nonpos__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_neg
thf(fact_3305_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_3306_add__nonpos__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_3307_add__nonneg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3308_add__nonneg__pos,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3309_add__nonneg__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3310_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3311_add__nonneg__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3312_add__neg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_3313_add__neg__nonpos,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ B2 @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) @ zero_z2226904508553997617atural ) ) ) ).

% add_neg_nonpos
thf(fact_3314_add__neg__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_nonpos
thf(fact_3315_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_3316_add__neg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_3317_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_3318_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_3319_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_3320_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C2
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) ) )
        = ( ( times_times_rat @ C2 @ B2 )
          = ( uminus_uminus_rat @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_3321_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) )
          = C2 )
        = ( ( uminus_uminus_rat @ A2 )
          = ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_3322_minus__divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) )
        = A2 )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B2 )
            = ( times_times_rat @ A2 @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_3323_eq__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ C2 )
            = ( uminus_uminus_rat @ B2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_3324_divide__eq__minus__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ B2 )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B2 != zero_zero_rat )
        & ( A2
          = ( uminus_uminus_rat @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_3325_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_3326_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_3327_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_3328_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_3329_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3330_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3331_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3332_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3333_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3334_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_3335_zero__less__two,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural ) ).

% zero_less_two
thf(fact_3336_le__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3337_le__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3338_le__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3339_le__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3340_le__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ C2 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3341_le__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3342_less__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3343_less__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3344_less__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3345_less__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3346_less__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3347_less__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E ) @ C2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ C2 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3348_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_3349_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_3350_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_3351_dbl__dec__def,axiom,
    ( neg_nu7757733837767384882nteger
    = ( ^ [X2: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).

% dbl_dec_def
thf(fact_3352_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_3353_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_3354_divide__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A2 ) @ ( divide_divide_rat @ C2 @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3355_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3356_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3357_pos__le__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_3358_pos__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3359_neg__le__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3360_neg__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_3361_divide__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A2 ) @ ( divide_divide_rat @ C2 @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_3362_le__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3363_divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3364_divide__le__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ A2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ A2 @ B2 ) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_3365_le__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% le_divide_eq_1
thf(fact_3366_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_3367_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_3368_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_3369_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_3370_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_3371_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_3372_minus__divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_3373_less__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_3374_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_3375_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B2 )
          = ( uminus_uminus_rat @ B2 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B2 )
          = ( divide_divide_rat @ ( minus_minus_rat @ A2 @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_3376_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B2 )
          = ( uminus_uminus_rat @ B2 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B2 )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_3377_convex__bound__le,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A2 )
     => ( ( ord_less_eq_rat @ Y @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3378_convex__bound__le,axiom,
    ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ A2 )
     => ( ( ord_le3102999989581377725nteger @ Y @ A2 )
       => ( ( 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3379_convex__bound__le,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ( ord_less_eq_int @ Y @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3380_div__mult__self4,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C2 ) @ A2 ) @ B2 )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_3381_div__mult__self4,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C2 ) @ A2 ) @ B2 )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_3382_div__mult__self4,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C2 ) @ A2 ) @ B2 )
        = ( plus_p5714425477246183910nteger @ C2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_3383_div__mult__self4,axiom,
    ! [B2: code_natural,C2: code_natural,A2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B2 @ C2 ) @ A2 ) @ B2 )
        = ( plus_p4538020629002901425atural @ C2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_3384_div__mult__self3,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B2 ) @ A2 ) @ B2 )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_3385_div__mult__self3,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B2 ) @ A2 ) @ B2 )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_3386_div__mult__self3,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C2 @ B2 ) @ A2 ) @ B2 )
        = ( plus_p5714425477246183910nteger @ C2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_3387_div__mult__self3,axiom,
    ! [B2: code_natural,C2: code_natural,A2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C2 @ B2 ) @ A2 ) @ B2 )
        = ( plus_p4538020629002901425atural @ C2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_3388_div__mult__self2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) @ B2 )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_3389_div__mult__self2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) @ B2 )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_3390_div__mult__self2,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) @ B2 )
        = ( plus_p5714425477246183910nteger @ C2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_3391_div__mult__self2,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A2 @ ( times_2397367101498566445atural @ B2 @ C2 ) ) @ B2 )
        = ( plus_p4538020629002901425atural @ C2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_3392_div__mult__self1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C2 @ B2 ) ) @ B2 )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_3393_div__mult__self1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C2 @ B2 ) ) @ B2 )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_3394_div__mult__self1,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) @ B2 )
        = ( plus_p5714425477246183910nteger @ C2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_3395_div__mult__self1,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A2 @ ( times_2397367101498566445atural @ C2 @ B2 ) ) @ B2 )
        = ( plus_p4538020629002901425atural @ C2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_3396_div__minus1__right,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A2 ) ) ).

% div_minus1_right
thf(fact_3397_div__minus1__right,axiom,
    ! [A2: code_integer] :
      ( ( divide6298287555418463151nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A2 ) ) ).

% div_minus1_right
thf(fact_3398_div__mult__mult1__if,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ( C2 = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
          = zero_zero_nat ) )
      & ( ( C2 != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
          = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_3399_div__mult__mult1__if,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ( C2 = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
          = zero_zero_int ) )
      & ( ( C2 != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
          = ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_3400_div__mult__mult1__if,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ( C2 = zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( C2 != zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
          = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_3401_div__mult__mult1__if,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( ( C2 = zero_z2226904508553997617atural )
       => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C2 @ A2 ) @ ( times_2397367101498566445atural @ C2 @ B2 ) )
          = zero_z2226904508553997617atural ) )
      & ( ( C2 != zero_z2226904508553997617atural )
       => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C2 @ A2 ) @ ( times_2397367101498566445atural @ C2 @ B2 ) )
          = ( divide5121882707175180666atural @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_3402_div__mult__mult2,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ A2 @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_3403_div__mult__mult2,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
        = ( divide_divide_int @ A2 @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_3404_div__mult__mult2,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( C2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
        = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_3405_div__mult__mult2,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( C2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ C2 ) )
        = ( divide5121882707175180666atural @ A2 @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_3406_div__mult__mult1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
        = ( divide_divide_nat @ A2 @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_3407_div__mult__mult1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( divide_divide_int @ A2 @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_3408_div__mult__mult1,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( C2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
        = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_3409_div__mult__mult1,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( C2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C2 @ A2 ) @ ( times_2397367101498566445atural @ C2 @ B2 ) )
        = ( divide5121882707175180666atural @ A2 @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_3410_bits__div__by__1,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ one_one_nat )
      = A2 ) ).

% bits_div_by_1
thf(fact_3411_bits__div__by__1,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ one_one_int )
      = A2 ) ).

% bits_div_by_1
thf(fact_3412_bits__div__by__1,axiom,
    ! [A2: code_integer] :
      ( ( divide6298287555418463151nteger @ A2 @ one_one_Code_integer )
      = A2 ) ).

% bits_div_by_1
thf(fact_3413_bits__div__by__1,axiom,
    ! [A2: code_natural] :
      ( ( divide5121882707175180666atural @ A2 @ one_one_Code_natural )
      = A2 ) ).

% bits_div_by_1
thf(fact_3414_div__add__self2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_3415_div__add__self2,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_3416_div__add__self2,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ B2 )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ one_one_Code_integer ) ) ) ).

% div_add_self2
thf(fact_3417_div__add__self2,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) @ B2 )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ one_one_Code_natural ) ) ) ).

% div_add_self2
thf(fact_3418_div__add__self1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_3419_div__add__self1,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_3420_div__add__self1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) @ B2 )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ one_one_Code_integer ) ) ) ).

% div_add_self1
thf(fact_3421_div__add__self1,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ B2 @ A2 ) @ B2 )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ one_one_Code_natural ) ) ) ).

% div_add_self1
thf(fact_3422_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_3423_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_3424_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_3425_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_3426_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_3427_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_3428_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_3429_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_3430_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_3431_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A2 = B2 )
          & ( C2 != D2 ) )
       => ( ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ R2 @ C2 ) )
         != ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3432_add__scale__eq__noteq,axiom,
    ! [R2: rat,A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A2 = B2 )
          & ( C2 != D2 ) )
       => ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R2 @ C2 ) )
         != ( plus_plus_rat @ B2 @ ( times_times_rat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3433_add__scale__eq__noteq,axiom,
    ! [R2: nat,A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A2 = B2 )
          & ( C2 != D2 ) )
       => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C2 ) )
         != ( plus_plus_nat @ B2 @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3434_add__scale__eq__noteq,axiom,
    ! [R2: int,A2: int,B2: int,C2: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A2 = B2 )
          & ( C2 != D2 ) )
       => ( ( plus_plus_int @ A2 @ ( times_times_int @ R2 @ C2 ) )
         != ( plus_plus_int @ B2 @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3435_crossproduct__eq,axiom,
    ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Y ) @ ( times_3573771949741848930nteger @ X @ Z ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Z ) @ ( times_3573771949741848930nteger @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3436_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3437_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3438_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3439_crossproduct__noteq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ( A2 != B2 )
        & ( C2 != D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) )
       != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ D2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_3440_crossproduct__noteq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( A2 != B2 )
        & ( C2 != D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) )
       != ( plus_plus_rat @ ( times_times_rat @ A2 @ D2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_3441_crossproduct__noteq,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ( A2 != B2 )
        & ( C2 != D2 ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) )
       != ( plus_plus_nat @ ( times_times_nat @ A2 @ D2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_3442_crossproduct__noteq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( A2 != B2 )
        & ( C2 != D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) )
       != ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_3443_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ 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 @ W ) ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_3444_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_3445_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ A2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3446_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A2: rat,B2: rat,C2: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A2 )
     => ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
        = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3447_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A2: int,B2: int,C2: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A2 )
     => ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
        = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3448_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ A2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3449_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A2: rat,B2: rat,C2: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A2 )
     => ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
        = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3450_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A2: int,B2: int,C2: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A2 )
     => ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
        = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3451_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ A2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3452_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A2: rat,B2: rat,C2: rat] :
      ( ( nO_MATCH_rat_rat @ ( divide_divide_rat @ X @ Y ) @ A2 )
     => ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
        = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3453_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A2: int,B2: int,C2: int] :
      ( ( nO_MATCH_rat_int @ ( divide_divide_rat @ X @ Y ) @ A2 )
     => ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
        = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3454_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ A2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3455_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3456_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3457_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3458_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Z ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3459_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ W ) @ Z ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3460_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3461_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3462_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3463_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3464_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ M2 ) @ ( numera5444537566228673987atural @ N ) )
      = ( numera5444537566228673987atural @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3465_distrib__left__numeral,axiom,
    ! [V: num,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3466_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3467_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3468_distrib__left__numeral,axiom,
    ! [V: num,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3469_distrib__left__numeral,axiom,
    ! [V: num,B2: code_natural,C2: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ B2 @ C2 ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ B2 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3470_distrib__right__numeral,axiom,
    ! [A2: nat,B2: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3471_distrib__right__numeral,axiom,
    ! [A2: int,B2: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3472_distrib__right__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3473_distrib__right__numeral,axiom,
    ! [A2: code_integer,B2: code_integer,V: num] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ ( numera6620942414471956472nteger @ V ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B2 @ ( numera6620942414471956472nteger @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3474_distrib__right__numeral,axiom,
    ! [A2: code_natural,B2: code_natural,V: num] :
      ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A2 @ B2 ) @ ( numera5444537566228673987atural @ V ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A2 @ ( numera5444537566228673987atural @ V ) ) @ ( times_2397367101498566445atural @ B2 @ ( numera5444537566228673987atural @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3475_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3476_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3477_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: code_integer,C2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( minus_8373710615458151222nteger @ B2 @ C2 ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3478_left__diff__distrib__numeral,axiom,
    ! [A2: int,B2: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3479_left__diff__distrib__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3480_left__diff__distrib__numeral,axiom,
    ! [A2: code_integer,B2: code_integer,V: num] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ ( numera6620942414471956472nteger @ V ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B2 @ ( numera6620942414471956472nteger @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3481_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3482_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3483_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3484_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3485_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3486_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3487_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3488_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3489_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3490_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A2 )
      = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_3491_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_3492_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) )
        = A2 )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_3493_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2
        = ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) )
            = B2 ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_3494_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A2 )
      = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_3495_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_3496_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_3497_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A2 )
      = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_3498_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2
        = ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B2 ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_3499_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A2 )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_3500_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A2 )
      = ( ord_less_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_3501_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_3502_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_3503_one__le__numeral,axiom,
    ! [N: num] : ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) ) ).

% one_le_numeral
thf(fact_3504_one__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).

% one_le_numeral
thf(fact_3505_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_3506_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_3507_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_3508_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_3509_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_3510_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_3511_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural ) ).

% not_numeral_less_one
thf(fact_3512_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_3513_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_3514_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_3515_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_3516_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_3517_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_3518_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_3519_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_3520_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3521_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3522_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3523_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_3524_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_3525_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_3526_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_3527_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_3528_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_3529_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_3530_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_3531_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_3532_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_3533_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_3534_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_3535_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3536_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3537_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3538_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ( divide_divide_rat @ B2 @ C2 )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3539_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3540_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_3541_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_3542_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_3543_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_3544_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_3545_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_3546_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_3547_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_3548_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_3549_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3550_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3551_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3552_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_3553_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_3554_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_3555_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3556_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3557_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ( divide_divide_rat @ B2 @ C2 )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_3558_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_3559_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ 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 @ W ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3560_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3561_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_3562_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ 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 @ W ) ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_3563_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ A2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3564_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A2: rat,B2: rat,C2: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A2 )
     => ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3565_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A2: nat,B2: nat,C2: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ A2 )
     => ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3566_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A2: int,B2: int,C2: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A2 )
     => ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3567_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ A2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3568_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A2: rat,B2: rat,C2: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A2 )
     => ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3569_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A2: nat,B2: nat,C2: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ A2 )
     => ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3570_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A2: int,B2: int,C2: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A2 )
     => ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3571_distrib__left__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ A2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C2 ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3572_distrib__left__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A2: rat,B2: rat,C2: rat] :
      ( ( nO_MATCH_rat_rat @ ( divide_divide_rat @ X @ Y ) @ A2 )
     => ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_3573_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3574_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: rat,A2: rat,B2: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3575_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: nat,A2: nat,B2: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
        = ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3576_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: int,A2: int,B2: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3577_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3578_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: rat,A2: rat,B2: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3579_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: nat,A2: nat,B2: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
        = ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3580_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: int,A2: int,B2: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3581_distrib__right__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ C2 )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3582_distrib__right__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C2: rat,A2: rat,B2: rat] :
      ( ( nO_MATCH_rat_rat @ ( divide_divide_rat @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_3583_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3584_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: rat,A2: rat,B2: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
        = ( minus_minus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3585_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C2: int,A2: int,B2: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C2 )
     => ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
        = ( minus_minus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3586_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3587_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: rat,A2: rat,B2: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
        = ( minus_minus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3588_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C2: int,A2: int,B2: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C2 )
     => ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
        = ( minus_minus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3589_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ C2 )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3590_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C2: rat,A2: rat,B2: rat] :
      ( ( nO_MATCH_rat_rat @ ( divide_divide_rat @ X @ Y ) @ C2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
        = ( minus_minus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3591_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C2: int,A2: int,B2: int] :
      ( ( nO_MATCH_rat_int @ ( divide_divide_rat @ X @ Y ) @ C2 )
     => ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
        = ( minus_minus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3592_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( nO_MAT8252062027627875367nteger @ ( divide6298287555418463151nteger @ X @ Y ) @ C2 )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3593_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_3594_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_3595_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_3596_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_3597_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_3598_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_3599_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_3600_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_3601_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_3602_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3603_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3604_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3605_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_3606_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( numeral_numeral_rat @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_3607_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( numera6620942414471956472nteger @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_3608_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3609_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3610_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3611_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_3612_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_3613_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_3614_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3615_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3616_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3617_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_3618_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_3619_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat @ ( inc @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_3620_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6620942414471956472nteger @ ( inc @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% numeral_inc
thf(fact_3621_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera5444537566228673987atural @ ( inc @ X ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).

% numeral_inc
thf(fact_3622_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3623_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3624_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3625_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3626_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3627_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3628_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3629_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3630_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3631_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3632_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3633_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3634_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 ) ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3635_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3636_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3637_power__decreasing__iff,axiom,
    ! [B2: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B2 )
     => ( ( ord_less_rat @ B2 @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ M2 ) @ ( power_power_rat @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3638_power__decreasing__iff,axiom,
    ! [B2: code_natural,M2: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ one_one_Code_natural )
       => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B2 @ M2 ) @ ( power_7079662738309270450atural @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3639_power__decreasing__iff,axiom,
    ! [B2: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B2 @ M2 ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3640_power__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3641_power__decreasing__iff,axiom,
    ! [B2: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3642_add_Ogroup__axioms,axiom,
    group_int @ plus_plus_int @ zero_zero_int @ uminus_uminus_int ).

% add.group_axioms
thf(fact_3643_add_Ogroup__axioms,axiom,
    group_rat @ plus_plus_rat @ zero_zero_rat @ uminus_uminus_rat ).

% add.group_axioms
thf(fact_3644_add_Ogroup__axioms,axiom,
    group_Code_integer @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ uminus1351360451143612070nteger ).

% add.group_axioms
thf(fact_3645_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_3646_power__minus_H,axiom,
    ! [X: int,N: nat] :
      ( ( nO_MATCH_int_int @ one_one_int @ X )
     => ( ( power_power_int @ ( uminus_uminus_int @ X ) @ N )
        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_3647_power__minus_H,axiom,
    ! [X: rat,N: nat] :
      ( ( nO_MATCH_rat_rat @ one_one_rat @ X )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ N )
        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_3648_power__minus_H,axiom,
    ! [X: code_integer,N: nat] :
      ( ( nO_MAT8252062027627875367nteger @ one_one_Code_integer @ X )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ N )
        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_3649_power__one,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N )
      = one_one_Code_integer ) ).

% power_one
thf(fact_3650_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_assn @ one_one_assn @ N )
      = one_one_assn ) ).

% power_one
thf(fact_3651_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_3652_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_3653_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_3654_inverse__mult__distrib,axiom,
    ! [A2: rat,B2: rat] :
      ( ( inverse_inverse_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) ) ) ).

% inverse_mult_distrib
thf(fact_3655_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( inverse_inverse_rat @ X )
        = one_one_rat )
      = ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
thf(fact_3656_inverse__1,axiom,
    ( ( inverse_inverse_rat @ one_one_rat )
    = one_one_rat ) ).

% inverse_1
thf(fact_3657_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3658_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3659_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3660_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3661_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_natural
        = ( numera5444537566228673987atural @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3662_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3663_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3664_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3665_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6620942414471956472nteger @ N )
        = one_one_Code_integer )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3666_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera5444537566228673987atural @ N )
        = one_one_Code_natural )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3667_power__inject__exp,axiom,
    ! [A2: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ( power_power_rat @ A2 @ M2 )
          = ( power_power_rat @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3668_power__inject__exp,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ( power_power_nat @ A2 @ M2 )
          = ( power_power_nat @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3669_power__inject__exp,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ( power_power_int @ A2 @ M2 )
          = ( power_power_int @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3670_power__inject__exp,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ( power_8256067586552552935nteger @ A2 @ M2 )
          = ( power_8256067586552552935nteger @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3671_power__inject__exp,axiom,
    ! [A2: code_natural,M2: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
     => ( ( ( power_7079662738309270450atural @ A2 @ M2 )
          = ( power_7079662738309270450atural @ A2 @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3672_power__strict__increasing__iff,axiom,
    ! [B2: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X ) @ ( power_power_rat @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3673_power__strict__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3674_power__strict__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3675_power__strict__increasing__iff,axiom,
    ! [B2: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B2 @ X ) @ ( power_8256067586552552935nteger @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3676_power__strict__increasing__iff,axiom,
    ! [B2: code_natural,X: nat,Y: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B2 )
     => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B2 @ X ) @ ( power_7079662738309270450atural @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3677_mod__mult__self2__is__0,axiom,
    ! [A2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_3678_mod__mult__self2__is__0,axiom,
    ! [A2: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_3679_mod__mult__self2__is__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_3680_mod__mult__self2__is__0,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ B2 )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self2_is_0
thf(fact_3681_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_3682_mod__mult__self1__is__0,axiom,
    ! [B2: int,A2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A2 ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_3683_mod__mult__self1__is__0,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_3684_mod__mult__self1__is__0,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B2 @ A2 ) @ B2 )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self1_is_0
thf(fact_3685_mod__by__1,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_3686_mod__by__1,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_3687_mod__by__1,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_3688_mod__by__1,axiom,
    ! [A2: code_natural] :
      ( ( modulo8411746178871703098atural @ A2 @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% mod_by_1
thf(fact_3689_bits__mod__by__1,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_3690_bits__mod__by__1,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_3691_bits__mod__by__1,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_3692_bits__mod__by__1,axiom,
    ! [A2: code_natural] :
      ( ( modulo8411746178871703098atural @ A2 @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% bits_mod_by_1
thf(fact_3693_mod__mult__self4,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C2 ) @ A2 ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mult_self4
thf(fact_3694_mod__mult__self4,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C2 ) @ A2 ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mult_self4
thf(fact_3695_mod__mult__self4,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C2 ) @ A2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mult_self4
thf(fact_3696_mod__mult__self4,axiom,
    ! [B2: code_natural,C2: code_natural,A2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B2 @ C2 ) @ A2 ) @ B2 )
      = ( modulo8411746178871703098atural @ A2 @ B2 ) ) ).

% mod_mult_self4
thf(fact_3697_mod__mult__self3,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B2 ) @ A2 ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mult_self3
thf(fact_3698_mod__mult__self3,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B2 ) @ A2 ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mult_self3
thf(fact_3699_mod__mult__self3,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C2 @ B2 ) @ A2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mult_self3
thf(fact_3700_mod__mult__self3,axiom,
    ! [C2: code_natural,B2: code_natural,A2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C2 @ B2 ) @ A2 ) @ B2 )
      = ( modulo8411746178871703098atural @ A2 @ B2 ) ) ).

% mod_mult_self3
thf(fact_3701_mod__mult__self2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mult_self2
thf(fact_3702_mod__mult__self2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mult_self2
thf(fact_3703_mod__mult__self2,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mult_self2
thf(fact_3704_mod__mult__self2,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A2 @ ( times_2397367101498566445atural @ B2 @ C2 ) ) @ B2 )
      = ( modulo8411746178871703098atural @ A2 @ B2 ) ) ).

% mod_mult_self2
thf(fact_3705_mod__mult__self1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C2 @ B2 ) ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mult_self1
thf(fact_3706_mod__mult__self1,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C2 @ B2 ) ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mult_self1
thf(fact_3707_mod__mult__self1,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mult_self1
thf(fact_3708_mod__mult__self1,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A2 @ ( times_2397367101498566445atural @ C2 @ B2 ) ) @ B2 )
      = ( modulo8411746178871703098atural @ A2 @ B2 ) ) ).

% mod_mult_self1
thf(fact_3709_power__add__numeral,axiom,
    ! [A2: code_integer,M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3710_power__add__numeral,axiom,
    ! [A2: assn,M2: num,N: num] :
      ( ( times_times_assn @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_assn @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3711_power__add__numeral,axiom,
    ! [A2: rat,M2: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3712_power__add__numeral,axiom,
    ! [A2: nat,M2: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3713_power__add__numeral,axiom,
    ! [A2: int,M2: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3714_power__add__numeral2,axiom,
    ! [A2: code_integer,M2: num,N: num,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_3715_power__add__numeral2,axiom,
    ! [A2: assn,M2: num,N: num,B2: assn] :
      ( ( times_times_assn @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_assn @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_assn @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_3716_power__add__numeral2,axiom,
    ! [A2: rat,M2: num,N: num,B2: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_3717_power__add__numeral2,axiom,
    ! [A2: nat,M2: num,N: num,B2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_3718_power__add__numeral2,axiom,
    ! [A2: int,M2: num,N: num,B2: int] :
      ( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_3719_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3720_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3721_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3722_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3723_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3724_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3725_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3726_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3727_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3728_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3729_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B2 )
     => ( ( ord_less_rat @ B2 @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B2 @ M2 ) @ ( power_power_rat @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3730_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3731_power__strict__decreasing__iff,axiom,
    ! [B2: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3732_power__strict__decreasing__iff,axiom,
    ! [B2: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B2 @ M2 ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3733_power__strict__decreasing__iff,axiom,
    ! [B2: code_natural,M2: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B2 )
     => ( ( ord_le5570908160329646204atural @ B2 @ one_one_Code_natural )
       => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B2 @ M2 ) @ ( power_7079662738309270450atural @ B2 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3734_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3735_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3736_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3737_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3738_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3739_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3740_mod__minus1__right,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_3741_mod__minus1__right,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_3742_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_3743_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_3744_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_3745_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_3746_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_3747_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_3748_left__inverse,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ A2 )
        = one_one_rat ) ) ).

% left_inverse
thf(fact_3749_right__inverse,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( times_times_rat @ A2 @ ( inverse_inverse_rat @ A2 ) )
        = one_one_rat ) ) ).

% right_inverse
thf(fact_3750_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_3751_power__increasing__iff,axiom,
    ! [B2: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X ) @ ( power_power_rat @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3752_power__increasing__iff,axiom,
    ! [B2: code_natural,X: nat,Y: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B2 )
     => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B2 @ X ) @ ( power_7079662738309270450atural @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3753_power__increasing__iff,axiom,
    ! [B2: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B2 @ X ) @ ( power_8256067586552552935nteger @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3754_power__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3755_power__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3756_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3757_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3758_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3759_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3760_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3761_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3762_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3763_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3764_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3765_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3766_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,X: rat] :
      ( ( ( times_times_rat @ Y @ X )
        = ( times_times_rat @ X @ Y ) )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ Y ) @ X )
        = ( times_times_rat @ X @ ( inverse_inverse_rat @ Y ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_3767_mod__mult__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_3768_mod__mult__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A2 @ C2 ) @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_3769_mod__mult__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_3770_mod__mult__eq,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A2 @ C2 ) @ ( modulo8411746178871703098atural @ B2 @ C2 ) ) @ C2 )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_3771_mod__mult__cong,axiom,
    ! [A2: nat,C2: nat,A4: nat,B2: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ C2 )
        = ( modulo_modulo_nat @ A4 @ C2 ) )
     => ( ( ( modulo_modulo_nat @ B2 @ C2 )
          = ( modulo_modulo_nat @ B4 @ C2 ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
          = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_3772_mod__mult__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( ( modulo_modulo_int @ A2 @ C2 )
        = ( modulo_modulo_int @ A4 @ C2 ) )
     => ( ( ( modulo_modulo_int @ B2 @ C2 )
          = ( modulo_modulo_int @ B4 @ C2 ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
          = ( modulo_modulo_int @ ( times_times_int @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_3773_mod__mult__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ C2 )
        = ( modulo364778990260209775nteger @ A4 @ C2 ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C2 )
          = ( modulo364778990260209775nteger @ B4 @ C2 ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_3774_mod__mult__cong,axiom,
    ! [A2: code_natural,C2: code_natural,A4: code_natural,B2: code_natural,B4: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A2 @ C2 )
        = ( modulo8411746178871703098atural @ A4 @ C2 ) )
     => ( ( ( modulo8411746178871703098atural @ B2 @ C2 )
          = ( modulo8411746178871703098atural @ B4 @ C2 ) )
       => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C2 )
          = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_3775_mod__mult__mult2,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_3776_mod__mult__mult2,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( times_times_int @ ( modulo_modulo_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_3777_mod__mult__mult2,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_3778_mod__mult__mult2,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ C2 ) )
      = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_3779_mult__mod__right,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( times_times_nat @ C2 @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ).

% mult_mod_right
thf(fact_3780_mult__mod__right,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( times_times_int @ C2 @ ( modulo_modulo_int @ A2 @ B2 ) )
      = ( modulo_modulo_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ).

% mult_mod_right
thf(fact_3781_mult__mod__right,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ C2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ).

% mult_mod_right
thf(fact_3782_mult__mod__right,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( times_2397367101498566445atural @ C2 @ ( modulo8411746178871703098atural @ A2 @ B2 ) )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ C2 @ A2 ) @ ( times_2397367101498566445atural @ C2 @ B2 ) ) ) ).

% mult_mod_right
thf(fact_3783_mod__mult__left__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_3784_mod__mult__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_3785_mod__mult__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_3786_mod__mult__left__eq,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_3787_mod__mult__right__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_3788_mod__mult__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_3789_mod__mult__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_3790_mod__mult__right__eq,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ ( modulo8411746178871703098atural @ B2 @ C2 ) ) @ C2 )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_3791_power__commutes,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ A2 )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_3792_power__commutes,axiom,
    ! [A2: assn,N: nat] :
      ( ( times_times_assn @ ( power_power_assn @ A2 @ N ) @ A2 )
      = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_3793_power__commutes,axiom,
    ! [A2: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ A2 )
      = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_3794_power__commutes,axiom,
    ! [A2: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
      = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_3795_power__commutes,axiom,
    ! [A2: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 )
      = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_3796_power__mult__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_3797_power__mult__distrib,axiom,
    ! [A2: assn,B2: assn,N: nat] :
      ( ( power_power_assn @ ( times_times_assn @ A2 @ B2 ) @ N )
      = ( times_times_assn @ ( power_power_assn @ A2 @ N ) @ ( power_power_assn @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_3798_power__mult__distrib,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A2 @ B2 ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_3799_power__mult__distrib,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A2 @ B2 ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_3800_power__mult__distrib,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A2 @ B2 ) @ N )
      = ( times_times_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_3801_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = ( times_3573771949741848930nteger @ Y @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ Y )
        = ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_3802_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = ( times_times_assn @ Y @ X ) )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ Y )
        = ( times_times_assn @ Y @ ( power_power_assn @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_3803_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_3804_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_3805_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_3806_power__minus__mult,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_3807_power__minus__mult,axiom,
    ! [N: nat,A2: assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_assn @ ( power_power_assn @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_assn @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_3808_power__minus__mult,axiom,
    ! [N: nat,A2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_rat @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_3809_power__minus__mult,axiom,
    ! [N: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_nat @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_3810_power__minus__mult,axiom,
    ! [N: nat,A2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_int @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_3811_power__add,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ M2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_add
thf(fact_3812_power__add,axiom,
    ! [A2: assn,M2: nat,N: nat] :
      ( ( power_power_assn @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A2 @ M2 ) @ ( power_power_assn @ A2 @ N ) ) ) ).

% power_add
thf(fact_3813_power__add,axiom,
    ! [A2: rat,M2: nat,N: nat] :
      ( ( power_power_rat @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ M2 ) @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_add
thf(fact_3814_power__add,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ M2 ) @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_add
thf(fact_3815_power__add,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( power_power_int @ A2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_int @ ( power_power_int @ A2 @ M2 ) @ ( power_power_int @ A2 @ N ) ) ) ).

% power_add
thf(fact_3816_power__0,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_3817_power__0,axiom,
    ! [A2: assn] :
      ( ( power_power_assn @ A2 @ zero_zero_nat )
      = one_one_assn ) ).

% power_0
thf(fact_3818_power__0,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ A2 @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_3819_power__0,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_3820_power__0,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_3821_nonzero__inverse__mult__distrib,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( inverse_inverse_rat @ ( times_times_rat @ A2 @ B2 ) )
          = ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_3822_inverse__unique,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
        = one_one_rat )
     => ( ( inverse_inverse_rat @ A2 )
        = B2 ) ) ).

% inverse_unique
thf(fact_3823_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_rat
    = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_3824_divide__inverse,axiom,
    ( divide_divide_rat
    = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).

% divide_inverse
thf(fact_3825_divide__inverse__commute,axiom,
    ( divide_divide_rat
    = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B3 ) @ A3 ) ) ) ).

% divide_inverse_commute
thf(fact_3826_inverse__eq__divide,axiom,
    ( inverse_inverse_rat
    = ( divide_divide_rat @ one_one_rat ) ) ).

% inverse_eq_divide
thf(fact_3827_mod__eqE,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ( modulo_modulo_int @ A2 @ C2 )
        = ( modulo_modulo_int @ B2 @ C2 ) )
     => ~ ! [D: int] :
            ( B2
           != ( plus_plus_int @ A2 @ ( times_times_int @ C2 @ D ) ) ) ) ).

% mod_eqE
thf(fact_3828_mod__eqE,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ C2 )
        = ( modulo364778990260209775nteger @ B2 @ C2 ) )
     => ~ ! [D: code_integer] :
            ( B2
           != ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ C2 @ D ) ) ) ) ).

% mod_eqE
thf(fact_3829_self__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3830_self__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le3102999989581377725nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3831_self__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3832_self__le__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3833_one__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A2 )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3834_one__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3835_one__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A2 )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3836_one__le__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A2 )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3837_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: rat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_3838_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_3839_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A2 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_3840_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: int] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_int @ one_one_int @ A2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_3841_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_3842_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_3843_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_3844_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_3845_one__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3846_one__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3847_one__less__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3848_one__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3849_one__less__power,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3850_power__less__imp__less__exp,axiom,
    ! [A2: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_rat @ ( power_power_rat @ A2 @ M2 ) @ ( power_power_rat @ A2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3851_power__less__imp__less__exp,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ ( power_power_nat @ A2 @ M2 ) @ ( power_power_nat @ A2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3852_power__less__imp__less__exp,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_int @ ( power_power_int @ A2 @ M2 ) @ ( power_power_int @ A2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3853_power__less__imp__less__exp,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ M2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3854_power__less__imp__less__exp,axiom,
    ! [A2: code_natural,M2: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
     => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ M2 ) @ ( power_7079662738309270450atural @ A2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3855_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: rat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_rat @ one_one_rat @ A2 )
       => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_3856_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: nat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_nat @ one_one_nat @ A2 )
       => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_3857_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: int] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_int @ one_one_int @ A2 )
       => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_3858_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_3859_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: code_natural] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
       => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N ) @ ( power_7079662738309270450atural @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_3860_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P7: code_integer,M: nat] : ( if_Code_integer @ ( M = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P7 @ ( power_8256067586552552935nteger @ P7 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3861_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P7: assn,M: nat] : ( if_assn @ ( M = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P7 @ ( power_power_assn @ P7 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3862_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P7: rat,M: nat] : ( if_rat @ ( M = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P7 @ ( power_power_rat @ P7 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3863_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P7: nat,M: nat] : ( if_nat @ ( M = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P7 @ ( power_power_nat @ P7 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3864_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P7: int,M: nat] : ( if_int @ ( M = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P7 @ ( power_power_int @ P7 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3865_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_3866_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ ( power_power_assn @ Y @ N ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_3867_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_3868_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_3869_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_3870_power__one__over,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ N )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_one_over
thf(fact_3871_mult__numeral__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_3872_mult__numeral__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_3873_mult__numeral__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_3874_mult__numeral__1,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_3875_mult__numeral__1,axiom,
    ! [A2: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_3876_mult__numeral__1__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_3877_mult__numeral__1__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_3878_mult__numeral__1__right,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ ( numeral_numeral_rat @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_3879_mult__numeral__1__right,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( numera6620942414471956472nteger @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_3880_mult__numeral__1__right,axiom,
    ! [A2: code_natural] :
      ( ( times_2397367101498566445atural @ A2 @ ( numera5444537566228673987atural @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_3881_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_3882_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_3883_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_3884_numeral__One,axiom,
    ( ( numera6620942414471956472nteger @ one )
    = one_one_Code_integer ) ).

% numeral_One
thf(fact_3885_numeral__One,axiom,
    ( ( numera5444537566228673987atural @ one )
    = one_one_Code_natural ) ).

% numeral_One
thf(fact_3886_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% inverse_le_1_iff
thf(fact_3887_one__less__inverse,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ A2 @ one_one_rat )
       => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A2 ) ) ) ) ).

% one_less_inverse
thf(fact_3888_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% one_less_inverse_iff
thf(fact_3889_division__ring__inverse__add,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_3890_inverse__add,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
          = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).

% inverse_add
thf(fact_3891_field__class_Ofield__inverse,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ A2 )
        = one_one_rat ) ) ).

% field_class.field_inverse
thf(fact_3892_division__ring__inverse__diff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ ( minus_minus_rat @ B2 @ A2 ) ) @ ( inverse_inverse_rat @ B2 ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_3893_nonzero__inverse__eq__divide,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A2 )
        = ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_3894_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) @ C2 )
      = ( plus_plus_nat @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3895_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) @ C2 )
      = ( plus_plus_int @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3896_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) @ C2 )
      = ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3897_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) @ ( modulo8411746178871703098atural @ A2 @ B2 ) ) @ C2 )
      = ( plus_p4538020629002901425atural @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3898_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) @ C2 )
      = ( plus_plus_nat @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3899_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A2 @ B2 ) ) @ C2 )
      = ( plus_plus_int @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3900_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) @ C2 )
      = ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3901_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ B2 ) @ ( modulo8411746178871703098atural @ A2 @ B2 ) ) @ C2 )
      = ( plus_p4538020629002901425atural @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3902_mod__div__decomp,axiom,
    ! [A2: nat,B2: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_3903_mod__div__decomp,axiom,
    ! [A2: int,B2: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_3904_mod__div__decomp,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( A2
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_3905_mod__div__decomp,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( A2
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ B2 ) @ ( modulo8411746178871703098atural @ A2 @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_3906_div__mult__mod__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = A2 ) ).

% div_mult_mod_eq
thf(fact_3907_div__mult__mod__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A2 @ B2 ) )
      = A2 ) ).

% div_mult_mod_eq
thf(fact_3908_div__mult__mod__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = A2 ) ).

% div_mult_mod_eq
thf(fact_3909_div__mult__mod__eq,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ B2 ) @ ( modulo8411746178871703098atural @ A2 @ B2 ) )
      = A2 ) ).

% div_mult_mod_eq
thf(fact_3910_mod__div__mult__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) )
      = A2 ) ).

% mod_div_mult_eq
thf(fact_3911_mod__div__mult__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A2 @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) )
      = A2 ) ).

% mod_div_mult_eq
thf(fact_3912_mod__div__mult__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) )
      = A2 ) ).

% mod_div_mult_eq
thf(fact_3913_mod__div__mult__eq,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A2 @ B2 ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ B2 ) )
      = A2 ) ).

% mod_div_mult_eq
thf(fact_3914_mod__mult__div__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) )
      = A2 ) ).

% mod_mult_div_eq
thf(fact_3915_mod__mult__div__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A2 @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) )
      = A2 ) ).

% mod_mult_div_eq
thf(fact_3916_mod__mult__div__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) )
      = A2 ) ).

% mod_mult_div_eq
thf(fact_3917_mod__mult__div__eq,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A2 @ B2 ) @ ( times_2397367101498566445atural @ B2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) )
      = A2 ) ).

% mod_mult_div_eq
thf(fact_3918_mult__div__mod__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = A2 ) ).

% mult_div_mod_eq
thf(fact_3919_mult__div__mod__eq,axiom,
    ! [B2: int,A2: int] :
      ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) @ ( modulo_modulo_int @ A2 @ B2 ) )
      = A2 ) ).

% mult_div_mod_eq
thf(fact_3920_mult__div__mod__eq,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = A2 ) ).

% mult_div_mod_eq
thf(fact_3921_mult__div__mod__eq,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) @ ( modulo8411746178871703098atural @ A2 @ B2 ) )
      = A2 ) ).

% mult_div_mod_eq
thf(fact_3922_div__mult1__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) ) @ ( divide_divide_nat @ ( times_times_nat @ A2 @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_3923_div__mult1__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) ) @ ( divide_divide_int @ ( times_times_int @ A2 @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_3924_div__mult1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ C2 ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_3925_div__mult1__eq,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C2 )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A2 @ ( divide5121882707175180666atural @ B2 @ C2 ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ ( modulo8411746178871703098atural @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_3926_minus__div__mult__eq__mod,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_3927_minus__div__mult__eq__mod,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_3928_minus__div__mult__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_3929_minus__div__mult__eq__mod,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( minus_7197305767214868737atural @ A2 @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ B2 ) )
      = ( modulo8411746178871703098atural @ A2 @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_3930_minus__mod__eq__div__mult,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_3931_minus__mod__eq__div__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( modulo_modulo_int @ A2 @ B2 ) )
      = ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_3932_minus__mod__eq__div__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_3933_minus__mod__eq__div__mult,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( minus_7197305767214868737atural @ A2 @ ( modulo8411746178871703098atural @ A2 @ B2 ) )
      = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_3934_minus__mod__eq__mult__div,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_3935_minus__mod__eq__mult__div,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( modulo_modulo_int @ A2 @ B2 ) )
      = ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_3936_minus__mod__eq__mult__div,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_3937_minus__mod__eq__mult__div,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( minus_7197305767214868737atural @ A2 @ ( modulo8411746178871703098atural @ A2 @ B2 ) )
      = ( times_2397367101498566445atural @ B2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_3938_minus__mult__div__eq__mod,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_3939_minus__mult__div__eq__mod,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_3940_minus__mult__div__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_3941_minus__mult__div__eq__mod,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( minus_7197305767214868737atural @ A2 @ ( times_2397367101498566445atural @ B2 @ ( divide5121882707175180666atural @ A2 @ B2 ) ) )
      = ( modulo8411746178871703098atural @ A2 @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_3942_power__le__one,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_3943_power__le__one,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_3944_power__le__one,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_3945_power__le__one,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ A2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_3946_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: rat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3947_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3948_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3949_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: int] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ A2 @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3950_power__le__imp__le__exp,axiom,
    ! [A2: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M2 ) @ ( power_power_rat @ A2 @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3951_power__le__imp__le__exp,axiom,
    ! [A2: code_natural,M2: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
     => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ M2 ) @ ( power_7079662738309270450atural @ A2 @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3952_power__le__imp__le__exp,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ M2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3953_power__le__imp__le__exp,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M2 ) @ ( power_power_nat @ A2 @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3954_power__le__imp__le__exp,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M2 ) @ ( power_power_int @ A2 @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3955_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: rat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_rat @ A2 @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3956_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: nat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_nat @ A2 @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3957_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: int] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ( ord_less_int @ A2 @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3958_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3959_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: code_natural] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
       => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
         => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N3 ) @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3960_power__gt1__lemma,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3961_power__gt1__lemma,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3962_power__gt1__lemma,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3963_power__gt1__lemma,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3964_power__gt1__lemma,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
     => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3965_power__less__power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3966_power__less__power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3967_power__less__power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3968_power__less__power__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3969_power__less__power__Suc,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
     => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N ) @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3970_power__diff__power__eq,axiom,
    ! [A2: nat,N: nat,M2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M2 ) @ ( power_power_nat @ A2 @ N ) )
            = ( power_power_nat @ A2 @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M2 ) @ ( power_power_nat @ A2 @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3971_power__diff__power__eq,axiom,
    ! [A2: int,N: nat,M2: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A2 @ M2 ) @ ( power_power_int @ A2 @ N ) )
            = ( power_power_int @ A2 @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A2 @ M2 ) @ ( power_power_int @ A2 @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3972_power__diff__power__eq,axiom,
    ! [A2: code_integer,N: nat,M2: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ M2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
            = ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ M2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3973_power__diff__power__eq,axiom,
    ! [A2: code_natural,N: nat,M2: nat] :
      ( ( A2 != zero_z2226904508553997617atural )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A2 @ M2 ) @ ( power_7079662738309270450atural @ A2 @ N ) )
            = ( power_7079662738309270450atural @ A2 @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A2 @ M2 ) @ ( power_7079662738309270450atural @ A2 @ N ) )
            = ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3974_power__minus,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A2 @ N ) ) ) ).

% power_minus
thf(fact_3975_power__minus,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_minus
thf(fact_3976_power__minus,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_minus
thf(fact_3977_mult__1s__ring__1_I1_J,axiom,
    ! [B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B2 )
      = ( uminus_uminus_int @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_3978_mult__1s__ring__1_I1_J,axiom,
    ! [B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B2 )
      = ( uminus_uminus_rat @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_3979_mult__1s__ring__1_I1_J,axiom,
    ! [B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B2 )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_3980_mult__1s__ring__1_I2_J,axiom,
    ! [B2: int] :
      ( ( times_times_int @ B2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_3981_mult__1s__ring__1_I2_J,axiom,
    ! [B2: rat] :
      ( ( times_times_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_3982_mult__1s__ring__1_I2_J,axiom,
    ! [B2: code_integer] :
      ( ( times_3573771949741848930nteger @ B2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_3983_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_3984_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_3985_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_3986_inverse__less__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_rat @ B2 @ A2 ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
         => ( ord_less_rat @ A2 @ B2 ) ) ) ) ).

% inverse_less_iff
thf(fact_3987_inverse__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_eq_rat @ B2 @ A2 ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
         => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ).

% inverse_le_iff
thf(fact_3988_one__le__inverse,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A2 ) ) ) ) ).

% one_le_inverse
thf(fact_3989_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% inverse_less_1_iff
thf(fact_3990_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X )
        & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).

% one_le_inverse_iff
thf(fact_3991_power__Suc__less,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ A2 @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3992_power__Suc__less,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3993_power__Suc__less,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3994_power__Suc__less,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3995_power__Suc__less,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N ) ) @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3996_power__diff__conv__inverse,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( X != zero_zero_rat )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M2 ) )
          = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M2 ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_3997_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3998_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3999_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4000_power__mult__inverse__distrib,axiom,
    ! [X: rat,M2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( inverse_inverse_rat @ X ) )
      = ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M2 ) ) ) ).

% power_mult_inverse_distrib
thf(fact_4001_power__mult__power__inverse__commute,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
      = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M2 ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_4002_divmod__digit__1_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4003_divmod__digit__1_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4004_divmod__digit__1_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
            = ( divide_divide_int @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4005_one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ N ) ) ) ).

% one_mod_numeral
thf(fact_4006_one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ).

% one_mod_numeral
thf(fact_4007_one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ one @ N ) ) ) ).

% one_mod_numeral
thf(fact_4008_one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ N ) ) ) ).

% one_div_numeral
thf(fact_4009_one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( product_fst_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ).

% one_div_numeral
thf(fact_4010_one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( produc8508995932063986495nteger @ ( unique3479559517661332726nteger @ one @ N ) ) ) ).

% one_div_numeral
thf(fact_4011_divmod__digit__1_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4012_divmod__digit__1_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_nat @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4013_divmod__digit__1_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4014_mod__double__modulus,axiom,
    ! [M2: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo364778990260209775nteger @ X @ M2 ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4015_mod__double__modulus,axiom,
    ! [M2: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_nat @ X @ M2 ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4016_mod__double__modulus,axiom,
    ! [M2: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_int @ X @ M2 ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4017_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_4018_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_4019_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_4020_one__add__one,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_4021_one__add__one,axiom,
    ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural )
    = ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_4022_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_mod_two_eq_one
thf(fact_4023_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_mod_two_eq_one
thf(fact_4024_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_4025_one__mod__two__eq__one,axiom,
    ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = one_one_Code_natural ) ).

% one_mod_two_eq_one
thf(fact_4026_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% bits_one_mod_two_eq_one
thf(fact_4027_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_one_mod_two_eq_one
thf(fact_4028_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_4029_bits__one__mod__two__eq__one,axiom,
    ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = one_one_Code_natural ) ).

% bits_one_mod_two_eq_one
thf(fact_4030_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5055182867167087721od_nat @ M2 @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_4031_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ M2 @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_4032_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique3479559517661332726nteger @ M2 @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M2 ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_4033_bits__1__div__2,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% bits_1_div_2
thf(fact_4034_bits__1__div__2,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% bits_1_div_2
thf(fact_4035_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_4036_bits__1__div__2,axiom,
    ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% bits_1_div_2
thf(fact_4037_one__div__two__eq__zero,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% one_div_two_eq_zero
thf(fact_4038_one__div__two__eq__zero,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% one_div_two_eq_zero
thf(fact_4039_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_4040_one__div__two__eq__zero,axiom,
    ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% one_div_two_eq_zero
thf(fact_4041_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_4042_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_4043_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_4044_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4045_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4046_diff__numeral__special_I10_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4047_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4048_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4049_diff__numeral__special_I11_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4050_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4051_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: int] :
      ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4052_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4053_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
       != zero_z2226904508553997617atural )
      = ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4054_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4055_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: int] :
      ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4056_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4057_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
       != one_one_Code_natural )
      = ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4058_minus__1__div__2__eq,axiom,
    ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_2_eq
thf(fact_4059_minus__1__div__2__eq,axiom,
    ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% minus_1_div_2_eq
thf(fact_4060_minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% minus_1_mod_2_eq
thf(fact_4061_minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% minus_1_mod_2_eq
thf(fact_4062_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_minus_1_mod_2_eq
thf(fact_4063_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_minus_1_mod_2_eq
thf(fact_4064_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_4065_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_4066_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_4067_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4068_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4069_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4070_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_4071_power4__eq__xxxx,axiom,
    ! [X: code_integer] :
      ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_4072_power4__eq__xxxx,axiom,
    ! [X: assn] :
      ( ( power_power_assn @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_4073_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_4074_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_4075_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_4076_power2__eq__square,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_4077_power2__eq__square,axiom,
    ! [A2: assn] :
      ( ( power_power_assn @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_assn @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_4078_power2__eq__square,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_4079_power2__eq__square,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_4080_power2__eq__square,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_4081_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_4082_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_4083_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_4084_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_4085_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_4086_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_4087_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_4088_mult__2,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2
thf(fact_4089_mult__2,axiom,
    ! [Z: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Z )
      = ( plus_p4538020629002901425atural @ Z @ Z ) ) ).

% mult_2
thf(fact_4090_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_4091_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_4092_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_4093_mult__2__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2_right
thf(fact_4094_mult__2__right,axiom,
    ! [Z: code_natural] :
      ( ( times_2397367101498566445atural @ Z @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ Z @ Z ) ) ).

% mult_2_right
thf(fact_4095_left__add__twice,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_4096_left__add__twice,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_4097_left__add__twice,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_4098_left__add__twice,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_4099_left__add__twice,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( plus_p4538020629002901425atural @ A2 @ ( plus_p4538020629002901425atural @ A2 @ B2 ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_4100_minus__power__mult__self,axiom,
    ! [A2: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N ) )
      = ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_4101_minus__power__mult__self,axiom,
    ! [A2: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N ) )
      = ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_4102_minus__power__mult__self,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N ) )
      = ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_4103_power2__eq__1__iff,axiom,
    ! [A2: int] :
      ( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A2 = one_one_int )
        | ( A2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4104_power2__eq__1__iff,axiom,
    ! [A2: rat] :
      ( ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A2 = one_one_rat )
        | ( A2
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4105_power2__eq__1__iff,axiom,
    ! [A2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A2 = one_one_Code_integer )
        | ( A2
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4106_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_4107_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_4108_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_4109_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_4110_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_4111_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_4112_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_4113_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_4114_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_4115_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_4116_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_4117_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_nat @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4118_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4119_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4120_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_4121_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_4122_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_4123_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4124_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A2: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4125_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4126_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A2: code_natural] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
        = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4127_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4128_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4129_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4130_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q5: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q5 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ 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 @ Q5 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4131_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q5: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q5 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ 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 @ Q5 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4132_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q5: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q5 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q5 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4133_set__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A2 )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4134_set__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A2 )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4135_set__bit__0,axiom,
    ! [A2: code_integer] :
      ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A2 )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4136_set__bit__0,axiom,
    ! [A2: code_natural] :
      ( ( bit_se1617098188084679374atural @ zero_zero_nat @ A2 )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4137_unset__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A2 )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_4138_unset__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A2 )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_4139_unset__bit__0,axiom,
    ! [A2: code_integer] :
      ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A2 )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_4140_unset__bit__0,axiom,
    ! [A2: code_natural] :
      ( ( bit_se7083795435491715335atural @ zero_zero_nat @ A2 )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_4141_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4142_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4143_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4144_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4145_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4146_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_4147_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_4148_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_4149_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4150_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4151_dbl__simps_I3_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4152_even__succ__mod__exp,axiom,
    ! [A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4153_even__succ__mod__exp,axiom,
    ! [A2: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4154_even__succ__mod__exp,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4155_even__succ__mod__exp,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A2 ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4156_dvd__times__right__cancel__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ A2 ) )
        = ( dvd_dvd_Code_integer @ B2 @ C2 ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_4157_dvd__times__right__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C2 @ A2 ) )
        = ( dvd_dvd_nat @ B2 @ C2 ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_4158_dvd__times__right__cancel__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C2 @ A2 ) )
        = ( dvd_dvd_int @ B2 @ C2 ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_4159_dvd__times__left__cancel__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C2 ) )
        = ( dvd_dvd_Code_integer @ B2 @ C2 ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_4160_dvd__times__left__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) )
        = ( dvd_dvd_nat @ B2 @ C2 ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_4161_dvd__times__left__cancel__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) )
        = ( dvd_dvd_int @ B2 @ C2 ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_4162_dvd__mult__cancel__right,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4163_dvd__mult__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( dvd_dvd_rat @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4164_dvd__mult__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4165_dvd__mult__cancel__left,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4166_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( dvd_dvd_rat @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4167_dvd__mult__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_int )
        | ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4168_dvd__add__times__triv__left__iff,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ B2 ) )
      = ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4169_dvd__add__times__triv__left__iff,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ ( times_times_rat @ C2 @ A2 ) @ B2 ) )
      = ( dvd_dvd_rat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4170_dvd__add__times__triv__left__iff,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C2 @ A2 ) @ B2 ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4171_dvd__add__times__triv__left__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C2 @ A2 ) @ B2 ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4172_dvd__add__times__triv__right__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ C2 @ A2 ) ) )
      = ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4173_dvd__add__times__triv__right__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C2 @ A2 ) ) )
      = ( dvd_dvd_rat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4174_dvd__add__times__triv__right__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C2 @ A2 ) ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4175_dvd__add__times__triv__right__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ ( times_times_int @ C2 @ A2 ) ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4176_unit__prod,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_4177_unit__prod,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_4178_unit__prod,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_4179_dvd__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_4180_dvd__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_4181_dvd__div__mult__self,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_4182_dvd__div__mult__self,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ B2 )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_4183_dvd__mult__div__cancel,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ A2 ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_4184_dvd__mult__div__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ A2 ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_4185_dvd__mult__div__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ A2 ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_4186_dvd__mult__div__cancel,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ B2 )
     => ( ( times_2397367101498566445atural @ A2 @ ( divide5121882707175180666atural @ B2 @ A2 ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_4187_unit__div__1__div__1,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A2 ) )
        = A2 ) ) ).

% unit_div_1_div_1
thf(fact_4188_unit__div__1__div__1,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A2 ) )
        = A2 ) ) ).

% unit_div_1_div_1
thf(fact_4189_unit__div__1__div__1,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A2 ) )
        = A2 ) ) ).

% unit_div_1_div_1
thf(fact_4190_unit__div__1__div__1,axiom,
    ! [A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
     => ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( divide5121882707175180666atural @ one_one_Code_natural @ A2 ) )
        = A2 ) ) ).

% unit_div_1_div_1
thf(fact_4191_unit__div__1__unit,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A2 ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_4192_unit__div__1__unit,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A2 ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_4193_unit__div__1__unit,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A2 ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_4194_unit__div__1__unit,axiom,
    ! [A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
     => ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ one_one_Code_natural @ A2 ) @ one_one_Code_natural ) ) ).

% unit_div_1_unit
thf(fact_4195_unit__div,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_4196_unit__div,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_4197_unit__div,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_4198_unit__div,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
       => ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ one_one_Code_natural ) ) ) ).

% unit_div
thf(fact_4199_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4200_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4201_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri6519982836138164636nteger @ ( numeral_numeral_nat @ K ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% signed_take_bit_numeral_of_1
thf(fact_4202_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_4203_unit__mult__div__div,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A2 ) )
        = ( divide_divide_nat @ B2 @ A2 ) ) ) ).

% unit_mult_div_div
thf(fact_4204_unit__mult__div__div,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A2 ) )
        = ( divide_divide_int @ B2 @ A2 ) ) ) ).

% unit_mult_div_div
thf(fact_4205_unit__mult__div__div,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A2 ) )
        = ( divide6298287555418463151nteger @ B2 @ A2 ) ) ) ).

% unit_mult_div_div
thf(fact_4206_unit__mult__div__div,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ B2 @ ( divide5121882707175180666atural @ one_one_Code_natural @ A2 ) )
        = ( divide5121882707175180666atural @ B2 @ A2 ) ) ) ).

% unit_mult_div_div
thf(fact_4207_unit__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_4208_unit__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_4209_unit__div__mult__self,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_4210_unit__div__mult__self,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_4211_even__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A2 @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_4212_even__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A2 @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_4213_even__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_4214_even__mult__iff,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( times_2397367101498566445atural @ A2 @ B2 ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_4215_even__plus__one__iff,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_4216_even__plus__one__iff,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_4217_even__plus__one__iff,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_4218_even__plus__one__iff,axiom,
    ! [A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) )
      = ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_4219_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_4220_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_4221_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_4222_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_4223_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_4224_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_4225_even__succ__div__2,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4226_even__succ__div__2,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4227_even__succ__div__2,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4228_even__succ__div__2,axiom,
    ! [A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A2 ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4229_even__succ__div__two,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4230_even__succ__div__two,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4231_even__succ__div__two,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4232_even__succ__div__two,axiom,
    ! [A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4233_odd__succ__div__two,axiom,
    ! [A2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_4234_odd__succ__div__two,axiom,
    ! [A2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_4235_odd__succ__div__two,axiom,
    ! [A2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_4236_odd__succ__div__two,axiom,
    ! [A2: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ) ).

% odd_succ_div_two
thf(fact_4237_odd__two__times__div__two__succ,axiom,
    ! [A2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A2 ) ) ).

% odd_two_times_div_two_succ
thf(fact_4238_odd__two__times__div__two__succ,axiom,
    ! [A2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A2 ) ) ).

% odd_two_times_div_two_succ
thf(fact_4239_odd__two__times__div__two__succ,axiom,
    ! [A2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A2 ) ) ).

% odd_two_times_div_two_succ
thf(fact_4240_odd__two__times__div__two__succ,axiom,
    ! [A2: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ one_one_Code_natural )
        = A2 ) ) ).

% odd_two_times_div_two_succ
thf(fact_4241_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4242_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4243_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4244_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4245_even__succ__div__exp,axiom,
    ! [A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4246_even__succ__div__exp,axiom,
    ! [A2: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4247_even__succ__div__exp,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4248_even__succ__div__exp,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A2 ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
          = ( divide5121882707175180666atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4249_dvd__triv__right,axiom,
    ! [A2: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_4250_dvd__triv__right,axiom,
    ! [A2: assn,B2: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_4251_dvd__triv__right,axiom,
    ! [A2: rat,B2: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_4252_dvd__triv__right,axiom,
    ! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_4253_dvd__triv__right,axiom,
    ! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_4254_dvd__mult__right,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_Code_integer @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_4255_dvd__mult__right,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_assn @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_4256_dvd__mult__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_rat @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_4257_dvd__mult__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_nat @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_4258_dvd__mult__right,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_int @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_4259_mult__dvd__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ C2 @ D2 )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_4260_mult__dvd__mono,axiom,
    ! [A2: assn,B2: assn,C2: assn,D2: assn] :
      ( ( dvd_dvd_assn @ A2 @ B2 )
     => ( ( dvd_dvd_assn @ C2 @ D2 )
       => ( dvd_dvd_assn @ ( times_times_assn @ A2 @ C2 ) @ ( times_times_assn @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_4261_mult__dvd__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( ( dvd_dvd_rat @ C2 @ D2 )
       => ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_4262_mult__dvd__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ C2 @ D2 )
       => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_4263_mult__dvd__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ C2 @ D2 )
       => ( dvd_dvd_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_4264_dvd__triv__left,axiom,
    ! [A2: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_4265_dvd__triv__left,axiom,
    ! [A2: assn,B2: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_4266_dvd__triv__left,axiom,
    ! [A2: rat,B2: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_4267_dvd__triv__left,axiom,
    ! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_4268_dvd__triv__left,axiom,
    ! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_4269_dvd__mult__left,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_Code_integer @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_4270_dvd__mult__left,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_assn @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_4271_dvd__mult__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_rat @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_4272_dvd__mult__left,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_nat @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_4273_dvd__mult__left,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_int @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_4274_dvd__mult2,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_4275_dvd__mult2,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( dvd_dvd_assn @ A2 @ B2 )
     => ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_4276_dvd__mult2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_4277_dvd__mult2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_4278_dvd__mult2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_4279_dvd__mult,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ C2 )
     => ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_4280_dvd__mult,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( ( dvd_dvd_assn @ A2 @ C2 )
     => ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_4281_dvd__mult,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ C2 )
     => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_4282_dvd__mult,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ C2 )
     => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_4283_dvd__mult,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ C2 )
     => ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_4284_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B3: code_integer,A3: code_integer] :
        ? [K4: code_integer] :
          ( A3
          = ( times_3573771949741848930nteger @ B3 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_4285_dvd__def,axiom,
    ( dvd_dvd_assn
    = ( ^ [B3: assn,A3: assn] :
        ? [K4: assn] :
          ( A3
          = ( times_times_assn @ B3 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_4286_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B3: rat,A3: rat] :
        ? [K4: rat] :
          ( A3
          = ( times_times_rat @ B3 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_4287_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B3: nat,A3: nat] :
        ? [K4: nat] :
          ( A3
          = ( times_times_nat @ B3 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_4288_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B3: int,A3: int] :
        ? [K4: int] :
          ( A3
          = ( times_times_int @ B3 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_4289_dvdI,axiom,
    ! [A2: code_integer,B2: code_integer,K: code_integer] :
      ( ( A2
        = ( times_3573771949741848930nteger @ B2 @ K ) )
     => ( dvd_dvd_Code_integer @ B2 @ A2 ) ) ).

% dvdI
thf(fact_4290_dvdI,axiom,
    ! [A2: assn,B2: assn,K: assn] :
      ( ( A2
        = ( times_times_assn @ B2 @ K ) )
     => ( dvd_dvd_assn @ B2 @ A2 ) ) ).

% dvdI
thf(fact_4291_dvdI,axiom,
    ! [A2: rat,B2: rat,K: rat] :
      ( ( A2
        = ( times_times_rat @ B2 @ K ) )
     => ( dvd_dvd_rat @ B2 @ A2 ) ) ).

% dvdI
thf(fact_4292_dvdI,axiom,
    ! [A2: nat,B2: nat,K: nat] :
      ( ( A2
        = ( times_times_nat @ B2 @ K ) )
     => ( dvd_dvd_nat @ B2 @ A2 ) ) ).

% dvdI
thf(fact_4293_dvdI,axiom,
    ! [A2: int,B2: int,K: int] :
      ( ( A2
        = ( times_times_int @ B2 @ K ) )
     => ( dvd_dvd_int @ B2 @ A2 ) ) ).

% dvdI
thf(fact_4294_dvdE,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ~ ! [K2: code_integer] :
            ( A2
           != ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_4295_dvdE,axiom,
    ! [B2: assn,A2: assn] :
      ( ( dvd_dvd_assn @ B2 @ A2 )
     => ~ ! [K2: assn] :
            ( A2
           != ( times_times_assn @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_4296_dvdE,axiom,
    ! [B2: rat,A2: rat] :
      ( ( dvd_dvd_rat @ B2 @ A2 )
     => ~ ! [K2: rat] :
            ( A2
           != ( times_times_rat @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_4297_dvdE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ~ ! [K2: nat] :
            ( A2
           != ( times_times_nat @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_4298_dvdE,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ~ ! [K2: int] :
            ( A2
           != ( times_times_int @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_4299_one__dvd,axiom,
    ! [A2: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A2 ) ).

% one_dvd
thf(fact_4300_one__dvd,axiom,
    ! [A2: assn] : ( dvd_dvd_assn @ one_one_assn @ A2 ) ).

% one_dvd
thf(fact_4301_one__dvd,axiom,
    ! [A2: rat] : ( dvd_dvd_rat @ one_one_rat @ A2 ) ).

% one_dvd
thf(fact_4302_one__dvd,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).

% one_dvd
thf(fact_4303_one__dvd,axiom,
    ! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).

% one_dvd
thf(fact_4304_unit__imp__dvd,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B2 @ A2 ) ) ).

% unit_imp_dvd
thf(fact_4305_unit__imp__dvd,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( dvd_dvd_nat @ B2 @ A2 ) ) ).

% unit_imp_dvd
thf(fact_4306_unit__imp__dvd,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( dvd_dvd_int @ B2 @ A2 ) ) ).

% unit_imp_dvd
thf(fact_4307_dvd__unit__imp__unit,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4308_dvd__unit__imp__unit,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4309_dvd__unit__imp__unit,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ A2 @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4310_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_4311_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_4312_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_4313_is__unit__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_4314_is__unit__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
        & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_4315_is__unit__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A2 @ one_one_int )
        & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_4316_dvd__mult__unit__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ C2 @ B2 ) )
        = ( dvd_dvd_Code_integer @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4317_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C2 @ B2 ) )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4318_dvd__mult__unit__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C2 @ B2 ) )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4319_mult__unit__dvd__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_Code_integer @ A2 @ C2 ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4320_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4321_mult__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4322_dvd__mult__unit__iff_H,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
        = ( dvd_dvd_Code_integer @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4323_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4324_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4325_mult__unit__dvd__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_Code_integer @ B2 @ C2 ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4326_mult__unit__dvd__iff_H,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_nat @ B2 @ C2 ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4327_mult__unit__dvd__iff_H,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_int @ B2 @ C2 ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4328_unit__mult__left__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A2 @ B2 )
          = ( times_3573771949741848930nteger @ A2 @ C2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
thf(fact_4329_unit__mult__left__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( times_times_nat @ A2 @ B2 )
          = ( times_times_nat @ A2 @ C2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
thf(fact_4330_unit__mult__left__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( times_times_int @ A2 @ B2 )
          = ( times_times_int @ A2 @ C2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
thf(fact_4331_unit__mult__right__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B2 @ A2 )
          = ( times_3573771949741848930nteger @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
thf(fact_4332_unit__mult__right__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( times_times_nat @ B2 @ A2 )
          = ( times_times_nat @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
thf(fact_4333_unit__mult__right__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( times_times_int @ B2 @ A2 )
          = ( times_times_int @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
thf(fact_4334_dvd__div__mult,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C2 ) @ A2 )
        = ( divide_divide_nat @ ( times_times_nat @ B2 @ A2 ) @ C2 ) ) ) ).

% dvd_div_mult
thf(fact_4335_dvd__div__mult,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ C2 ) @ A2 )
        = ( divide_divide_int @ ( times_times_int @ B2 @ A2 ) @ C2 ) ) ) ).

% dvd_div_mult
thf(fact_4336_dvd__div__mult,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ B2 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ C2 ) @ A2 )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ C2 ) ) ) ).

% dvd_div_mult
thf(fact_4337_dvd__div__mult,axiom,
    ! [C2: code_natural,B2: code_natural,A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ B2 )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B2 @ C2 ) @ A2 )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ B2 @ A2 ) @ C2 ) ) ) ).

% dvd_div_mult
thf(fact_4338_div__mult__swap,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ) ).

% div_mult_swap
thf(fact_4339_div__mult__swap,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ) ).

% div_mult_swap
thf(fact_4340_div__mult__swap,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ B2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ C2 ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 ) ) ) ).

% div_mult_swap
thf(fact_4341_div__mult__swap,axiom,
    ! [C2: code_natural,B2: code_natural,A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ B2 )
     => ( ( times_2397367101498566445atural @ A2 @ ( divide5121882707175180666atural @ B2 @ C2 ) )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C2 ) ) ) ).

% div_mult_swap
thf(fact_4342_div__div__eq__right,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( divide_divide_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) )
          = ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_div_eq_right
thf(fact_4343_div__div__eq__right,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( divide_divide_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) )
          = ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_div_eq_right
thf(fact_4344_div__div__eq__right,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ A2 )
       => ( ( divide6298287555418463151nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ C2 ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_div_eq_right
thf(fact_4345_div__div__eq__right,axiom,
    ! [C2: code_natural,B2: code_natural,A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ B2 )
     => ( ( dvd_dvd_Code_natural @ B2 @ A2 )
       => ( ( divide5121882707175180666atural @ A2 @ ( divide5121882707175180666atural @ B2 @ C2 ) )
          = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_div_eq_right
thf(fact_4346_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C2 ) @ A2 )
     => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4347_dvd__div__mult2__eq,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C2 ) @ A2 )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4348_dvd__div__mult2__eq,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ C2 ) @ A2 )
     => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4349_dvd__div__mult2__eq,axiom,
    ! [B2: code_natural,C2: code_natural,A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ B2 @ C2 ) @ A2 )
     => ( ( divide5121882707175180666atural @ A2 @ ( times_2397367101498566445atural @ B2 @ C2 ) )
        = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ C2 ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4350_dvd__mult__imp__div,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C2 ) @ B2 )
     => ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) ) ) ).

% dvd_mult_imp_div
thf(fact_4351_dvd__mult__imp__div,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ C2 ) @ B2 )
     => ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) ) ) ).

% dvd_mult_imp_div
thf(fact_4352_dvd__mult__imp__div,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ B2 )
     => ( dvd_dvd_Code_integer @ A2 @ ( divide6298287555418463151nteger @ B2 @ C2 ) ) ) ).

% dvd_mult_imp_div
thf(fact_4353_dvd__mult__imp__div,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ B2 )
     => ( dvd_dvd_Code_natural @ A2 @ ( divide5121882707175180666atural @ B2 @ C2 ) ) ) ).

% dvd_mult_imp_div
thf(fact_4354_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A2: nat,D2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ( ( dvd_dvd_nat @ D2 @ C2 )
       => ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ ( divide_divide_nat @ C2 @ D2 ) )
          = ( divide_divide_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4355_div__mult__div__if__dvd,axiom,
    ! [B2: int,A2: int,D2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ( ( dvd_dvd_int @ D2 @ C2 )
       => ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ C2 @ D2 ) )
          = ( divide_divide_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4356_div__mult__div__if__dvd,axiom,
    ! [B2: code_integer,A2: code_integer,D2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ( ( dvd_dvd_Code_integer @ D2 @ C2 )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ ( divide6298287555418463151nteger @ C2 @ D2 ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4357_div__mult__div__if__dvd,axiom,
    ! [B2: code_natural,A2: code_natural,D2: code_natural,C2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ A2 )
     => ( ( dvd_dvd_Code_natural @ D2 @ C2 )
       => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ ( divide5121882707175180666atural @ C2 @ D2 ) )
          = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ ( times_2397367101498566445atural @ B2 @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4358_dvd__div__unit__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ C2 @ B2 ) )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_4359_dvd__div__unit__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ C2 @ B2 ) )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_4360_dvd__div__unit__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A2 @ ( divide6298287555418463151nteger @ C2 @ B2 ) )
        = ( dvd_dvd_Code_integer @ A2 @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_4361_dvd__div__unit__iff,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ A2 @ ( divide5121882707175180666atural @ C2 @ B2 ) )
        = ( dvd_dvd_Code_natural @ A2 @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_4362_div__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_4363_div__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_4364_div__unit__dvd__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_Code_integer @ A2 @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_4365_div__unit__dvd__iff,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_Code_natural @ A2 @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_4366_unit__div__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ B2 @ A2 )
          = ( divide_divide_nat @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_div_cancel
thf(fact_4367_unit__div__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( divide_divide_int @ B2 @ A2 )
          = ( divide_divide_int @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_div_cancel
thf(fact_4368_unit__div__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B2 @ A2 )
          = ( divide6298287555418463151nteger @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_div_cancel
thf(fact_4369_unit__div__cancel,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ B2 @ A2 )
          = ( divide5121882707175180666atural @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_div_cancel
thf(fact_4370_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
      = ( ? [X2: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4371_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L @ X2 ) ) )
      = ( ? [X2: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4372_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4373_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4374_unit__dvdE,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ~ ( ( A2 != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] :
              ( B2
             != ( times_3573771949741848930nteger @ A2 @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_4375_unit__dvdE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ~ ( ( A2 != zero_zero_nat )
         => ! [C3: nat] :
              ( B2
             != ( times_times_nat @ A2 @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_4376_unit__dvdE,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ~ ( ( A2 != zero_zero_int )
         => ! [C3: int] :
              ( B2
             != ( times_times_int @ A2 @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_4377_inf__period_I3_J,axiom,
    ! [D2: code_integer,D3: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D2 @ D3 )
     => ! [X4: code_integer,K3: code_integer] :
          ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ T ) )
          = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4378_inf__period_I3_J,axiom,
    ! [D2: rat,D3: rat,T: rat] :
      ( ( dvd_dvd_rat @ D2 @ D3 )
     => ! [X4: rat,K3: rat] :
          ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ T ) )
          = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4379_inf__period_I3_J,axiom,
    ! [D2: int,D3: int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X4: int,K3: int] :
          ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T ) )
          = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4380_inf__period_I4_J,axiom,
    ! [D2: code_integer,D3: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D2 @ D3 )
     => ! [X4: code_integer,K3: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4381_inf__period_I4_J,axiom,
    ! [D2: rat,D3: rat,T: rat] :
      ( ( dvd_dvd_rat @ D2 @ D3 )
     => ! [X4: rat,K3: rat] :
          ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4382_inf__period_I4_J,axiom,
    ! [D2: int,D3: int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X4: int,K3: int] :
          ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4383_dvd__div__eq__mult,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A2 @ B2 )
       => ( ( ( divide_divide_nat @ B2 @ A2 )
            = C2 )
          = ( B2
            = ( times_times_nat @ C2 @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4384_dvd__div__eq__mult,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ A2 @ B2 )
       => ( ( ( divide_divide_int @ B2 @ A2 )
            = C2 )
          = ( B2
            = ( times_times_int @ C2 @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4385_dvd__div__eq__mult,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A2 @ B2 )
       => ( ( ( divide6298287555418463151nteger @ B2 @ A2 )
            = C2 )
          = ( B2
            = ( times_3573771949741848930nteger @ C2 @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4386_dvd__div__eq__mult,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( A2 != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ A2 @ B2 )
       => ( ( ( divide5121882707175180666atural @ B2 @ A2 )
            = C2 )
          = ( B2
            = ( times_2397367101498566445atural @ C2 @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4387_div__dvd__iff__mult,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 )
          = ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C2 @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4388_div__dvd__iff__mult,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 )
          = ( dvd_dvd_int @ A2 @ ( times_times_int @ C2 @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4389_div__dvd__iff__mult,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ A2 )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 )
          = ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4390_div__dvd__iff__mult,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B2 @ A2 )
       => ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ C2 )
          = ( dvd_dvd_Code_natural @ A2 @ ( times_2397367101498566445atural @ C2 @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4391_dvd__div__iff__mult,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C2 @ B2 )
       => ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C2 ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4392_dvd__div__iff__mult,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( dvd_dvd_int @ C2 @ B2 )
       => ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) )
          = ( dvd_dvd_int @ ( times_times_int @ A2 @ C2 ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4393_dvd__div__iff__mult,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( C2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C2 @ B2 )
       => ( ( dvd_dvd_Code_integer @ A2 @ ( divide6298287555418463151nteger @ B2 @ C2 ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4394_dvd__div__iff__mult,axiom,
    ! [C2: code_natural,B2: code_natural,A2: code_natural] :
      ( ( C2 != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ C2 @ B2 )
       => ( ( dvd_dvd_Code_natural @ A2 @ ( divide5121882707175180666atural @ B2 @ C2 ) )
          = ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4395_dvd__div__div__eq__mult,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( C2 != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A2 @ B2 )
         => ( ( dvd_dvd_nat @ C2 @ D2 )
           => ( ( ( divide_divide_nat @ B2 @ A2 )
                = ( divide_divide_nat @ D2 @ C2 ) )
              = ( ( times_times_nat @ B2 @ C2 )
                = ( times_times_nat @ A2 @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4396_dvd__div__div__eq__mult,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( C2 != zero_zero_int )
       => ( ( dvd_dvd_int @ A2 @ B2 )
         => ( ( dvd_dvd_int @ C2 @ D2 )
           => ( ( ( divide_divide_int @ B2 @ A2 )
                = ( divide_divide_int @ D2 @ C2 ) )
              = ( ( times_times_int @ B2 @ C2 )
                = ( times_times_int @ A2 @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4397_dvd__div__div__eq__mult,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( C2 != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A2 @ B2 )
         => ( ( dvd_dvd_Code_integer @ C2 @ D2 )
           => ( ( ( divide6298287555418463151nteger @ B2 @ A2 )
                = ( divide6298287555418463151nteger @ D2 @ C2 ) )
              = ( ( times_3573771949741848930nteger @ B2 @ C2 )
                = ( times_3573771949741848930nteger @ A2 @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4398_dvd__div__div__eq__mult,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural,D2: code_natural] :
      ( ( A2 != zero_z2226904508553997617atural )
     => ( ( C2 != zero_z2226904508553997617atural )
       => ( ( dvd_dvd_Code_natural @ A2 @ B2 )
         => ( ( dvd_dvd_Code_natural @ C2 @ D2 )
           => ( ( ( divide5121882707175180666atural @ B2 @ A2 )
                = ( divide5121882707175180666atural @ D2 @ C2 ) )
              = ( ( times_2397367101498566445atural @ B2 @ C2 )
                = ( times_2397367101498566445atural @ A2 @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4399_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A2 @ B2 )
          = zero_zero_nat )
        = ( A2 = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4400_unit__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A2 @ B2 )
          = zero_zero_int )
        = ( A2 = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4401_unit__div__eq__0__iff,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A2 @ B2 )
          = zero_z3403309356797280102nteger )
        = ( A2 = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4402_unit__div__eq__0__iff,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ A2 @ B2 )
          = zero_z2226904508553997617atural )
        = ( A2 = zero_z2226904508553997617atural ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4403_unit__eq__div1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A2 @ B2 )
          = C2 )
        = ( A2
          = ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_4404_unit__eq__div1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A2 @ B2 )
          = C2 )
        = ( A2
          = ( times_times_int @ C2 @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_4405_unit__eq__div1,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A2 @ B2 )
          = C2 )
        = ( A2
          = ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_4406_unit__eq__div1,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ A2 @ B2 )
          = C2 )
        = ( A2
          = ( times_2397367101498566445atural @ C2 @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_4407_unit__eq__div2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( A2
          = ( divide_divide_nat @ C2 @ B2 ) )
        = ( ( times_times_nat @ A2 @ B2 )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_4408_unit__eq__div2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( A2
          = ( divide_divide_int @ C2 @ B2 ) )
        = ( ( times_times_int @ A2 @ B2 )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_4409_unit__eq__div2,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( A2
          = ( divide6298287555418463151nteger @ C2 @ B2 ) )
        = ( ( times_3573771949741848930nteger @ A2 @ B2 )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_4410_unit__eq__div2,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
     => ( ( A2
          = ( divide5121882707175180666atural @ C2 @ B2 ) )
        = ( ( times_2397367101498566445atural @ A2 @ B2 )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_4411_div__mult__unit2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_4412_div__mult__unit2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
          = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_4413_div__mult__unit2,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ A2 )
       => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_4414_div__mult__unit2,axiom,
    ! [C2: code_natural,B2: code_natural,A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ B2 @ A2 )
       => ( ( divide5121882707175180666atural @ A2 @ ( times_2397367101498566445atural @ B2 @ C2 ) )
          = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_4415_unit__div__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ C2 ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_4416_unit__div__commute,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 )
        = ( divide_divide_int @ ( times_times_int @ A2 @ C2 ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_4417_unit__div__commute,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_4418_unit__div__commute,axiom,
    ! [B2: code_natural,A2: code_natural,C2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ C2 )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ C2 ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_4419_unit__div__mult__swap,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C2 @ one_one_nat )
     => ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_4420_unit__div__mult__swap,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C2 @ one_one_int )
     => ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_4421_unit__div__mult__swap,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ C2 ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_4422_unit__div__mult__swap,axiom,
    ! [C2: code_natural,A2: code_natural,B2: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ A2 @ ( divide5121882707175180666atural @ B2 @ C2 ) )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_4423_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ C2 @ one_one_nat )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4424_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ C2 @ one_one_int )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
          = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4425_is__unit__div__mult2__eq,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4426_is__unit__div__mult2__eq,axiom,
    ! [B2: code_natural,C2: code_natural,A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ C2 @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A2 @ ( times_2397367101498566445atural @ B2 @ C2 ) )
          = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4427_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( modulo_modulo_nat @ A2 @ B2 )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_4428_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( modulo_modulo_int @ A2 @ B2 )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_4429_unit__imp__mod__eq__0,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_4430_unit__imp__mod__eq__0,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
     => ( ( modulo8411746178871703098atural @ A2 @ B2 )
        = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
thf(fact_4431_is__unit__power__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4432_is__unit__power__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4433_is__unit__power__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ one_one_int )
      = ( ( dvd_dvd_int @ A2 @ one_one_int )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4434_is__unit__div__mult__cancel__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4435_is__unit__div__mult__cancel__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ A2 ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4436_is__unit__div__mult__cancel__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ A2 ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4437_is__unit__div__mult__cancel__right,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( A2 != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A2 @ ( times_2397367101498566445atural @ B2 @ A2 ) )
          = ( divide5121882707175180666atural @ one_one_Code_natural @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4438_is__unit__div__mult__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4439_is__unit__div__mult__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ A2 @ B2 ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4440_is__unit__div__mult__cancel__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4441_is__unit__div__mult__cancel__left,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( A2 != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A2 @ ( times_2397367101498566445atural @ A2 @ B2 ) )
          = ( divide5121882707175180666atural @ one_one_Code_natural @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4442_is__unitE,axiom,
    ! [A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ~ ( ( A2 != zero_zero_nat )
         => ! [B5: nat] :
              ( ( B5 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B5 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A2 )
                    = B5 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B5 )
                      = A2 )
                   => ( ( ( times_times_nat @ A2 @ B5 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C2 @ A2 )
                       != ( times_times_nat @ C2 @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4443_is__unitE,axiom,
    ! [A2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ~ ( ( A2 != zero_zero_int )
         => ! [B5: int] :
              ( ( B5 != zero_zero_int )
             => ( ( dvd_dvd_int @ B5 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A2 )
                    = B5 )
                 => ( ( ( divide_divide_int @ one_one_int @ B5 )
                      = A2 )
                   => ( ( ( times_times_int @ A2 @ B5 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C2 @ A2 )
                       != ( times_times_int @ C2 @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4444_is__unitE,axiom,
    ! [A2: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ~ ( ( A2 != zero_z3403309356797280102nteger )
         => ! [B5: code_integer] :
              ( ( B5 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A2 )
                    = B5 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
                      = A2 )
                   => ( ( ( times_3573771949741848930nteger @ A2 @ B5 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C2 @ A2 )
                       != ( times_3573771949741848930nteger @ C2 @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4445_is__unitE,axiom,
    ! [A2: code_natural,C2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
     => ~ ( ( A2 != zero_z2226904508553997617atural )
         => ! [B5: code_natural] :
              ( ( B5 != zero_z2226904508553997617atural )
             => ( ( dvd_dvd_Code_natural @ B5 @ one_one_Code_natural )
               => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ A2 )
                    = B5 )
                 => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ B5 )
                      = A2 )
                   => ( ( ( times_2397367101498566445atural @ A2 @ B5 )
                        = one_one_Code_natural )
                     => ( ( divide5121882707175180666atural @ C2 @ A2 )
                       != ( times_2397367101498566445atural @ C2 @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4446_evenE,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B5: nat] :
            ( A2
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_4447_evenE,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B5: int] :
            ( A2
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_4448_evenE,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B5: code_integer] :
            ( A2
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_4449_evenE,axiom,
    ! [A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B5: code_natural] :
            ( A2
           != ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_4450_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_4451_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_4452_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_4453_odd__one,axiom,
    ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ one_one_Code_natural ) ).

% odd_one
thf(fact_4454_dvd__power__iff,axiom,
    ! [X: code_integer,M2: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M2 ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4455_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M2 ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4456_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4457_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).

% dvd_power
thf(fact_4458_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4459_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4460_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).

% dvd_power
thf(fact_4461_even__two__times__div__two,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A2 ) ) ).

% even_two_times_div_two
thf(fact_4462_even__two__times__div__two,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A2 ) ) ).

% even_two_times_div_two
thf(fact_4463_even__two__times__div__two,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A2 ) ) ).

% even_two_times_div_two
thf(fact_4464_even__two__times__div__two,axiom,
    ! [A2: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
        = A2 ) ) ).

% even_two_times_div_two
thf(fact_4465_odd__iff__mod__2__eq__one,axiom,
    ! [A2: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4466_odd__iff__mod__2__eq__one,axiom,
    ! [A2: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4467_odd__iff__mod__2__eq__one,axiom,
    ! [A2: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4468_odd__iff__mod__2__eq__one,axiom,
    ! [A2: code_natural] :
      ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = one_one_Code_natural ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4469_oddE,axiom,
    ! [A2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B5: nat] :
            ( A2
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_4470_oddE,axiom,
    ! [A2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B5: int] :
            ( A2
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_4471_oddE,axiom,
    ! [A2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B5: code_integer] :
            ( A2
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_4472_oddE,axiom,
    ! [A2: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B5: code_natural] :
            ( A2
           != ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_natural ) ) ) ).

% oddE
thf(fact_4473_mod2__eq__if,axiom,
    ! [A2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_4474_mod2__eq__if,axiom,
    ! [A2: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_4475_mod2__eq__if,axiom,
    ! [A2: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_4476_mod2__eq__if,axiom,
    ! [A2: code_natural] :
      ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
          = zero_z2226904508553997617atural ) )
      & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
          = one_one_Code_natural ) ) ) ).

% mod2_eq_if
thf(fact_4477_parity__cases,axiom,
    ! [A2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_4478_parity__cases,axiom,
    ! [A2: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_4479_parity__cases,axiom,
    ! [A2: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_4480_parity__cases,axiom,
    ! [A2: code_natural] :
      ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
         != zero_z2226904508553997617atural ) )
     => ~ ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
           != one_one_Code_natural ) ) ) ).

% parity_cases
thf(fact_4481_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_4482_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_4483_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_4484_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4485_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4486_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4487_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4488_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4489_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4490_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4491_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
          = zero_z2226904508553997617atural )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4492_even__mult__exp__div__exp__iff,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4493_even__mult__exp__div__exp__iff,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4494_even__mult__exp__div__exp__iff,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4495_even__mult__exp__div__exp__iff,axiom,
    ! [A2: code_natural,M2: nat,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
          = zero_z2226904508553997617atural )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4496_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,A3: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4497_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4498_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4499_take__bit__rec,axiom,
    ( bit_se569199155075624693atural
    = ( ^ [N2: nat,A3: code_natural] : ( if_Code_natural @ ( N2 = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide5121882707175180666atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4500_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4501_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4502_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4503_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4504_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4505_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4506_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4507_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4508_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4509_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4510_push__bit__numeral__minus__1,axiom,
    ! [N: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_4511_push__bit__numeral__minus__1,axiom,
    ! [N: num] :
      ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_4512_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4513_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4514_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural
        @ ( zero_n8403883297036319079atural
          @ ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 )
             != zero_z2226904508553997617atural )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4515_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4516_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M @ N2 ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).

% divmod_nat_def
thf(fact_4517_set__bit__Suc,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4518_set__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4519_set__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4520_set__bit__Suc,axiom,
    ! [N: nat,A2: code_natural] :
      ( ( bit_se1617098188084679374atural @ ( suc @ N ) @ A2 )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4521_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4522_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4523_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4524_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4525_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_4526_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_4527_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_4528_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_4529_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% take_bit_Suc_1
thf(fact_4530_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_4531_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_4532_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4533_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4534_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4535_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le5570908160329646204atural @ ( zero_n8403883297036319079atural @ P ) @ one_one_Code_natural )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4536_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4537_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4538_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4539_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4540_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% take_bit_numeral_1
thf(fact_4541_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_4542_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_4543_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% signed_take_bit_Suc_1
thf(fact_4544_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_4545_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
        = zero_z3403309356797280102nteger )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4546_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4547_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4548_sgn__mult__self__eq,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A2 ) @ ( sgn_sgn_Code_integer @ A2 ) )
      = ( zero_n356916108424825756nteger @ ( A2 != zero_z3403309356797280102nteger ) ) ) ).

% sgn_mult_self_eq
thf(fact_4549_sgn__mult__self__eq,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A2 ) @ ( sgn_sgn_rat @ A2 ) )
      = ( zero_n2052037380579107095ol_rat @ ( A2 != zero_zero_rat ) ) ) ).

% sgn_mult_self_eq
thf(fact_4550_sgn__mult__self__eq,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A2 ) @ ( sgn_sgn_int @ A2 ) )
      = ( zero_n2684676970156552555ol_int @ ( A2 != zero_zero_int ) ) ) ).

% sgn_mult_self_eq
thf(fact_4551_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4552_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4553_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4554_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4555_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4556_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4557_push__bit__Suc,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A2 )
      = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_4558_push__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A2 )
      = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_4559_push__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ A2 )
      = ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_4560_push__bit__Suc,axiom,
    ! [N: nat,A2: code_natural] :
      ( ( bit_se6611745700429515170atural @ ( suc @ N ) @ A2 )
      = ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_4561_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_4562_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ one_one_int )
      = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_4563_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer )
      = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_4564_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se6611745700429515170atural @ N @ one_one_Code_natural )
      = ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_4565_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4566_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4567_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4568_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4569_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4570_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4571_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4572_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4573_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4574_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4575_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4576_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4577_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4578_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4579_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se2925701944663578781it_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4580_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se1745604003318907178nteger @ M2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4581_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se569199155075624693atural @ M2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ N @ M2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4582_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se2923211474154528505it_int @ M2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4583_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4584_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4585_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4586_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4587_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          & Q ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_conj
thf(fact_4588_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_4589_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_4590_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_4591_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_4592_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_4593_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_bit1
thf(fact_4594_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_Suc_bit1
thf(fact_4595_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4596_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4597_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4598_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4599_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_4600_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_4601_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_4602_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_4603_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P5: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P5
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4604_split__of__bool__asm,axiom,
    ! [P: rat > $o,P5: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_rat ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4605_split__of__bool__asm,axiom,
    ! [P: nat > $o,P5: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_nat ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4606_split__of__bool__asm,axiom,
    ! [P: int > $o,P5: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_int ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4607_split__of__bool,axiom,
    ! [P: code_integer > $o,P5: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
      = ( ( P5
         => ( P @ one_one_Code_integer ) )
        & ( ~ P5
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_4608_split__of__bool,axiom,
    ! [P: rat > $o,P5: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
      = ( ( P5
         => ( P @ one_one_rat ) )
        & ( ~ P5
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_4609_split__of__bool,axiom,
    ! [P: nat > $o,P5: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ( P5
         => ( P @ one_one_nat ) )
        & ( ~ P5
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_4610_split__of__bool,axiom,
    ! [P: int > $o,P5: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ( P5
         => ( P @ one_one_int ) )
        & ( ~ P5
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_4611_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P7: $o] : ( if_Code_integer @ P7 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_4612_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P7: $o] : ( if_rat @ P7 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_4613_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P7: $o] : ( if_nat @ P7 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_4614_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P7: $o] : ( if_int @ P7 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_4615_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N4: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
       => ( ! [N4: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
         => ( ! [M3: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
           => ( ! [M3: num,N4: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) )
             => ( ! [M3: num,N4: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) )
               => ( ! [M3: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
                 => ( ! [M3: num,N4: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) )
                   => ~ ! [M3: num,N4: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_4616_power__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_4617_power__Suc,axiom,
    ! [A2: assn,N: nat] :
      ( ( power_power_assn @ A2 @ ( suc @ N ) )
      = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_4618_power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_4619_power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_4620_power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_4621_power__Suc2,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_4622_power__Suc2,axiom,
    ! [A2: assn,N: nat] :
      ( ( power_power_assn @ A2 @ ( suc @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_4623_power__Suc2,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_4624_power__Suc2,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_4625_power__Suc2,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_4626_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_4627_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_4628_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_4629_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_Bit1
thf(fact_4630_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera5444537566228673987atural @ ( bit1 @ N ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ ( numera5444537566228673987atural @ N ) ) @ one_one_Code_natural ) ) ).

% numeral_Bit1
thf(fact_4631_power__gt1,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4632_power__gt1,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4633_power__gt1,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4634_power__gt1,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4635_power__gt1,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
     => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_4636_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_4637_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_4638_take__bit__Suc,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A2 )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4639_take__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4640_take__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4641_take__bit__Suc,axiom,
    ! [N: nat,A2: code_natural] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ A2 )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4642_power__Suc__le__self,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_4643_power__Suc__le__self,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_4644_power__Suc__le__self,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_4645_power__Suc__le__self,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ A2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_4646_power3__eq__cube,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_4647_power3__eq__cube,axiom,
    ! [A2: assn] :
      ( ( power_power_assn @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_assn @ ( times_times_assn @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_4648_power3__eq__cube,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_4649_power3__eq__cube,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_4650_power3__eq__cube,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_4651_power__Suc__less__one,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ A2 @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_4652_power__Suc__less__one,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_4653_power__Suc__less__one,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_4654_power__Suc__less__one,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_4655_power__Suc__less__one,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
     => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
       => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N ) ) @ one_one_Code_natural ) ) ) ).

% power_Suc_less_one
thf(fact_4656_push__bit__double,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_4657_push__bit__double,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_4658_push__bit__double,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se7788150548672797655nteger @ N @ A2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_4659_push__bit__double,axiom,
    ! [N: nat,A2: code_natural] :
      ( ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
      = ( times_2397367101498566445atural @ ( bit_se6611745700429515170atural @ N @ A2 ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_4660_push__bit__eq__mult,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N2: nat,A3: nat] : ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_4661_push__bit__eq__mult,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N2: nat,A3: int] : ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_4662_push__bit__eq__mult,axiom,
    ( bit_se7788150548672797655nteger
    = ( ^ [N2: nat,A3: code_integer] : ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_4663_push__bit__eq__mult,axiom,
    ( bit_se6611745700429515170atural
    = ( ^ [N2: nat,A3: code_natural] : ( times_2397367101498566445atural @ A3 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_4664_bits__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [A5: nat] :
          ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: nat,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A2 ) ) ) ).

% bits_induct
thf(fact_4665_bits__induct,axiom,
    ! [P: code_integer > $o,A2: code_integer] :
      ( ! [A5: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: code_integer,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A2 ) ) ) ).

% bits_induct
thf(fact_4666_bits__induct,axiom,
    ! [P: code_natural > $o,A2: code_natural] :
      ( ! [A5: code_natural] :
          ( ( ( divide5121882707175180666atural @ A5 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: code_natural,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B5 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B5 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A2 ) ) ) ).

% bits_induct
thf(fact_4667_bits__induct,axiom,
    ! [P: int > $o,A2: int] :
      ( ! [A5: int] :
          ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: int,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A2 ) ) ) ).

% bits_induct
thf(fact_4668_power__odd__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_4669_power__odd__eq,axiom,
    ! [A2: assn,N: nat] :
      ( ( power_power_assn @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_assn @ A2 @ ( power_power_assn @ ( power_power_assn @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_4670_power__odd__eq,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_rat @ A2 @ ( power_power_rat @ ( power_power_rat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_4671_power__odd__eq,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_nat @ A2 @ ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_4672_power__odd__eq,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_int @ A2 @ ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_4673_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4674_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M2 @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4675_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M2 @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4676_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4677_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_4678_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_4679_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_4680_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_4681_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_4682_stable__imp__take__bit__eq,axiom,
    ! [A2: nat,N: nat] :
      ( ( ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
         => ( ( bit_se2925701944663578781it_nat @ N @ A2 )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
         => ( ( bit_se2925701944663578781it_nat @ N @ A2 )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4683_stable__imp__take__bit__eq,axiom,
    ! [A2: int,N: nat] :
      ( ( ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
         => ( ( bit_se2923211474154528505it_int @ N @ A2 )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
         => ( ( bit_se2923211474154528505it_int @ N @ A2 )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4684_stable__imp__take__bit__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
         => ( ( bit_se1745604003318907178nteger @ N @ A2 )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
         => ( ( bit_se1745604003318907178nteger @ N @ A2 )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4685_stable__imp__take__bit__eq,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
         => ( ( bit_se569199155075624693atural @ N @ A2 )
            = zero_z2226904508553997617atural ) )
        & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
         => ( ( bit_se569199155075624693atural @ N @ A2 )
            = ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4686_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4687_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4688_unset__bit__Suc,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4689_unset__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4690_unset__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4691_unset__bit__Suc,axiom,
    ! [N: nat,A2: code_natural] :
      ( ( bit_se7083795435491715335atural @ ( suc @ N ) @ A2 )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4692_flip__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A2 )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4693_flip__bit__0,axiom,
    ! [A2: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A2 )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4694_flip__bit__0,axiom,
    ! [A2: code_natural] :
      ( ( bit_se168947363167071951atural @ zero_zero_nat @ A2 )
      = ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4695_flip__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A2 )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4696_flip__bit__Suc,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4697_flip__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4698_flip__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4699_flip__bit__Suc,axiom,
    ! [N: nat,A2: code_natural] :
      ( ( bit_se168947363167071951atural @ ( suc @ N ) @ A2 )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4700_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_4701_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_4702_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_bit1
thf(fact_4703_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_numeral_bit1
thf(fact_4704_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_4705_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_4706_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_4707_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_4708_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_4709_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_4710_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_4711_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_4712_one__xor__eq,axiom,
    ! [A2: nat] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A2 )
      = ( minus_minus_nat @ ( plus_plus_nat @ A2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% one_xor_eq
thf(fact_4713_one__xor__eq,axiom,
    ! [A2: code_integer] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A2 )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% one_xor_eq
thf(fact_4714_one__xor__eq,axiom,
    ! [A2: code_natural] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ A2 )
      = ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A2 @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) ) )
        @ ( zero_n8403883297036319079atural
          @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% one_xor_eq
thf(fact_4715_one__xor__eq,axiom,
    ! [A2: int] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ A2 )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% one_xor_eq
thf(fact_4716_xor__one__eq,axiom,
    ! [A2: nat] :
      ( ( bit_se6528837805403552850or_nat @ A2 @ one_one_nat )
      = ( minus_minus_nat @ ( plus_plus_nat @ A2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% xor_one_eq
thf(fact_4717_xor__one__eq,axiom,
    ! [A2: code_integer] :
      ( ( bit_se3222712562003087583nteger @ A2 @ one_one_Code_integer )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% xor_one_eq
thf(fact_4718_xor__one__eq,axiom,
    ! [A2: code_natural] :
      ( ( bit_se2046307713759805098atural @ A2 @ one_one_Code_natural )
      = ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A2 @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) ) )
        @ ( zero_n8403883297036319079atural
          @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% xor_one_eq
thf(fact_4719_xor__one__eq,axiom,
    ! [A2: int] :
      ( ( bit_se6526347334894502574or_int @ A2 @ one_one_int )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% xor_one_eq
thf(fact_4720_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_4721_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_4722_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_4723_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_4724_bit_Odisj__one__left,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_left
thf(fact_4725_bit_Odisj__one__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_left
thf(fact_4726_bit_Odisj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_right
thf(fact_4727_bit_Odisj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_right
thf(fact_4728_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4729_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4730_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4731_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4732_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4733_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4734_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4735_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4736_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_4737_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_4738_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_4739_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_4740_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4741_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4742_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4743_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4744_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4745_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4746_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4747_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4748_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4749_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4750_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4751_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4752_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4753_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4754_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4755_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4756_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4757_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4758_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4759_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4760_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4761_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4762_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4763_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4764_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4765_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4766_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4767_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4768_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4769_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4770_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4771_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4772_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_4773_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_4774_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_4775_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_4776_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_4777_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_4778_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_4779_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_4780_flip__bit__eq__xor,axiom,
    ( bit_se1345352211410354436nteger
    = ( ^ [N2: nat,A3: code_integer] : ( bit_se3222712562003087583nteger @ A3 @ ( bit_se7788150548672797655nteger @ N2 @ one_one_Code_integer ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4781_flip__bit__eq__xor,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [N2: nat,A3: nat] : ( bit_se6528837805403552850or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4782_flip__bit__eq__xor,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N2: nat,A3: int] : ( bit_se6526347334894502574or_int @ A3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4783_or_Ocomm__monoid__axioms,axiom,
    comm_m1654649897431166824nteger @ bit_se1080825931792720795nteger @ zero_z3403309356797280102nteger ).

% or.comm_monoid_axioms
thf(fact_4784_or_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ bit_se1412395901928357646or_nat @ zero_zero_nat ).

% or.comm_monoid_axioms
thf(fact_4785_or_Ocomm__monoid__axioms,axiom,
    comm_monoid_int @ bit_se1409905431419307370or_int @ zero_zero_int ).

% or.comm_monoid_axioms
thf(fact_4786_or_Omonoid__axioms,axiom,
    monoid_Code_integer @ bit_se1080825931792720795nteger @ zero_z3403309356797280102nteger ).

% or.monoid_axioms
thf(fact_4787_or_Omonoid__axioms,axiom,
    monoid_nat @ bit_se1412395901928357646or_nat @ zero_zero_nat ).

% or.monoid_axioms
thf(fact_4788_or_Omonoid__axioms,axiom,
    monoid_int @ bit_se1409905431419307370or_int @ zero_zero_int ).

% or.monoid_axioms
thf(fact_4789_or_Osemilattice__neutr__axioms,axiom,
    semila6106414119144251899nteger @ bit_se1080825931792720795nteger @ zero_z3403309356797280102nteger ).

% or.semilattice_neutr_axioms
thf(fact_4790_or_Osemilattice__neutr__axioms,axiom,
    semila9079005292280841162tr_int @ bit_se1409905431419307370or_int @ zero_zero_int ).

% or.semilattice_neutr_axioms
thf(fact_4791_or_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ bit_se1412395901928357646or_nat @ zero_zero_nat ).

% or.semilattice_neutr_axioms
thf(fact_4792_xor_Ocomm__monoid__axioms,axiom,
    comm_m1654649897431166824nteger @ bit_se3222712562003087583nteger @ zero_z3403309356797280102nteger ).

% xor.comm_monoid_axioms
thf(fact_4793_xor_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ bit_se6528837805403552850or_nat @ zero_zero_nat ).

% xor.comm_monoid_axioms
thf(fact_4794_xor_Ocomm__monoid__axioms,axiom,
    comm_monoid_int @ bit_se6526347334894502574or_int @ zero_zero_int ).

% xor.comm_monoid_axioms
thf(fact_4795_xor_Omonoid__axioms,axiom,
    monoid_Code_integer @ bit_se3222712562003087583nteger @ zero_z3403309356797280102nteger ).

% xor.monoid_axioms
thf(fact_4796_xor_Omonoid__axioms,axiom,
    monoid_nat @ bit_se6528837805403552850or_nat @ zero_zero_nat ).

% xor.monoid_axioms
thf(fact_4797_xor_Omonoid__axioms,axiom,
    monoid_int @ bit_se6526347334894502574or_int @ zero_zero_int ).

% xor.monoid_axioms
thf(fact_4798_set__bit__eq__or,axiom,
    ( bit_se2793503036327961859nteger
    = ( ^ [N2: nat,A3: code_integer] : ( bit_se1080825931792720795nteger @ A3 @ ( bit_se7788150548672797655nteger @ N2 @ one_one_Code_integer ) ) ) ) ).

% set_bit_eq_or
thf(fact_4799_set__bit__eq__or,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [N2: nat,A3: nat] : ( bit_se1412395901928357646or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) ) ) ) ).

% set_bit_eq_or
thf(fact_4800_set__bit__eq__or,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,A3: int] : ( bit_se1409905431419307370or_int @ A3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% set_bit_eq_or
thf(fact_4801_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4802_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4803_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4804_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4805_or__one__eq,axiom,
    ! [A2: nat] :
      ( ( bit_se1412395901928357646or_nat @ A2 @ one_one_nat )
      = ( plus_plus_nat @ A2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% or_one_eq
thf(fact_4806_or__one__eq,axiom,
    ! [A2: code_integer] :
      ( ( bit_se1080825931792720795nteger @ A2 @ one_one_Code_integer )
      = ( plus_p5714425477246183910nteger @ A2 @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% or_one_eq
thf(fact_4807_or__one__eq,axiom,
    ! [A2: code_natural] :
      ( ( bit_se9127793120404214118atural @ A2 @ one_one_Code_natural )
      = ( plus_p4538020629002901425atural @ A2 @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% or_one_eq
thf(fact_4808_or__one__eq,axiom,
    ! [A2: int] :
      ( ( bit_se1409905431419307370or_int @ A2 @ one_one_int )
      = ( plus_plus_int @ A2 @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% or_one_eq
thf(fact_4809_one__or__eq,axiom,
    ! [A2: nat] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A2 )
      = ( plus_plus_nat @ A2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% one_or_eq
thf(fact_4810_one__or__eq,axiom,
    ! [A2: code_integer] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A2 )
      = ( plus_p5714425477246183910nteger @ A2 @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% one_or_eq
thf(fact_4811_one__or__eq,axiom,
    ! [A2: code_natural] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ A2 )
      = ( plus_p4538020629002901425atural @ A2 @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% one_or_eq
thf(fact_4812_one__or__eq,axiom,
    ! [A2: int] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ A2 )
      = ( plus_plus_int @ A2 @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ) ) ).

% one_or_eq
thf(fact_4813_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4814_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4815_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2119862282449309892nteger @ ( numeral_numeral_nat @ N ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4816_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se943457434206027407atural @ ( numeral_numeral_nat @ N ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4817_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
      = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_4818_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
      = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_4819_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2119862282449309892nteger @ ( suc @ N ) )
      = ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_4820_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se943457434206027407atural @ ( suc @ N ) )
      = ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_4821_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_4822_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_4823_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_4824_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_4825_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4826_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4827_mask__eq__exp__minus__1,axiom,
    ( bit_se2119862282449309892nteger
    = ( ^ [N2: nat] : ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4828_mask__eq__exp__minus__1,axiom,
    ( bit_se943457434206027407atural
    = ( ^ [N2: nat] : ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_natural ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4829_bit__sum__mult__2__cases,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1148574629649215175it_nat @ A2 @ ( suc @ J2 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_4830_bit__sum__mult__2__cases,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1146084159140164899it_int @ A2 @ ( suc @ J2 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_4831_bit__sum__mult__2__cases,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se9216721137139052372nteger @ A2 @ ( suc @ J2 ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_4832_bit__sum__mult__2__cases,axiom,
    ! [A2: code_natural,B2: code_natural,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se8040316288895769887atural @ A2 @ ( suc @ J2 ) )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ A2 @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B2 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se8040316288895769887atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B2 ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_4833_diff__numeral__special_I8_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( neg_numeral_sub_int @ one @ M2 ) ) ).

% diff_numeral_special(8)
thf(fact_4834_diff__numeral__special_I8_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( neg_numeral_sub_rat @ one @ M2 ) ) ).

% diff_numeral_special(8)
thf(fact_4835_diff__numeral__special_I8_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( neg_nu5755505904847501662nteger @ one @ M2 ) ) ).

% diff_numeral_special(8)
thf(fact_4836_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( neg_numeral_sub_int @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_4837_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( neg_numeral_sub_rat @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_4838_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_4839_minus__sub__one__diff__one,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M2 @ one ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% minus_sub_one_diff_one
thf(fact_4840_minus__sub__one__diff__one,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( neg_numeral_sub_rat @ M2 @ one ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% minus_sub_one_diff_one
thf(fact_4841_minus__sub__one__diff__one,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( neg_nu5755505904847501662nteger @ M2 @ one ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% minus_sub_one_diff_one
thf(fact_4842_bit__0__eq,axiom,
    ( ( bit_se9216721137139052372nteger @ zero_z3403309356797280102nteger )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_4843_bit__0__eq,axiom,
    ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_4844_bit__0__eq,axiom,
    ( ( bit_se1146084159140164899it_int @ zero_zero_int )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_4845_bit_Oconj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X ) ).

% bit.conj_one_right
thf(fact_4846_bit_Oconj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = X ) ).

% bit.conj_one_right
thf(fact_4847_and_Oright__neutral,axiom,
    ! [A2: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A2 ) ).

% and.right_neutral
thf(fact_4848_and_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( bit_se725231765392027082nd_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
      = A2 ) ).

% and.right_neutral
thf(fact_4849_and_Oleft__neutral,axiom,
    ! [A2: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A2 )
      = A2 ) ).

% and.left_neutral
thf(fact_4850_and_Oleft__neutral,axiom,
    ! [A2: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A2 )
      = A2 ) ).

% and.left_neutral
thf(fact_4851_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_4852_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% and_numerals(8)
thf(fact_4853_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% and_numerals(8)
thf(fact_4854_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_4855_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_numerals(2)
thf(fact_4856_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = one_one_Code_integer ) ).

% and_numerals(2)
thf(fact_4857_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = one_one_Code_natural ) ).

% and_numerals(2)
thf(fact_4858_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = one_one_int ) ).

% and_numerals(2)
thf(fact_4859_mask__Suc__0,axiom,
    ( ( bit_se2119862282449309892nteger @ ( suc @ zero_zero_nat ) )
    = one_one_Code_integer ) ).

% mask_Suc_0
thf(fact_4860_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_4861_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_4862_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_4863_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_se2119862282449309892nteger @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_4864_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_4865_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(1)
thf(fact_4866_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = zero_z2226904508553997617atural ) ).

% and_numerals(1)
thf(fact_4867_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_4868_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_4869_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(5)
thf(fact_4870_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% and_numerals(5)
thf(fact_4871_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_4872_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_4873_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_4874_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_4875_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_4876_diff__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int )
      = ( neg_numeral_sub_int @ M2 @ one ) ) ).

% diff_numeral_special(2)
thf(fact_4877_diff__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ one_one_rat )
      = ( neg_numeral_sub_rat @ M2 @ one ) ) ).

% diff_numeral_special(2)
thf(fact_4878_diff__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ one_one_Code_integer )
      = ( neg_nu5755505904847501662nteger @ M2 @ one ) ) ).

% diff_numeral_special(2)
thf(fact_4879_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( neg_numeral_sub_int @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_4880_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( neg_numeral_sub_rat @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_4881_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( neg_nu5755505904847501662nteger @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_4882_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_4883_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_4884_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_4885_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_4886_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_4887_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_4888_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_4889_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_4890_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) )
      = ( neg_numeral_sub_int @ one @ M2 ) ) ).

% add_neg_numeral_special(1)
thf(fact_4891_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) )
      = ( neg_numeral_sub_rat @ one @ M2 ) ) ).

% add_neg_numeral_special(1)
thf(fact_4892_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) )
      = ( neg_nu5755505904847501662nteger @ one @ M2 ) ) ).

% add_neg_numeral_special(1)
thf(fact_4893_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
      = ( neg_numeral_sub_int @ one @ M2 ) ) ).

% add_neg_numeral_special(2)
thf(fact_4894_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat )
      = ( neg_numeral_sub_rat @ one @ M2 ) ) ).

% add_neg_numeral_special(2)
thf(fact_4895_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer )
      = ( neg_nu5755505904847501662nteger @ one @ M2 ) ) ).

% add_neg_numeral_special(2)
thf(fact_4896_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( neg_numeral_sub_int @ M2 @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_4897_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( neg_numeral_sub_rat @ M2 @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_4898_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( neg_nu5755505904847501662nteger @ M2 @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_4899_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( neg_numeral_sub_int @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_4900_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( neg_numeral_sub_rat @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_4901_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_4902_and_Osemigroup__axioms,axiom,
    semigroup_int @ bit_se725231765392027082nd_int ).

% and.semigroup_axioms
thf(fact_4903_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A3: code_integer,N2: nat] :
          ( ( bit_se3949692690581998587nteger @ A3 @ ( bit_se7788150548672797655nteger @ N2 @ one_one_Code_integer ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_4904_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A3: nat,N2: nat] :
          ( ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) )
         != zero_zero_nat ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_4905_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A3: int,N2: nat] :
          ( ( bit_se725231765392027082nd_int @ A3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) )
         != zero_zero_int ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_4906_and__eq__minus__1__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A2 @ B2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ( A2
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
        & ( B2
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_4907_and__eq__minus__1__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( bit_se725231765392027082nd_int @ A2 @ B2 )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ( A2
          = ( uminus_uminus_int @ one_one_int ) )
        & ( B2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_4908_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se9216721137139052372nteger @ one_one_Code_integer @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_4909_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_4910_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_4911_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se9216721137139052372nteger @ one_one_Code_integer @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_4912_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_4913_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_4914_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se9216721137139052372nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_4915_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_4916_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_4917_and_Ocomm__monoid__axioms,axiom,
    comm_m1654649897431166824nteger @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% and.comm_monoid_axioms
thf(fact_4918_and_Ocomm__monoid__axioms,axiom,
    comm_monoid_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).

% and.comm_monoid_axioms
thf(fact_4919_and_Omonoid__axioms,axiom,
    monoid_Code_integer @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% and.monoid_axioms
thf(fact_4920_and_Omonoid__axioms,axiom,
    monoid_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).

% and.monoid_axioms
thf(fact_4921_and_Osemilattice__neutr__axioms,axiom,
    semila6106414119144251899nteger @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% and.semilattice_neutr_axioms
thf(fact_4922_and_Osemilattice__neutr__axioms,axiom,
    semila9079005292280841162tr_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).

% and.semilattice_neutr_axioms
thf(fact_4923_bit_Ocomplement__unique,axiom,
    ! [A2: code_integer,X: code_integer,Y: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A2 @ X )
        = zero_z3403309356797280102nteger )
     => ( ( ( bit_se1080825931792720795nteger @ A2 @ X )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
       => ( ( ( bit_se3949692690581998587nteger @ A2 @ Y )
            = zero_z3403309356797280102nteger )
         => ( ( ( bit_se1080825931792720795nteger @ A2 @ Y )
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_4924_bit_Ocomplement__unique,axiom,
    ! [A2: int,X: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ A2 @ X )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ A2 @ X )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( ( bit_se725231765392027082nd_int @ A2 @ Y )
            = zero_zero_int )
         => ( ( ( bit_se1409905431419307370or_int @ A2 @ Y )
              = ( uminus_uminus_int @ one_one_int ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_4925_one__and__eq,axiom,
    ! [A2: nat] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A2 )
      = ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4926_one__and__eq,axiom,
    ! [A2: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A2 )
      = ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4927_one__and__eq,axiom,
    ! [A2: code_natural] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ A2 )
      = ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4928_one__and__eq,axiom,
    ! [A2: int] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ A2 )
      = ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4929_and__one__eq,axiom,
    ! [A2: nat] :
      ( ( bit_se727722235901077358nd_nat @ A2 @ one_one_nat )
      = ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4930_and__one__eq,axiom,
    ! [A2: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A2 @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4931_and__one__eq,axiom,
    ! [A2: code_natural] :
      ( ( bit_se2773287842338716102atural @ A2 @ one_one_Code_natural )
      = ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4932_and__one__eq,axiom,
    ! [A2: int] :
      ( ( bit_se725231765392027082nd_int @ A2 @ one_one_int )
      = ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4933_even__bit__succ__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ N )
        = ( ( bit_se1148574629649215175it_nat @ A2 @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_4934_even__bit__succ__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A2 ) @ N )
        = ( ( bit_se1146084159140164899it_int @ A2 @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_4935_even__bit__succ__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ N )
        = ( ( bit_se9216721137139052372nteger @ A2 @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_4936_even__bit__succ__iff,axiom,
    ! [A2: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A2 ) @ N )
        = ( ( bit_se8040316288895769887atural @ A2 @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_4937_odd__bit__iff__bit__pred,axiom,
    ! [A2: nat,N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( bit_se1148574629649215175it_nat @ A2 @ N )
        = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A2 @ one_one_nat ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_4938_odd__bit__iff__bit__pred,axiom,
    ! [A2: int,N: nat] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( bit_se1146084159140164899it_int @ A2 @ N )
        = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A2 @ one_one_int ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_4939_odd__bit__iff__bit__pred,axiom,
    ! [A2: code_integer,N: nat] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( bit_se9216721137139052372nteger @ A2 @ N )
        = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A2 @ one_one_Code_integer ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_4940_odd__bit__iff__bit__pred,axiom,
    ! [A2: code_natural,N: nat] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 )
     => ( ( bit_se8040316288895769887atural @ A2 @ N )
        = ( ( bit_se8040316288895769887atural @ ( minus_7197305767214868737atural @ A2 @ one_one_Code_natural ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_4941_signed__take__bit__def,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N2: nat,A3: code_integer] : ( bit_se1080825931792720795nteger @ ( bit_se1745604003318907178nteger @ N2 @ A3 ) @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( bit_se9216721137139052372nteger @ A3 @ N2 ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N2 ) ) ) ) ) ) ).

% signed_take_bit_def
thf(fact_4942_signed__take__bit__def,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,A3: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ A3 ) @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ A3 @ N2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ) ) ).

% signed_take_bit_def
thf(fact_4943_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
        = one_one_int )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_4944_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
        = one_one_rat )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_4945_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
        = one_one_Code_integer )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_4946_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_int
        = ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_4947_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_rat
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y ) ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ one @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_4948_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_Code_integer
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y ) ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_4949_dvd__productE,axiom,
    ! [P5: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ P5 @ ( times_times_nat @ A2 @ B2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P5
              = ( times_times_nat @ X3 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X3 @ A2 )
             => ~ ( dvd_dvd_nat @ Y3 @ B2 ) ) ) ) ).

% dvd_productE
thf(fact_4950_dvd__productE,axiom,
    ! [P5: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ P5 @ ( times_times_int @ A2 @ B2 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P5
              = ( times_times_int @ X3 @ Y3 ) )
           => ( ( dvd_dvd_int @ X3 @ A2 )
             => ~ ( dvd_dvd_int @ Y3 @ B2 ) ) ) ) ).

% dvd_productE
thf(fact_4951_division__decomp,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
     => ? [B6: nat,C5: nat] :
          ( ( A2
            = ( times_times_nat @ B6 @ C5 ) )
          & ( dvd_dvd_nat @ B6 @ B2 )
          & ( dvd_dvd_nat @ C5 @ C2 ) ) ) ).

% division_decomp
thf(fact_4952_division__decomp,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
     => ? [B6: int,C5: int] :
          ( ( A2
            = ( times_times_int @ B6 @ C5 ) )
          & ( dvd_dvd_int @ B6 @ B2 )
          & ( dvd_dvd_int @ C5 @ C2 ) ) ) ).

% division_decomp
thf(fact_4953_mask__mod__exp,axiom,
    ! [N: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( ord_min_nat @ M2 @ N ) ) @ one_one_nat ) ) ).

% mask_mod_exp
thf(fact_4954_mask__mod__exp,axiom,
    ! [N: nat,M2: nat] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ord_min_nat @ M2 @ N ) ) @ one_one_int ) ) ).

% mask_mod_exp
thf(fact_4955_mask__mod__exp,axiom,
    ! [N: nat,M2: nat] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ord_min_nat @ M2 @ N ) ) @ one_one_Code_integer ) ) ).

% mask_mod_exp
thf(fact_4956_mask__mod__exp,axiom,
    ! [N: nat,M2: nat] :
      ( ( modulo8411746178871703098atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) )
      = ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( ord_min_nat @ M2 @ N ) ) @ one_one_Code_natural ) ) ).

% mask_mod_exp
thf(fact_4957_min_Obounded__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( ord_min_Code_integer @ B2 @ C2 ) )
      = ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
        & ( ord_le3102999989581377725nteger @ A2 @ C2 ) ) ) ).

% min.bounded_iff
thf(fact_4958_min_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C2 ) )
      = ( ( ord_less_eq_nat @ A2 @ B2 )
        & ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% min.bounded_iff
thf(fact_4959_min_Obounded__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ ( ord_min_int @ B2 @ C2 ) )
      = ( ( ord_less_eq_int @ A2 @ B2 )
        & ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% min.bounded_iff
thf(fact_4960_min_Oabsorb2,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( ord_min_Code_integer @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb2
thf(fact_4961_min_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb2
thf(fact_4962_min_Oabsorb2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb2
thf(fact_4963_min_Oabsorb1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_min_Code_integer @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb1
thf(fact_4964_min_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb1
thf(fact_4965_min_Oabsorb1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb1
thf(fact_4966_min__less__iff__conj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        & ( ord_less_nat @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_4967_min__less__iff__conj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_min_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        & ( ord_less_int @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_4968_min__less__iff__conj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z @ ( ord_min_Code_integer @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Z @ X )
        & ( ord_le6747313008572928689nteger @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_4969_min__less__iff__conj,axiom,
    ! [Z: code_natural,X: code_natural,Y: code_natural] :
      ( ( ord_le5570908160329646204atural @ Z @ ( ord_min_Code_natural @ X @ Y ) )
      = ( ( ord_le5570908160329646204atural @ Z @ X )
        & ( ord_le5570908160329646204atural @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_4970_min_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_4971_min_Oabsorb4,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_4972_min_Oabsorb4,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( ord_min_Code_integer @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_4973_min_Oabsorb4,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( ord_le5570908160329646204atural @ B2 @ A2 )
     => ( ( ord_min_Code_natural @ A2 @ B2 )
        = B2 ) ) ).

% min.absorb4
thf(fact_4974_min_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb3
thf(fact_4975_min_Oabsorb3,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_min_int @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb3
thf(fact_4976_min_Oabsorb3,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_min_Code_integer @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb3
thf(fact_4977_min_Oabsorb3,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ B2 )
     => ( ( ord_min_Code_natural @ A2 @ B2 )
        = A2 ) ) ).

% min.absorb3
thf(fact_4978_min__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_mi4194197829855655752at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% min_bot
thf(fact_4979_min__bot,axiom,
    ! [X: set_o] :
      ( ( ord_min_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% min_bot
thf(fact_4980_min__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% min_bot
thf(fact_4981_min__bot,axiom,
    ! [X: set_int] :
      ( ( ord_min_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% min_bot
thf(fact_4982_min__bot,axiom,
    ! [X: nat] :
      ( ( ord_min_nat @ bot_bot_nat @ X )
      = bot_bot_nat ) ).

% min_bot
thf(fact_4983_min__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_mi4194197829855655752at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% min_bot2
thf(fact_4984_min__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_min_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% min_bot2
thf(fact_4985_min__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% min_bot2
thf(fact_4986_min__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_min_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% min_bot2
thf(fact_4987_min__bot2,axiom,
    ! [X: nat] :
      ( ( ord_min_nat @ X @ bot_bot_nat )
      = bot_bot_nat ) ).

% min_bot2
thf(fact_4988_min__top,axiom,
    ! [X: assn] :
      ( ( ord_min_assn @ top_top_assn @ X )
      = X ) ).

% min_top
thf(fact_4989_min__top,axiom,
    ! [X: set_o] :
      ( ( ord_min_set_o @ top_top_set_o @ X )
      = X ) ).

% min_top
thf(fact_4990_min__top,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ top_top_set_nat @ X )
      = X ) ).

% min_top
thf(fact_4991_min__top,axiom,
    ! [X: set_Product_unit] :
      ( ( ord_mi7693666415311043668t_unit @ top_to1996260823553986621t_unit @ X )
      = X ) ).

% min_top
thf(fact_4992_min__top2,axiom,
    ! [X: assn] :
      ( ( ord_min_assn @ X @ top_top_assn )
      = X ) ).

% min_top2
thf(fact_4993_min__top2,axiom,
    ! [X: set_o] :
      ( ( ord_min_set_o @ X @ top_top_set_o )
      = X ) ).

% min_top2
thf(fact_4994_min__top2,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ X @ top_top_set_nat )
      = X ) ).

% min_top2
thf(fact_4995_min__top2,axiom,
    ! [X: set_Product_unit] :
      ( ( ord_mi7693666415311043668t_unit @ X @ top_to1996260823553986621t_unit )
      = X ) ).

% min_top2
thf(fact_4996_min__0__1_I2_J,axiom,
    ( ( ord_min_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% min_0_1(2)
thf(fact_4997_min__0__1_I2_J,axiom,
    ( ( ord_min_rat @ one_one_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% min_0_1(2)
thf(fact_4998_min__0__1_I2_J,axiom,
    ( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% min_0_1(2)
thf(fact_4999_min__0__1_I2_J,axiom,
    ( ( ord_min_int @ one_one_int @ zero_zero_int )
    = zero_zero_int ) ).

% min_0_1(2)
thf(fact_5000_min__0__1_I1_J,axiom,
    ( ( ord_min_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% min_0_1(1)
thf(fact_5001_min__0__1_I1_J,axiom,
    ( ( ord_min_rat @ zero_zero_rat @ one_one_rat )
    = zero_zero_rat ) ).

% min_0_1(1)
thf(fact_5002_min__0__1_I1_J,axiom,
    ( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
    = zero_zero_nat ) ).

% min_0_1(1)
thf(fact_5003_min__0__1_I1_J,axiom,
    ( ( ord_min_int @ zero_zero_int @ one_one_int )
    = zero_zero_int ) ).

% min_0_1(1)
thf(fact_5004_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = one_one_nat ) ).

% min_0_1(5)
thf(fact_5005_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = one_one_int ) ).

% min_0_1(5)
thf(fact_5006_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = one_one_rat ) ).

% min_0_1(5)
thf(fact_5007_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = one_one_Code_integer ) ).

% min_0_1(5)
thf(fact_5008_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_natural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
      = one_one_Code_natural ) ).

% min_0_1(5)
thf(fact_5009_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = one_one_nat ) ).

% min_0_1(6)
thf(fact_5010_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = one_one_int ) ).

% min_0_1(6)
thf(fact_5011_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = one_one_rat ) ).

% min_0_1(6)
thf(fact_5012_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% min_0_1(6)
thf(fact_5013_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_natural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% min_0_1(6)
thf(fact_5014_bit_Ocompl__one,axiom,
    ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% bit.compl_one
thf(fact_5015_bit_Ocompl__one,axiom,
    ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% bit.compl_one
thf(fact_5016_bit_Ocompl__zero,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.compl_zero
thf(fact_5017_bit_Ocompl__zero,axiom,
    ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.compl_zero
thf(fact_5018_bit_Odisj__cancel__left,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_cancel_left
thf(fact_5019_bit_Odisj__cancel__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_cancel_left
thf(fact_5020_bit_Odisj__cancel__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_cancel_right
thf(fact_5021_bit_Odisj__cancel__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_cancel_right
thf(fact_5022_bit_Oxor__one__left,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
      = ( bit_ri7919022796975470100ot_int @ X ) ) ).

% bit.xor_one_left
thf(fact_5023_bit_Oxor__one__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
      = ( bit_ri7632146776885996613nteger @ X ) ) ).

% bit.xor_one_left
thf(fact_5024_bit_Oxor__one__right,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ X ) ) ).

% bit.xor_one_right
thf(fact_5025_bit_Oxor__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ X ) ) ).

% bit.xor_one_right
thf(fact_5026_bit_Oxor__cancel__left,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_left
thf(fact_5027_bit_Oxor__cancel__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_left
thf(fact_5028_bit_Oxor__cancel__right,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_right
thf(fact_5029_bit_Oxor__cancel__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_right
thf(fact_5030_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_5031_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] :
      ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_5032_not__one__eq,axiom,
    ( ( bit_ri7919022796975470100ot_int @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_5033_not__one__eq,axiom,
    ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_5034_min__absorb2,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y @ X )
     => ( ( ord_min_Code_integer @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_5035_min__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_min_assn @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_5036_min__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_min_set_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_5037_min__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_min_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_5038_min__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_min_int @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_5039_min__absorb1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_min_Code_integer @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_5040_min__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_min_assn @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_5041_min__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_min_set_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_5042_min__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_min_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_5043_min__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_min_int @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_5044_min__def,axiom,
    ( ord_min_Code_integer
    = ( ^ [A3: code_integer,B3: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_5045_min__def,axiom,
    ( ord_min_assn
    = ( ^ [A3: assn,B3: assn] : ( if_assn @ ( ord_less_eq_assn @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_5046_min__def,axiom,
    ( ord_min_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_5047_min__def,axiom,
    ( ord_min_nat
    = ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_5048_min__def,axiom,
    ( ord_min_int
    = ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).

% min_def
thf(fact_5049_min__le__iff__disj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ X @ Y ) @ Z )
      = ( ( ord_le3102999989581377725nteger @ X @ Z )
        | ( ord_le3102999989581377725nteger @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_5050_min__le__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        | ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_5051_min__le__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        | ( ord_less_eq_int @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_5052_min_OcoboundedI2,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ C2 )
     => ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ C2 ) ) ).

% min.coboundedI2
thf(fact_5053_min_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ C2 )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C2 ) ) ).

% min.coboundedI2
thf(fact_5054_min_OcoboundedI2,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ C2 )
     => ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ C2 ) ) ).

% min.coboundedI2
thf(fact_5055_min_OcoboundedI1,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ C2 )
     => ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ C2 ) ) ).

% min.coboundedI1
thf(fact_5056_min_OcoboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C2 ) ) ).

% min.coboundedI1
thf(fact_5057_min_OcoboundedI1,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ C2 )
     => ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ C2 ) ) ).

% min.coboundedI1
thf(fact_5058_min_Oabsorb__iff2,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B3: code_integer,A3: code_integer] :
          ( ( ord_min_Code_integer @ A3 @ B3 )
          = B3 ) ) ) ).

% min.absorb_iff2
thf(fact_5059_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_min_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% min.absorb_iff2
thf(fact_5060_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_min_int @ A3 @ B3 )
          = B3 ) ) ) ).

% min.absorb_iff2
thf(fact_5061_min_Oabsorb__iff1,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ord_min_Code_integer @ A3 @ B3 )
          = A3 ) ) ) ).

% min.absorb_iff1
thf(fact_5062_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_min_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% min.absorb_iff1
thf(fact_5063_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_min_int @ A3 @ B3 )
          = A3 ) ) ) ).

% min.absorb_iff1
thf(fact_5064_min_Ocobounded2,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ B2 ) ).

% min.cobounded2
thf(fact_5065_min_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 ) ).

% min.cobounded2
thf(fact_5066_min_Ocobounded2,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ B2 ) ).

% min.cobounded2
thf(fact_5067_min_Ocobounded1,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ A2 ) ).

% min.cobounded1
thf(fact_5068_min_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ A2 ) ).

% min.cobounded1
thf(fact_5069_min_Ocobounded1,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ A2 ) ).

% min.cobounded1
thf(fact_5070_min_Oorder__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( A3
          = ( ord_min_Code_integer @ A3 @ B3 ) ) ) ) ).

% min.order_iff
thf(fact_5071_min_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( A3
          = ( ord_min_nat @ A3 @ B3 ) ) ) ) ).

% min.order_iff
thf(fact_5072_min_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( A3
          = ( ord_min_int @ A3 @ B3 ) ) ) ) ).

% min.order_iff
thf(fact_5073_min_OboundedI,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ C2 )
       => ( ord_le3102999989581377725nteger @ A2 @ ( ord_min_Code_integer @ B2 @ C2 ) ) ) ) ).

% min.boundedI
thf(fact_5074_min_OboundedI,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ C2 )
       => ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C2 ) ) ) ) ).

% min.boundedI
thf(fact_5075_min_OboundedI,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ A2 @ C2 )
       => ( ord_less_eq_int @ A2 @ ( ord_min_int @ B2 @ C2 ) ) ) ) ).

% min.boundedI
thf(fact_5076_min_OboundedE,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( ord_min_Code_integer @ B2 @ C2 ) )
     => ~ ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
         => ~ ( ord_le3102999989581377725nteger @ A2 @ C2 ) ) ) ).

% min.boundedE
thf(fact_5077_min_OboundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% min.boundedE
thf(fact_5078_min_OboundedE,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ ( ord_min_int @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_int @ A2 @ B2 )
         => ~ ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% min.boundedE
thf(fact_5079_min_OorderI,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( ord_min_Code_integer @ A2 @ B2 ) )
     => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% min.orderI
thf(fact_5080_min_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( ord_min_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% min.orderI
thf(fact_5081_min_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( ord_min_int @ A2 @ B2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% min.orderI
thf(fact_5082_min_OorderE,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( A2
        = ( ord_min_Code_integer @ A2 @ B2 ) ) ) ).

% min.orderE
thf(fact_5083_min_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( A2
        = ( ord_min_nat @ A2 @ B2 ) ) ) ).

% min.orderE
thf(fact_5084_min_OorderE,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( A2
        = ( ord_min_int @ A2 @ B2 ) ) ) ).

% min.orderE
thf(fact_5085_min_Omono,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ C2 )
     => ( ( ord_le3102999989581377725nteger @ B2 @ D2 )
       => ( ord_le3102999989581377725nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ ( ord_min_Code_integer @ C2 @ D2 ) ) ) ) ).

% min.mono
thf(fact_5086_min_Omono,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ ( ord_min_nat @ C2 @ D2 ) ) ) ) ).

% min.mono
thf(fact_5087_min_Omono,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ C2 )
     => ( ( ord_less_eq_int @ B2 @ D2 )
       => ( ord_less_eq_int @ ( ord_min_int @ A2 @ B2 ) @ ( ord_min_int @ C2 @ D2 ) ) ) ) ).

% min.mono
thf(fact_5088_min_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ C2 )
     => ( ord_less_nat @ ( ord_min_nat @ A2 @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI2
thf(fact_5089_min_Ostrict__coboundedI2,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_int @ B2 @ C2 )
     => ( ord_less_int @ ( ord_min_int @ A2 @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI2
thf(fact_5090_min_Ostrict__coboundedI2,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ C2 )
     => ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI2
thf(fact_5091_min_Ostrict__coboundedI2,axiom,
    ! [B2: code_natural,C2: code_natural,A2: code_natural] :
      ( ( ord_le5570908160329646204atural @ B2 @ C2 )
     => ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ A2 @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI2
thf(fact_5092_min_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ C2 )
     => ( ord_less_nat @ ( ord_min_nat @ A2 @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI1
thf(fact_5093_min_Ostrict__coboundedI1,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ A2 @ C2 )
     => ( ord_less_int @ ( ord_min_int @ A2 @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI1
thf(fact_5094_min_Ostrict__coboundedI1,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ C2 )
     => ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A2 @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI1
thf(fact_5095_min_Ostrict__coboundedI1,axiom,
    ! [A2: code_natural,C2: code_natural,B2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ C2 )
     => ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ A2 @ B2 ) @ C2 ) ) ).

% min.strict_coboundedI1
thf(fact_5096_min_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( A3
            = ( ord_min_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_5097_min_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( A3
            = ( ord_min_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_5098_min_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( A3
            = ( ord_min_Code_integer @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_5099_min_Ostrict__order__iff,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [A3: code_natural,B3: code_natural] :
          ( ( A3
            = ( ord_min_Code_natural @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
thf(fact_5100_min_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ ( ord_min_nat @ B2 @ C2 ) )
     => ~ ( ( ord_less_nat @ A2 @ B2 )
         => ~ ( ord_less_nat @ A2 @ C2 ) ) ) ).

% min.strict_boundedE
thf(fact_5101_min_Ostrict__boundedE,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ ( ord_min_int @ B2 @ C2 ) )
     => ~ ( ( ord_less_int @ A2 @ B2 )
         => ~ ( ord_less_int @ A2 @ C2 ) ) ) ).

% min.strict_boundedE
thf(fact_5102_min_Ostrict__boundedE,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( ord_min_Code_integer @ B2 @ C2 ) )
     => ~ ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
         => ~ ( ord_le6747313008572928689nteger @ A2 @ C2 ) ) ) ).

% min.strict_boundedE
thf(fact_5103_min_Ostrict__boundedE,axiom,
    ! [A2: code_natural,B2: code_natural,C2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A2 @ ( ord_min_Code_natural @ B2 @ C2 ) )
     => ~ ( ( ord_le5570908160329646204atural @ A2 @ B2 )
         => ~ ( ord_le5570908160329646204atural @ A2 @ C2 ) ) ) ).

% min.strict_boundedE
thf(fact_5104_min__less__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        | ( ord_less_nat @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_5105_min__less__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        | ( ord_less_int @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_5106_min__less__iff__disj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ X @ Y ) @ Z )
      = ( ( ord_le6747313008572928689nteger @ X @ Z )
        | ( ord_le6747313008572928689nteger @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_5107_min__less__iff__disj,axiom,
    ! [X: code_natural,Y: code_natural,Z: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ X @ Y ) @ Z )
      = ( ( ord_le5570908160329646204atural @ X @ Z )
        | ( ord_le5570908160329646204atural @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_5108_min__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_min_Code_integer @ Y @ Z ) )
      = ( ord_min_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5109_min__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_min_rat @ Y @ Z ) )
      = ( ord_min_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5110_min__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z ) )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5111_min__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_min_int @ Y @ Z ) )
      = ( ord_min_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5112_min__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_min_Code_integer @ X @ Y ) @ Z )
      = ( ord_min_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5113_min__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_min_rat @ X @ Y ) @ Z )
      = ( ord_min_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5114_min__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5115_min__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5116_min__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_min_rat @ X @ Y ) @ Z )
      = ( ord_min_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_5117_min__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_5118_not__iszero__1,axiom,
    ~ ( ring_16219924574208605041nteger @ one_one_Code_integer ) ).

% not_iszero_1
thf(fact_5119_not__iszero__1,axiom,
    ~ ( ring_1_iszero_rat @ one_one_rat ) ).

% not_iszero_1
thf(fact_5120_not__iszero__1,axiom,
    ~ ( ring_1_iszero_int @ one_one_int ) ).

% not_iszero_1
thf(fact_5121_not__iszero__neg__1,axiom,
    ~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_iszero_neg_1
thf(fact_5122_not__iszero__neg__1,axiom,
    ~ ( ring_1_iszero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_iszero_neg_1
thf(fact_5123_not__iszero__neg__1,axiom,
    ~ ( ring_16219924574208605041nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_iszero_neg_1
thf(fact_5124_minus__eq__not__plus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A3: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ one_one_int ) ) ) ).

% minus_eq_not_plus_1
thf(fact_5125_minus__eq__not__plus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A3: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A3 ) @ one_one_Code_integer ) ) ) ).

% minus_eq_not_plus_1
thf(fact_5126_not__eq__complement,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).

% not_eq_complement
thf(fact_5127_not__eq__complement,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).

% not_eq_complement
thf(fact_5128_minus__eq__not__minus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A3: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A3 @ one_one_int ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_5129_minus__eq__not__minus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A3: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A3 @ one_one_Code_integer ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_5130_unset__bit__eq__and__not,axiom,
    ( bit_se8260200283734997820nteger
    = ( ^ [N2: nat,A3: code_integer] : ( bit_se3949692690581998587nteger @ A3 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N2 @ one_one_Code_integer ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_5131_unset__bit__eq__and__not,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_5132_bit_Ocompl__unique,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ X @ Y )
        = zero_z3403309356797280102nteger )
     => ( ( ( bit_se1080825931792720795nteger @ X @ Y )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
       => ( ( bit_ri7632146776885996613nteger @ X )
          = Y ) ) ) ).

% bit.compl_unique
thf(fact_5133_bit_Ocompl__unique,axiom,
    ! [X: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Y )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ X @ Y )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( bit_ri7919022796975470100ot_int @ X )
          = Y ) ) ) ).

% bit.compl_unique
thf(fact_5134_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numeral_numeral_int @ X )
        = one_one_int )
      = ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_5135_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numeral_numeral_rat @ X )
        = one_one_rat )
      = ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_5136_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numera6620942414471956472nteger @ X )
        = one_one_Code_integer )
      = ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_5137_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ Y ) )
      = ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_5138_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ Y ) )
      = ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ one @ Y ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_5139_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ Y ) )
      = ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ one @ Y ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_5140_bit_Oabstract__boolean__algebra__axioms,axiom,
    boolea5643492148547655015nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% bit.abstract_boolean_algebra_axioms
thf(fact_5141_bit_Oabstract__boolean__algebra__axioms,axiom,
    boolea5469654029284665142ra_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ).

% bit.abstract_boolean_algebra_axioms
thf(fact_5142_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    boolea2445317508997433345nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ bit_se3222712562003087583nteger ).

% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_5143_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    boolea8527374999097803216ff_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) @ bit_se6526347334894502574or_int ).

% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_5144_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_5145_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_5146_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_5147_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_5148_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_5149_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_5150_euclidean__size__times__nonunit,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( B2 != zero_z3403309356797280102nteger )
       => ( ~ ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
         => ( ord_less_nat @ ( euclid6377331345833325938nteger @ B2 ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ) ) ).

% euclidean_size_times_nonunit
thf(fact_5151_euclidean__size__times__nonunit,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( ~ ( dvd_dvd_nat @ A2 @ one_one_nat )
         => ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ B2 ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ) ) ).

% euclidean_size_times_nonunit
thf(fact_5152_euclidean__size__times__nonunit,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( ~ ( dvd_dvd_int @ A2 @ one_one_int )
         => ( ord_less_nat @ ( euclid4774559944035922753ze_int @ B2 ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ) ) ).

% euclidean_size_times_nonunit
thf(fact_5153_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A2 @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_5154_odd__card__imp__not__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite711546835091564841at_nat @ A ) )
     => ( A != bot_bo2099793752762293965at_nat ) ) ).

% odd_card_imp_not_empty
thf(fact_5155_odd__card__imp__not__empty,axiom,
    ! [A: set_o] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_o @ A ) )
     => ( A != bot_bot_set_o ) ) ).

% odd_card_imp_not_empty
thf(fact_5156_odd__card__imp__not__empty,axiom,
    ! [A: set_nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ A ) )
     => ( A != bot_bot_set_nat ) ) ).

% odd_card_imp_not_empty
thf(fact_5157_odd__card__imp__not__empty,axiom,
    ! [A: set_int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_int @ A ) )
     => ( A != bot_bot_set_int ) ) ).

% odd_card_imp_not_empty
thf(fact_5158_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_5159_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_5160_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_5161_abs__idempotent,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_idempotent
thf(fact_5162_abs__idempotent,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_idempotent
thf(fact_5163_abs__idempotent,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_idempotent
thf(fact_5164_abs__0__eq,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_5165_abs__0__eq,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_5166_abs__0__eq,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_5167_abs__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_5168_abs__eq__0,axiom,
    ! [A2: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_5169_abs__eq__0,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_5170_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_5171_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_5172_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_5173_abs__add__abs,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_add_abs
thf(fact_5174_abs__add__abs,axiom,
    ! [A2: rat,B2: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_add_abs
thf(fact_5175_abs__add__abs,axiom,
    ! [A2: int,B2: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_add_abs
thf(fact_5176_abs__mult__self__eq,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ A2 ) )
      = ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).

% abs_mult_self_eq
thf(fact_5177_abs__mult__self__eq,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ A2 ) )
      = ( times_times_rat @ A2 @ A2 ) ) ).

% abs_mult_self_eq
thf(fact_5178_abs__mult__self__eq,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ A2 ) )
      = ( times_times_int @ A2 @ A2 ) ) ).

% abs_mult_self_eq
thf(fact_5179_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_5180_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_5181_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_5182_abs__minus__cancel,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_minus_cancel
thf(fact_5183_abs__minus__cancel,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_minus_cancel
thf(fact_5184_abs__minus__cancel,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_minus_cancel
thf(fact_5185_abs__of__nonneg,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( abs_abs_rat @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_5186_abs__of__nonneg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( abs_abs_Code_integer @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_5187_abs__of__nonneg,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_5188_abs__le__self__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ A2 )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% abs_le_self_iff
thf(fact_5189_abs__le__self__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ A2 )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% abs_le_self_iff
thf(fact_5190_abs__le__self__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% abs_le_self_iff
thf(fact_5191_abs__le__zero__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_5192_abs__le__zero__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_5193_abs__le__zero__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_5194_zero__less__abs__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) )
      = ( A2 != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_5195_zero__less__abs__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
      = ( A2 != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_5196_zero__less__abs__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) )
      = ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_5197_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_5198_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_5199_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_5200_gbinomial__0_I1_J,axiom,
    ! [A2: code_integer] :
      ( ( gbinom8545251970709558553nteger @ A2 @ zero_zero_nat )
      = one_one_Code_integer ) ).

% gbinomial_0(1)
thf(fact_5201_gbinomial__0_I1_J,axiom,
    ! [A2: rat] :
      ( ( gbinomial_rat @ A2 @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_5202_gbinomial__0_I1_J,axiom,
    ! [A2: nat] :
      ( ( gbinomial_nat @ A2 @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_5203_gbinomial__0_I1_J,axiom,
    ! [A2: int] :
      ( ( gbinomial_int @ A2 @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_5204_euclidean__size__1,axiom,
    ( ( euclid6377331345833325938nteger @ one_one_Code_integer )
    = one_one_nat ) ).

% euclidean_size_1
thf(fact_5205_euclidean__size__1,axiom,
    ( ( euclid4777050414544973029ze_nat @ one_one_nat )
    = one_one_nat ) ).

% euclidean_size_1
thf(fact_5206_euclidean__size__1,axiom,
    ( ( euclid4774559944035922753ze_int @ one_one_int )
    = one_one_nat ) ).

% euclidean_size_1
thf(fact_5207_abs__of__nonpos,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( abs_abs_rat @ A2 )
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_5208_abs__of__nonpos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_5209_abs__of__nonpos,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( abs_abs_int @ A2 )
        = ( uminus_uminus_int @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_5210_abs__sgn__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
        = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
thf(fact_5211_abs__sgn__eq__1,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
        = one_one_rat ) ) ).

% abs_sgn_eq_1
thf(fact_5212_abs__sgn__eq__1,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
        = one_one_int ) ) ).

% abs_sgn_eq_1
thf(fact_5213_abs__le__D1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B2 )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% abs_le_D1
thf(fact_5214_abs__le__D1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 )
     => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% abs_le_D1
thf(fact_5215_abs__le__D1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% abs_le_D1
thf(fact_5216_abs__ge__self,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_self
thf(fact_5217_abs__ge__self,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ A2 @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_self
thf(fact_5218_abs__ge__self,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_self
thf(fact_5219_abs__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_mult
thf(fact_5220_abs__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_mult
thf(fact_5221_abs__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( abs_abs_int @ ( times_times_int @ A2 @ B2 ) )
      = ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_mult
thf(fact_5222_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_5223_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_5224_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_5225_abs__minus__commute,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_5226_abs__minus__commute,axiom,
    ! [A2: rat,B2: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_5227_abs__minus__commute,axiom,
    ! [A2: int,B2: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_5228_abs__ge__zero,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_zero
thf(fact_5229_abs__ge__zero,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_zero
thf(fact_5230_abs__ge__zero,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_zero
thf(fact_5231_abs__not__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_5232_abs__not__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_5233_abs__not__less__zero,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_5234_abs__of__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( abs_abs_rat @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_5235_abs__of__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_5236_abs__of__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( abs_abs_Code_integer @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_5237_abs__triangle__ineq,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A2 @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_5238_abs__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_5239_abs__triangle__ineq,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A2 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_5240_abs__triangle__ineq2,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_5241_abs__triangle__ineq2,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_5242_abs__triangle__ineq2,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_5243_abs__triangle__ineq3,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_5244_abs__triangle__ineq3,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_5245_abs__triangle__ineq3,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_5246_abs__triangle__ineq2__sym,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5247_abs__triangle__ineq2__sym,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5248_abs__triangle__ineq2__sym,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5249_abs__mult__less,axiom,
    ! [A2: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ C2 )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D2 )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5250_abs__mult__less,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C2 )
     => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D2 )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5251_abs__mult__less,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ C2 )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_5252_abs__leI,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B2 ) ) ) ).

% abs_leI
thf(fact_5253_abs__leI,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 ) ) ) ).

% abs_leI
thf(fact_5254_abs__leI,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
       => ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 ) ) ) ).

% abs_leI
thf(fact_5255_abs__le__D2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B2 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) ) ).

% abs_le_D2
thf(fact_5256_abs__le__D2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ).

% abs_le_D2
thf(fact_5257_abs__le__D2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).

% abs_le_D2
thf(fact_5258_abs__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B2 )
      = ( ( ord_less_eq_rat @ A2 @ B2 )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_5259_abs__le__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 )
      = ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_5260_abs__le__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
      = ( ( ord_less_eq_int @ A2 @ B2 )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_5261_abs__ge__minus__self,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_5262_abs__ge__minus__self,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_5263_abs__ge__minus__self,axiom,
    ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_5264_gbinomial__Suc__Suc,axiom,
    ! [A2: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A2 @ K ) @ ( gbinomial_rat @ A2 @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_5265_mult__sgn__abs,axiom,
    ! [X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_5266_mult__sgn__abs,axiom,
    ! [X: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_5267_mult__sgn__abs,axiom,
    ! [X: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_5268_sgn__mult__abs,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ A2 ) )
      = A2 ) ).

% sgn_mult_abs
thf(fact_5269_sgn__mult__abs,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A2 ) @ ( abs_abs_rat @ A2 ) )
      = A2 ) ).

% sgn_mult_abs
thf(fact_5270_sgn__mult__abs,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A2 ) @ ( abs_abs_int @ A2 ) )
      = A2 ) ).

% sgn_mult_abs
thf(fact_5271_abs__mult__sgn,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( sgn_sgn_Code_integer @ A2 ) )
      = A2 ) ).

% abs_mult_sgn
thf(fact_5272_abs__mult__sgn,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( sgn_sgn_rat @ A2 ) )
      = A2 ) ).

% abs_mult_sgn
thf(fact_5273_abs__mult__sgn,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( sgn_sgn_int @ A2 ) )
      = A2 ) ).

% abs_mult_sgn
thf(fact_5274_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K4: code_integer] : ( times_3573771949741848930nteger @ K4 @ ( sgn_sgn_Code_integer @ K4 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5275_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_rat
    = ( ^ [K4: rat] : ( times_times_rat @ K4 @ ( sgn_sgn_rat @ K4 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5276_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_int
    = ( ^ [K4: int] : ( times_times_int @ K4 @ ( sgn_sgn_int @ K4 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5277_euclidean__size__unit,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( euclid6377331345833325938nteger @ A2 )
        = ( euclid6377331345833325938nteger @ one_one_Code_integer ) ) ) ).

% euclidean_size_unit
thf(fact_5278_euclidean__size__unit,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( euclid4777050414544973029ze_nat @ A2 )
        = ( euclid4777050414544973029ze_nat @ one_one_nat ) ) ) ).

% euclidean_size_unit
thf(fact_5279_euclidean__size__unit,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( euclid4774559944035922753ze_int @ A2 )
        = ( euclid4774559944035922753ze_int @ one_one_int ) ) ) ).

% euclidean_size_unit
thf(fact_5280_euclidean__size__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( times_times_nat @ ( euclid6377331345833325938nteger @ A2 ) @ ( euclid6377331345833325938nteger @ B2 ) ) ) ).

% euclidean_size_mult
thf(fact_5281_euclidean__size__mult,axiom,
    ! [A2: nat,B2: nat] :
      ( ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A2 @ B2 ) )
      = ( times_times_nat @ ( euclid4777050414544973029ze_nat @ A2 ) @ ( euclid4777050414544973029ze_nat @ B2 ) ) ) ).

% euclidean_size_mult
thf(fact_5282_euclidean__size__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( euclid4774559944035922753ze_int @ ( times_times_int @ A2 @ B2 ) )
      = ( times_times_nat @ ( euclid4774559944035922753ze_int @ A2 ) @ ( euclid4774559944035922753ze_int @ B2 ) ) ) ).

% euclidean_size_mult
thf(fact_5283_gbinomial__addition__formula,axiom,
    ! [A2: rat,K: nat] :
      ( ( gbinomial_rat @ A2 @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_5284_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_5285_abs__eq__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          | ( ord_less_eq_rat @ A2 @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
          | ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A2 @ B2 ) )
        = ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_5286_abs__eq__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          | ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
          | ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_5287_abs__eq__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          | ( ord_less_eq_int @ A2 @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B2 )
          | ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A2 @ B2 ) )
        = ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_5288_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_5289_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_5290_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_5291_abs__minus__le__zero,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A2 ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_5292_abs__minus__le__zero,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A2 ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_5293_abs__minus__le__zero,axiom,
    ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A2 ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_5294_abs__triangle__ineq4,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_5295_abs__triangle__ineq4,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_5296_abs__triangle__ineq4,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_5297_abs__diff__triangle__ineq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( plus_plus_rat @ C2 @ D2 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ C2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5298_abs__diff__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ ( plus_p5714425477246183910nteger @ C2 @ D2 ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ C2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5299_abs__diff__triangle__ineq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ ( plus_plus_int @ C2 @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5300_abs__of__neg,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( abs_abs_int @ A2 )
        = ( uminus_uminus_int @ A2 ) ) ) ).

% abs_of_neg
thf(fact_5301_abs__of__neg,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( abs_abs_rat @ A2 )
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% abs_of_neg
thf(fact_5302_abs__of__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% abs_of_neg
thf(fact_5303_abs__sgn__eq,axiom,
    ! [A2: code_integer] :
      ( ( ( A2 = zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( A2 != zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
          = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
thf(fact_5304_abs__sgn__eq,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
          = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
          = one_one_rat ) ) ) ).

% abs_sgn_eq
thf(fact_5305_abs__sgn__eq,axiom,
    ! [A2: int] :
      ( ( ( A2 = zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
          = zero_zero_int ) )
      & ( ( A2 != zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
          = one_one_int ) ) ) ).

% abs_sgn_eq
thf(fact_5306_unit__iff__euclidean__size,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
      = ( ( ( euclid6377331345833325938nteger @ A2 )
          = ( euclid6377331345833325938nteger @ one_one_Code_integer ) )
        & ( A2 != zero_z3403309356797280102nteger ) ) ) ).

% unit_iff_euclidean_size
thf(fact_5307_unit__iff__euclidean__size,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
      = ( ( ( euclid4777050414544973029ze_nat @ A2 )
          = ( euclid4777050414544973029ze_nat @ one_one_nat ) )
        & ( A2 != zero_zero_nat ) ) ) ).

% unit_iff_euclidean_size
thf(fact_5308_unit__iff__euclidean__size,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
      = ( ( ( euclid4774559944035922753ze_int @ A2 )
          = ( euclid4774559944035922753ze_int @ one_one_int ) )
        & ( A2 != zero_zero_int ) ) ) ).

% unit_iff_euclidean_size
thf(fact_5309_size__mult__mono,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ord_less_eq_nat @ ( euclid6377331345833325938nteger @ A2 ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ) ).

% size_mult_mono
thf(fact_5310_size__mult__mono,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A2 ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).

% size_mult_mono
thf(fact_5311_size__mult__mono,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ A2 ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% size_mult_mono
thf(fact_5312_size__mult__mono_H,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ord_less_eq_nat @ ( euclid6377331345833325938nteger @ A2 ) @ ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) ) ) ) ).

% size_mult_mono'
thf(fact_5313_size__mult__mono_H,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A2 ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ B2 @ A2 ) ) ) ) ).

% size_mult_mono'
thf(fact_5314_size__mult__mono_H,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ord_less_eq_nat @ ( euclid4774559944035922753ze_int @ A2 ) @ ( euclid4774559944035922753ze_int @ ( times_times_int @ B2 @ A2 ) ) ) ) ).

% size_mult_mono'
thf(fact_5315_euclidean__size__times__unit,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( euclid6377331345833325938nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
        = ( euclid6377331345833325938nteger @ B2 ) ) ) ).

% euclidean_size_times_unit
thf(fact_5316_euclidean__size__times__unit,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A2 @ B2 ) )
        = ( euclid4777050414544973029ze_nat @ B2 ) ) ) ).

% euclidean_size_times_unit
thf(fact_5317_euclidean__size__times__unit,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( euclid4774559944035922753ze_int @ ( times_times_int @ A2 @ B2 ) )
        = ( euclid4774559944035922753ze_int @ B2 ) ) ) ).

% euclidean_size_times_unit
thf(fact_5318_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_5319_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_5320_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_5321_card_Oempty,axiom,
    ( ( finite711546835091564841at_nat @ bot_bo2099793752762293965at_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_5322_card_Oempty,axiom,
    ( ( finite_card_o @ bot_bot_set_o )
    = zero_zero_nat ) ).

% card.empty
thf(fact_5323_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_5324_card_Oempty,axiom,
    ( ( finite_card_int @ bot_bot_set_int )
    = zero_zero_nat ) ).

% card.empty
thf(fact_5325_divmod__cases,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ( B2 != zero_zero_nat )
       => ( ( ( modulo_modulo_nat @ A2 @ B2 )
            = zero_zero_nat )
         => ( A2
           != ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) ) ) )
     => ( ( ( B2 != zero_zero_nat )
         => ! [Q6: nat,R3: nat] :
              ( ( ( euclid3398187327856392827nt_nat @ R3 )
                = ( euclid3398187327856392827nt_nat @ B2 ) )
             => ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R3 ) @ ( euclid4777050414544973029ze_nat @ B2 ) )
               => ( ( R3 != zero_zero_nat )
                 => ( ( ( divide_divide_nat @ A2 @ B2 )
                      = Q6 )
                   => ( ( ( modulo_modulo_nat @ A2 @ B2 )
                        = R3 )
                     => ( A2
                       != ( plus_plus_nat @ ( times_times_nat @ Q6 @ B2 ) @ R3 ) ) ) ) ) ) ) )
       => ( B2 = zero_zero_nat ) ) ) ).

% divmod_cases
thf(fact_5326_divmod__cases,axiom,
    ! [B2: int,A2: int] :
      ( ( ( B2 != zero_zero_int )
       => ( ( ( modulo_modulo_int @ A2 @ B2 )
            = zero_zero_int )
         => ( A2
           != ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) ) ) )
     => ( ( ( B2 != zero_zero_int )
         => ! [Q6: int,R3: int] :
              ( ( ( euclid3395696857347342551nt_int @ R3 )
                = ( euclid3395696857347342551nt_int @ B2 ) )
             => ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R3 ) @ ( euclid4774559944035922753ze_int @ B2 ) )
               => ( ( R3 != zero_zero_int )
                 => ( ( ( divide_divide_int @ A2 @ B2 )
                      = Q6 )
                   => ( ( ( modulo_modulo_int @ A2 @ B2 )
                        = R3 )
                     => ( A2
                       != ( plus_plus_int @ ( times_times_int @ Q6 @ B2 ) @ R3 ) ) ) ) ) ) ) )
       => ( B2 = zero_zero_int ) ) ) ).

% divmod_cases
thf(fact_5327_divmod__cases,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ( B2 != zero_z3403309356797280102nteger )
       => ( ( ( modulo364778990260209775nteger @ A2 @ B2 )
            = zero_z3403309356797280102nteger )
         => ( A2
           != ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) ) ) )
     => ( ( ( B2 != zero_z3403309356797280102nteger )
         => ! [Q6: code_integer,R3: code_integer] :
              ( ( ( euclid6289375185220004616nteger @ R3 )
                = ( euclid6289375185220004616nteger @ B2 ) )
             => ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ R3 ) @ ( euclid6377331345833325938nteger @ B2 ) )
               => ( ( R3 != zero_z3403309356797280102nteger )
                 => ( ( ( divide6298287555418463151nteger @ A2 @ B2 )
                      = Q6 )
                   => ( ( ( modulo364778990260209775nteger @ A2 @ B2 )
                        = R3 )
                     => ( A2
                       != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ Q6 @ B2 ) @ R3 ) ) ) ) ) ) ) )
       => ( B2 = zero_z3403309356797280102nteger ) ) ) ).

% divmod_cases
thf(fact_5328_divmod__cases,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( ( B2 != zero_z2226904508553997617atural )
       => ( ( ( modulo8411746178871703098atural @ A2 @ B2 )
            = zero_z2226904508553997617atural )
         => ( A2
           != ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A2 @ B2 ) @ B2 ) ) ) )
     => ( ( ( B2 != zero_z2226904508553997617atural )
         => ! [Q6: code_natural,R3: code_natural] :
              ( ( ( euclid5112970336976722131atural @ R3 )
                = ( euclid5112970336976722131atural @ B2 ) )
             => ( ( ord_less_nat @ ( euclid5200926497590043453atural @ R3 ) @ ( euclid5200926497590043453atural @ B2 ) )
               => ( ( R3 != zero_z2226904508553997617atural )
                 => ( ( ( divide5121882707175180666atural @ A2 @ B2 )
                      = Q6 )
                   => ( ( ( modulo8411746178871703098atural @ A2 @ B2 )
                        = R3 )
                     => ( A2
                       != ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ Q6 @ B2 ) @ R3 ) ) ) ) ) ) ) )
       => ( B2 = zero_z2226904508553997617atural ) ) ) ).

% divmod_cases
thf(fact_5329_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A2 @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_5330_mod__eqI,axiom,
    ! [B2: nat,R2: nat,Q5: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( ( euclid3398187327856392827nt_nat @ R2 )
          = ( euclid3398187327856392827nt_nat @ B2 ) )
       => ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R2 ) @ ( euclid4777050414544973029ze_nat @ B2 ) )
         => ( ( ( plus_plus_nat @ ( times_times_nat @ Q5 @ B2 ) @ R2 )
              = A2 )
           => ( ( modulo_modulo_nat @ A2 @ B2 )
              = R2 ) ) ) ) ) ).

% mod_eqI
thf(fact_5331_mod__eqI,axiom,
    ! [B2: int,R2: int,Q5: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( euclid3395696857347342551nt_int @ R2 )
          = ( euclid3395696857347342551nt_int @ B2 ) )
       => ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R2 ) @ ( euclid4774559944035922753ze_int @ B2 ) )
         => ( ( ( plus_plus_int @ ( times_times_int @ Q5 @ B2 ) @ R2 )
              = A2 )
           => ( ( modulo_modulo_int @ A2 @ B2 )
              = R2 ) ) ) ) ) ).

% mod_eqI
thf(fact_5332_mod__eqI,axiom,
    ! [B2: code_integer,R2: code_integer,Q5: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( ( euclid6289375185220004616nteger @ R2 )
          = ( euclid6289375185220004616nteger @ B2 ) )
       => ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ R2 ) @ ( euclid6377331345833325938nteger @ B2 ) )
         => ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ Q5 @ B2 ) @ R2 )
              = A2 )
           => ( ( modulo364778990260209775nteger @ A2 @ B2 )
              = R2 ) ) ) ) ) ).

% mod_eqI
thf(fact_5333_mod__eqI,axiom,
    ! [B2: code_natural,R2: code_natural,Q5: code_natural,A2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( ( euclid5112970336976722131atural @ R2 )
          = ( euclid5112970336976722131atural @ B2 ) )
       => ( ( ord_less_nat @ ( euclid5200926497590043453atural @ R2 ) @ ( euclid5200926497590043453atural @ B2 ) )
         => ( ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ Q5 @ B2 ) @ R2 )
              = A2 )
           => ( ( modulo8411746178871703098atural @ A2 @ B2 )
              = R2 ) ) ) ) ) ).

% mod_eqI
thf(fact_5334_div__bounded,axiom,
    ! [B2: nat,R2: nat,Q5: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( ( euclid3398187327856392827nt_nat @ R2 )
          = ( euclid3398187327856392827nt_nat @ B2 ) )
       => ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R2 ) @ ( euclid4777050414544973029ze_nat @ B2 ) )
         => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ Q5 @ B2 ) @ R2 ) @ B2 )
            = Q5 ) ) ) ) ).

% div_bounded
thf(fact_5335_div__bounded,axiom,
    ! [B2: int,R2: int,Q5: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( euclid3395696857347342551nt_int @ R2 )
          = ( euclid3395696857347342551nt_int @ B2 ) )
       => ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R2 ) @ ( euclid4774559944035922753ze_int @ B2 ) )
         => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ Q5 @ B2 ) @ R2 ) @ B2 )
            = Q5 ) ) ) ) ).

% div_bounded
thf(fact_5336_div__bounded,axiom,
    ! [B2: code_integer,R2: code_integer,Q5: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( ( euclid6289375185220004616nteger @ R2 )
          = ( euclid6289375185220004616nteger @ B2 ) )
       => ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ R2 ) @ ( euclid6377331345833325938nteger @ B2 ) )
         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ Q5 @ B2 ) @ R2 ) @ B2 )
            = Q5 ) ) ) ) ).

% div_bounded
thf(fact_5337_div__bounded,axiom,
    ! [B2: code_natural,R2: code_natural,Q5: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( ( euclid5112970336976722131atural @ R2 )
          = ( euclid5112970336976722131atural @ B2 ) )
       => ( ( ord_less_nat @ ( euclid5200926497590043453atural @ R2 ) @ ( euclid5200926497590043453atural @ B2 ) )
         => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ Q5 @ B2 ) @ R2 ) @ B2 )
            = Q5 ) ) ) ) ).

% div_bounded
thf(fact_5338_division__segment__1,axiom,
    ( ( euclid6289375185220004616nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% division_segment_1
thf(fact_5339_division__segment__1,axiom,
    ( ( euclid3398187327856392827nt_nat @ one_one_nat )
    = one_one_nat ) ).

% division_segment_1
thf(fact_5340_division__segment__1,axiom,
    ( ( euclid3395696857347342551nt_int @ one_one_int )
    = one_one_int ) ).

% division_segment_1
thf(fact_5341_division__segment__numeral,axiom,
    ! [K: num] :
      ( ( euclid3398187327856392827nt_nat @ ( numeral_numeral_nat @ K ) )
      = one_one_nat ) ).

% division_segment_numeral
thf(fact_5342_division__segment__numeral,axiom,
    ! [K: num] :
      ( ( euclid3395696857347342551nt_int @ ( numeral_numeral_int @ K ) )
      = one_one_int ) ).

% division_segment_numeral
thf(fact_5343_division__segment__numeral,axiom,
    ! [K: num] :
      ( ( euclid6289375185220004616nteger @ ( numera6620942414471956472nteger @ K ) )
      = one_one_Code_integer ) ).

% division_segment_numeral
thf(fact_5344_division__segment__numeral,axiom,
    ! [K: num] :
      ( ( euclid5112970336976722131atural @ ( numera5444537566228673987atural @ K ) )
      = one_one_Code_natural ) ).

% division_segment_numeral
thf(fact_5345_division__segment__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid6289375185220004616nteger @ ( semiri4939895301339042750nteger @ N ) )
      = one_one_Code_integer ) ).

% division_segment_of_nat
thf(fact_5346_division__segment__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid3398187327856392827nt_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = one_one_nat ) ).

% division_segment_of_nat
thf(fact_5347_division__segment__of__nat,axiom,
    ! [N: nat] :
      ( ( euclid3395696857347342551nt_int @ ( semiri1314217659103216013at_int @ N ) )
      = one_one_int ) ).

% division_segment_of_nat
thf(fact_5348_division__segment__euclidean__size,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( euclid6289375185220004616nteger @ A2 ) @ ( semiri4939895301339042750nteger @ ( euclid6377331345833325938nteger @ A2 ) ) )
      = A2 ) ).

% division_segment_euclidean_size
thf(fact_5349_division__segment__euclidean__size,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ ( euclid3398187327856392827nt_nat @ A2 ) @ ( semiri1316708129612266289at_nat @ ( euclid4777050414544973029ze_nat @ A2 ) ) )
      = A2 ) ).

% division_segment_euclidean_size
thf(fact_5350_division__segment__euclidean__size,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( euclid3395696857347342551nt_int @ A2 ) @ ( semiri1314217659103216013at_int @ ( euclid4774559944035922753ze_int @ A2 ) ) )
      = A2 ) ).

% division_segment_euclidean_size
thf(fact_5351_div__mult2__eq_H,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( divide_divide_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% div_mult2_eq'
thf(fact_5352_div__mult2__eq_H,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ ( semiri4939895301339042750nteger @ M2 ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% div_mult2_eq'
thf(fact_5353_div__mult2__eq_H,axiom,
    ! [A2: code_natural,M2: nat,N: nat] :
      ( ( divide5121882707175180666atural @ A2 @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M2 ) @ ( semiri3763490453095760265atural @ N ) ) )
      = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A2 @ ( semiri3763490453095760265atural @ M2 ) ) @ ( semiri3763490453095760265atural @ N ) ) ) ).

% div_mult2_eq'
thf(fact_5354_div__mult2__eq_H,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( divide_divide_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A2 @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% div_mult2_eq'
thf(fact_5355_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X: rat] :
      ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) @ X )
      = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_5356_division__segment__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( B2 != zero_z3403309356797280102nteger )
       => ( ( euclid6289375185220004616nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
          = ( times_3573771949741848930nteger @ ( euclid6289375185220004616nteger @ A2 ) @ ( euclid6289375185220004616nteger @ B2 ) ) ) ) ) ).

% division_segment_mult
thf(fact_5357_division__segment__mult,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( ( euclid3398187327856392827nt_nat @ ( times_times_nat @ A2 @ B2 ) )
          = ( times_times_nat @ ( euclid3398187327856392827nt_nat @ A2 ) @ ( euclid3398187327856392827nt_nat @ B2 ) ) ) ) ) ).

% division_segment_mult
thf(fact_5358_division__segment__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( ( euclid3395696857347342551nt_int @ ( times_times_int @ A2 @ B2 ) )
          = ( times_times_int @ ( euclid3395696857347342551nt_int @ A2 ) @ ( euclid3395696857347342551nt_int @ B2 ) ) ) ) ) ).

% division_segment_mult
thf(fact_5359_is__unit__division__segment,axiom,
    ! [A2: code_integer] : ( dvd_dvd_Code_integer @ ( euclid6289375185220004616nteger @ A2 ) @ one_one_Code_integer ) ).

% is_unit_division_segment
thf(fact_5360_is__unit__division__segment,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ ( euclid3398187327856392827nt_nat @ A2 ) @ one_one_nat ) ).

% is_unit_division_segment
thf(fact_5361_is__unit__division__segment,axiom,
    ! [A2: int] : ( dvd_dvd_int @ ( euclid3395696857347342551nt_int @ A2 ) @ one_one_int ) ).

% is_unit_division_segment
thf(fact_5362_mod__mult2__eq_H,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A2 @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5363_mod__mult2__eq_H,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A2 @ ( semiri4939895301339042750nteger @ M2 ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A2 @ ( semiri4939895301339042750nteger @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5364_mod__mult2__eq_H,axiom,
    ! [A2: code_natural,M2: nat,N: nat] :
      ( ( modulo8411746178871703098atural @ A2 @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M2 ) @ ( semiri3763490453095760265atural @ N ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M2 ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A2 @ ( semiri3763490453095760265atural @ M2 ) ) @ ( semiri3763490453095760265atural @ N ) ) ) @ ( modulo8411746178871703098atural @ A2 @ ( semiri3763490453095760265atural @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5365_mod__mult2__eq_H,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( ( modulo_modulo_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5366_gbinomial__absorb__comp,axiom,
    ! [A2: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A2 @ K ) )
      = ( times_times_rat @ A2 @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_5367_gbinomial__mult__1,axiom,
    ! [A2: rat,K: nat] :
      ( ( times_times_rat @ A2 @ ( gbinomial_rat @ A2 @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A2 @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_5368_gbinomial__mult__1_H,axiom,
    ! [A2: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A2 @ K ) @ A2 )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A2 @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_5369_Suc__times__gbinomial,axiom,
    ! [K: nat,A2: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( gbinomial_rat @ A2 @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_5370_gbinomial__absorption,axiom,
    ! [K: nat,A2: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K ) ) )
      = ( times_times_rat @ A2 @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_5371_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M2: nat,A2: rat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( times_times_rat @ ( gbinomial_rat @ A2 @ M2 ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M2 ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A2 @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M2 @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_5372_gbinomial__rec,axiom,
    ! [A2: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A2 @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_5373_gbinomial__factors,axiom,
    ! [A2: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A2 @ K ) ) ) ).

% gbinomial_factors
thf(fact_5374_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_5375_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A3: rat,K4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K4 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K4 ) @ A3 ) @ one_one_rat ) @ K4 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_5376_gbinomial__minus,axiom,
    ! [A2: rat,K: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A2 ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_5377_div__eqI,axiom,
    ! [B2: nat,R2: nat,Q5: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( ( euclid3398187327856392827nt_nat @ R2 )
          = ( euclid3398187327856392827nt_nat @ B2 ) )
       => ( ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R2 ) @ ( euclid4777050414544973029ze_nat @ B2 ) )
         => ( ( ( plus_plus_nat @ ( times_times_nat @ Q5 @ B2 ) @ R2 )
              = A2 )
           => ( ( divide_divide_nat @ A2 @ B2 )
              = Q5 ) ) ) ) ) ).

% div_eqI
thf(fact_5378_div__eqI,axiom,
    ! [B2: int,R2: int,Q5: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( euclid3395696857347342551nt_int @ R2 )
          = ( euclid3395696857347342551nt_int @ B2 ) )
       => ( ( ord_less_nat @ ( euclid4774559944035922753ze_int @ R2 ) @ ( euclid4774559944035922753ze_int @ B2 ) )
         => ( ( ( plus_plus_int @ ( times_times_int @ Q5 @ B2 ) @ R2 )
              = A2 )
           => ( ( divide_divide_int @ A2 @ B2 )
              = Q5 ) ) ) ) ) ).

% div_eqI
thf(fact_5379_div__eqI,axiom,
    ! [B2: code_integer,R2: code_integer,Q5: code_integer,A2: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( ( euclid6289375185220004616nteger @ R2 )
          = ( euclid6289375185220004616nteger @ B2 ) )
       => ( ( ord_less_nat @ ( euclid6377331345833325938nteger @ R2 ) @ ( euclid6377331345833325938nteger @ B2 ) )
         => ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ Q5 @ B2 ) @ R2 )
              = A2 )
           => ( ( divide6298287555418463151nteger @ A2 @ B2 )
              = Q5 ) ) ) ) ) ).

% div_eqI
thf(fact_5380_div__eqI,axiom,
    ! [B2: code_natural,R2: code_natural,Q5: code_natural,A2: code_natural] :
      ( ( B2 != zero_z2226904508553997617atural )
     => ( ( ( euclid5112970336976722131atural @ R2 )
          = ( euclid5112970336976722131atural @ B2 ) )
       => ( ( ord_less_nat @ ( euclid5200926497590043453atural @ R2 ) @ ( euclid5200926497590043453atural @ B2 ) )
         => ( ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ Q5 @ B2 ) @ R2 )
              = A2 )
           => ( ( divide5121882707175180666atural @ A2 @ B2 )
              = Q5 ) ) ) ) ) ).

% div_eqI
thf(fact_5381_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M2 ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5382_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M2 ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5383_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5384_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_5385_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_5386_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_5387_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_5388_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_5389_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5390_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5391_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5392_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5393_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri4939895301339042750nteger @ N )
        = one_one_Code_integer )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5394_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri681578069525770553at_rat @ N )
        = one_one_rat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5395_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5396_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5397_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M2 @ N ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mult
thf(fact_5398_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_5399_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_5400_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_5401_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N4: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_5402_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N4: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_5403_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_5404_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_5405_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_5406_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_5407_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_5408_pochhammer__double,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_5409_pochhammer__minus_H,axiom,
    ! [B2: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_5410_pochhammer__minus_H,axiom,
    ! [B2: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_5411_pochhammer__minus_H,axiom,
    ! [B2: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_5412_pochhammer__minus,axiom,
    ! [B2: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_5413_pochhammer__minus,axiom,
    ! [B2: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_5414_pochhammer__minus,axiom,
    ! [B2: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_5415_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_5416_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_5417_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_5418_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri3624122377584611663nteger @ N )
        = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5419_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri773545260158071498ct_rat @ N )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5420_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1408675320244567234ct_nat @ N )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5421_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1406184849735516958ct_int @ N )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_5422_fact__num__eq__if,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [M: nat] : ( if_Code_integer @ ( M = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5423_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M: nat] : ( if_rat @ ( M = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5424_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M: nat] : ( if_nat @ ( M = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5425_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M: nat] : ( if_int @ ( M = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_5426_fact__0,axiom,
    ( ( semiri3624122377584611663nteger @ zero_zero_nat )
    = one_one_Code_integer ) ).

% fact_0
thf(fact_5427_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_5428_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_5429_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_5430_pochhammer__0,axiom,
    ! [A2: code_integer] :
      ( ( comm_s8582702949713902594nteger @ A2 @ zero_zero_nat )
      = one_one_Code_integer ) ).

% pochhammer_0
thf(fact_5431_pochhammer__0,axiom,
    ! [A2: rat] :
      ( ( comm_s4028243227959126397er_rat @ A2 @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_5432_pochhammer__0,axiom,
    ! [A2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A2 @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_5433_pochhammer__0,axiom,
    ! [A2: int] :
      ( ( comm_s4660882817536571857er_int @ A2 @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_5434_fact__1,axiom,
    ( ( semiri3624122377584611663nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% fact_1
thf(fact_5435_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_5436_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_5437_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_5438_fact__Suc__0,axiom,
    ( ( semiri3624122377584611663nteger @ ( suc @ zero_zero_nat ) )
    = one_one_Code_integer ) ).

% fact_Suc_0
thf(fact_5439_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_5440_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_5441_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_5442_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri3624122377584611663nteger @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% fact_Suc
thf(fact_5443_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_5444_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_5445_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_5446_pochhammer__fact,axiom,
    ( semiri3624122377584611663nteger
    = ( comm_s8582702949713902594nteger @ one_one_Code_integer ) ) ).

% pochhammer_fact
thf(fact_5447_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_5448_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_5449_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_5450_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_5451_fact__ge__1,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( semiri3624122377584611663nteger @ N ) ) ).

% fact_ge_1
thf(fact_5452_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_5453_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_5454_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% pochhammer_same
thf(fact_5455_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% pochhammer_same
thf(fact_5456_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% pochhammer_same
thf(fact_5457_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A3: rat,K4: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K4 ) ) @ one_one_rat ) @ K4 ) @ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_5458_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A3: rat,K4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K4 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A3 ) @ K4 ) ) @ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_5459_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_5460_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_5461_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_5462_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_5463_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% pochhammer_0_left
thf(fact_5464_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_5465_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_5466_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_5467_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% choose_dvd
thf(fact_5468_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% choose_dvd
thf(fact_5469_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% choose_dvd
thf(fact_5470_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% choose_dvd
thf(fact_5471_pochhammer__rec,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_5472_pochhammer__rec,axiom,
    ! [A2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ A2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_5473_pochhammer__rec,axiom,
    ! [A2: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ A2 @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_5474_pochhammer__rec,axiom,
    ! [A2: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ A2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A2 @ one_one_int ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_5475_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N ) ) @ ( comm_s8582702949713902594nteger @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_5476_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_5477_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_5478_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_5479_pochhammer__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A2 @ N ) @ ( plus_p5714425477246183910nteger @ A2 @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_5480_pochhammer__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A2 @ N ) @ ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_5481_pochhammer__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A2 @ N ) @ ( plus_plus_nat @ A2 @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_5482_pochhammer__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A2 @ N ) @ ( plus_plus_int @ A2 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_5483_pochhammer__product_H,axiom,
    ! [Z: code_integer,N: nat,M2: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ N ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_5484_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_5485_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_5486_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_5487_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_5488_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
      = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_5489_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_5490_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri3624122377584611663nteger @ ( numeral_numeral_nat @ K ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ K ) @ ( semiri3624122377584611663nteger @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_5491_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri2447717529341329178atural @ ( numeral_numeral_nat @ K ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ K ) @ ( semiri2447717529341329178atural @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_5492_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_double
thf(fact_5493_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s8582702949713902594nteger @ Z @ N )
        = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ M2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_5494_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_5495_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_5496_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4660882817536571857er_int @ Z @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_5497_card__doubleton__eq__2__iff,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5498_card__doubleton__eq__2__iff,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5499_card__doubleton__eq__2__iff,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( finite_card_o @ ( insert_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5500_card__doubleton__eq__2__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( finite_card_nat @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5501_card__doubleton__eq__2__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( finite_card_int @ ( insert_int @ A2 @ ( insert_int @ B2 @ bot_bot_set_int ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5502_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_5503_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_5504_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_5505_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_5506_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_5507_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_5508_singletonI,axiom,
    ! [A2: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A2 @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ).

% singletonI
thf(fact_5509_singletonI,axiom,
    ! [A2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ).

% singletonI
thf(fact_5510_singletonI,axiom,
    ! [A2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_5511_singletonI,axiom,
    ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_5512_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_5513_singletonI,axiom,
    ! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_5514_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_5515_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_5516_singleton__insert__inj__eq,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_5517_singleton__insert__inj__eq,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_5518_singleton__insert__inj__eq,axiom,
    ! [B2: $o,A2: $o,A: set_o] :
      ( ( ( insert_o @ B2 @ bot_bot_set_o )
        = ( insert_o @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_5519_singleton__insert__inj__eq,axiom,
    ! [B2: int,A2: int,A: set_int] :
      ( ( ( insert_int @ B2 @ bot_bot_set_int )
        = ( insert_int @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_int @ A @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_5520_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat @ B2 @ bot_bot_set_nat )
        = ( insert_nat @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_5521_singleton__insert__inj__eq_H,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A2 @ A )
        = ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
      = ( ( A2 = B2 )
        & ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_5522_singleton__insert__inj__eq_H,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ A )
        = ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
      = ( ( A2 = B2 )
        & ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_5523_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A: set_o,B2: $o] :
      ( ( ( insert_o @ A2 @ A )
        = ( insert_o @ B2 @ bot_bot_set_o ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_5524_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A: set_int,B2: int] :
      ( ( ( insert_int @ A2 @ A )
        = ( insert_int @ B2 @ bot_bot_set_int ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_int @ A @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_5525_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B2: nat] :
      ( ( ( insert_nat @ A2 @ A )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_5526_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ B ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B2 @ A )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_5527_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B2: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ B2 @ B ) ) )
      = ( ~ ( member6576561426505652726_nat_o @ B2 @ A )
        & ( bot_bo7824918357723582233_nat_o
          = ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_5528_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ B ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B2 @ A )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_5529_disjoint__insert_I2_J,axiom,
    ! [A: set_o,B2: $o,B: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A @ ( insert_o @ B2 @ B ) ) )
      = ( ~ ( member_o @ B2 @ A )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_5530_disjoint__insert_I2_J,axiom,
    ! [A: set_nat,B2: nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
      = ( ~ ( member_nat @ B2 @ A )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_5531_disjoint__insert_I2_J,axiom,
    ! [A: set_int,B2: int,B: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ A @ ( insert_int @ B2 @ B ) ) )
      = ( ~ ( member_int @ B2 @ A )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_5532_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ B @ ( insert9069300056098147895at_nat @ A2 @ A ) )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A2 @ B )
        & ( ( inf_in7913087082777306421at_nat @ B @ A )
          = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_5533_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ B @ ( insert5175938949040314269_nat_o @ A2 @ A ) )
        = bot_bo7824918357723582233_nat_o )
      = ( ~ ( member6576561426505652726_nat_o @ A2 @ B )
        & ( ( inf_in1906310914598751387_nat_o @ B @ A )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% disjoint_insert(1)
thf(fact_5534_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ B @ ( insert8211810215607154385at_nat @ A2 @ A ) )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B )
        & ( ( inf_in2572325071724192079at_nat @ B @ A )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_5535_disjoint__insert_I1_J,axiom,
    ! [B: set_o,A2: $o,A: set_o] :
      ( ( ( inf_inf_set_o @ B @ ( insert_o @ A2 @ A ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A2 @ B )
        & ( ( inf_inf_set_o @ B @ A )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_5536_disjoint__insert_I1_J,axiom,
    ! [B: set_nat,A2: nat,A: set_nat] :
      ( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B )
        & ( ( inf_inf_set_nat @ B @ A )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_5537_disjoint__insert_I1_J,axiom,
    ! [B: set_int,A2: int,A: set_int] :
      ( ( ( inf_inf_set_int @ B @ ( insert_int @ A2 @ A ) )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A2 @ B )
        & ( ( inf_inf_set_int @ B @ A )
          = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
thf(fact_5538_insert__disjoint_I2_J,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ A ) @ B ) )
      = ( ~ ( member8757157785044589968at_nat @ A2 @ B )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_5539_insert__disjoint_I2_J,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B ) )
      = ( ~ ( member6576561426505652726_nat_o @ A2 @ B )
        & ( bot_bo7824918357723582233_nat_o
          = ( inf_in1906310914598751387_nat_o @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_5540_insert__disjoint_I2_J,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A ) @ B ) )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_5541_insert__disjoint_I2_J,axiom,
    ! [A2: $o,A: set_o,B: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B ) )
      = ( ~ ( member_o @ A2 @ B )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_5542_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
      = ( ~ ( member_nat @ A2 @ B )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_5543_insert__disjoint_I2_J,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B ) )
      = ( ~ ( member_int @ A2 @ B )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_5544_insert__disjoint_I1_J,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A2 @ A ) @ B )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A2 @ B )
        & ( ( inf_in7913087082777306421at_nat @ A @ B )
          = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_5545_insert__disjoint_I1_J,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B )
        = bot_bo7824918357723582233_nat_o )
      = ( ~ ( member6576561426505652726_nat_o @ A2 @ B )
        & ( ( inf_in1906310914598751387_nat_o @ A @ B )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% insert_disjoint(1)
thf(fact_5546_insert__disjoint_I1_J,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A ) @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B )
        & ( ( inf_in2572325071724192079at_nat @ A @ B )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_5547_insert__disjoint_I1_J,axiom,
    ! [A2: $o,A: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A2 @ B )
        & ( ( inf_inf_set_o @ A @ B )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_5548_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B )
        & ( ( inf_inf_set_nat @ A @ B )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_5549_insert__disjoint_I1_J,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A2 @ B )
        & ( ( inf_inf_set_int @ A @ B )
          = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
thf(fact_5550_insert__Diff__single,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A2 @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) )
      = ( insert9069300056098147895at_nat @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_5551_insert__Diff__single,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      = ( insert8211810215607154385at_nat @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_5552_insert__Diff__single,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
      = ( insert_o @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_5553_insert__Diff__single,axiom,
    ! [A2: int,A: set_int] :
      ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
      = ( insert_int @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_5554_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_5555_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_5556_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_5557_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_5558_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_5559_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_5560_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_5561_subset__Compl__singleton,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_5562_subset__Compl__singleton,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B2: produc3658429121746597890et_nat > $o] :
      ( ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ ( insert5175938949040314269_nat_o @ B2 @ bot_bo7824918357723582233_nat_o ) ) )
      = ( ~ ( member6576561426505652726_nat_o @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_5563_subset__Compl__singleton,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_5564_subset__Compl__singleton,axiom,
    ! [A: set_o,B2: $o] :
      ( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_5565_subset__Compl__singleton,axiom,
    ! [A: set_int,B2: int] :
      ( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ ( insert_int @ B2 @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_5566_subset__Compl__singleton,axiom,
    ! [A: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_5567_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_5568_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_5569_floor__diff__one,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_5570_ceiling__diff__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_5571_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_5572_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_5573_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_5574_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_5575_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_5576_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_5577_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_5578_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_int_def
thf(fact_5579_singleton__inject,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_5580_singleton__inject,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_5581_singleton__inject,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( insert_o @ A2 @ bot_bot_set_o )
        = ( insert_o @ B2 @ bot_bot_set_o ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_5582_singleton__inject,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_5583_singleton__inject,axiom,
    ! [A2: int,B2: int] :
      ( ( ( insert_int @ A2 @ bot_bot_set_int )
        = ( insert_int @ B2 @ bot_bot_set_int ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_5584_insert__not__empty,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A2 @ A )
     != bot_bo228742789529271731at_nat ) ).

% insert_not_empty
thf(fact_5585_insert__not__empty,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A2 @ A )
     != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
thf(fact_5586_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o @ A2 @ A )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_5587_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_5588_insert__not__empty,axiom,
    ! [A2: int,A: set_int] :
      ( ( insert_int @ A2 @ A )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_5589_doubleton__eq__iff,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,C2: produc3843707927480180839at_nat,D2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ C2 @ ( insert9069300056098147895at_nat @ D2 @ bot_bo228742789529271731at_nat ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5590_doubleton__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,C2: product_prod_nat_nat,D2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C2 @ ( insert8211810215607154385at_nat @ D2 @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5591_doubleton__eq__iff,axiom,
    ! [A2: $o,B2: $o,C2: $o,D2: $o] :
      ( ( ( insert_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) )
        = ( insert_o @ C2 @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5592_doubleton__eq__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
        = ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5593_doubleton__eq__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( insert_int @ A2 @ ( insert_int @ B2 @ bot_bot_set_int ) )
        = ( insert_int @ C2 @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5594_singleton__iff,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B2 @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_5595_singleton__iff,axiom,
    ! [B2: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ B2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_5596_singleton__iff,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_5597_singleton__iff,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_5598_singleton__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_5599_singleton__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_5600_singletonD,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B2 @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_5601_singletonD,axiom,
    ! [B2: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ B2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_5602_singletonD,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_5603_singletonD,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_5604_singletonD,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_5605_singletonD,axiom,
    ! [B2: int,A2: int] :
      ( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_5606_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_5607_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_5608_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K4: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K4 @ ( 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 ) ) @ K4 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_5609_subset__singletonD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
     => ( ( A = bot_bo228742789529271731at_nat )
        | ( A
          = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_5610_subset__singletonD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
     => ( ( A = bot_bo2099793752762293965at_nat )
        | ( A
          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_5611_subset__singletonD,axiom,
    ! [A: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
     => ( ( A = bot_bot_set_o )
        | ( A
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_5612_subset__singletonD,axiom,
    ! [A: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A = bot_bot_set_int )
        | ( A
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_5613_subset__singletonD,axiom,
    ! [A: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A = bot_bot_set_nat )
        | ( A
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_5614_subset__singleton__iff,axiom,
    ! [X6: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X6 @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) )
      = ( ( X6 = bot_bo228742789529271731at_nat )
        | ( X6
          = ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_5615_subset__singleton__iff,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X6 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
      = ( ( X6 = bot_bo2099793752762293965at_nat )
        | ( X6
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_5616_subset__singleton__iff,axiom,
    ! [X6: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( ( X6 = bot_bot_set_o )
        | ( X6
          = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_5617_subset__singleton__iff,axiom,
    ! [X6: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ X6 @ ( insert_int @ A2 @ bot_bot_set_int ) )
      = ( ( X6 = bot_bot_set_int )
        | ( X6
          = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_5618_subset__singleton__iff,axiom,
    ! [X6: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( ( X6 = bot_bot_set_nat )
        | ( X6
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_5619_insert__is__Un,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_5620_insert__is__Un,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_5621_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A3: $o] : ( sup_sup_set_o @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_5622_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_5623_insert__is__Un,axiom,
    ( insert_int
    = ( ^ [A3: int] : ( sup_sup_set_int @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ).

% insert_is_Un
thf(fact_5624_Un__singleton__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A @ B )
        = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = ( ( ( A = bot_bo228742789529271731at_nat )
          & ( B
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B = bot_bo228742789529271731at_nat ) )
        | ( ( A
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5625_Un__singleton__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A @ B )
        = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = ( ( ( A = bot_bo2099793752762293965at_nat )
          & ( B
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B = bot_bo2099793752762293965at_nat ) )
        | ( ( A
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5626_Un__singleton__iff,axiom,
    ! [A: set_o,B: set_o,X: $o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = ( insert_o @ X @ bot_bot_set_o ) )
      = ( ( ( A = bot_bot_set_o )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B = bot_bot_set_o ) )
        | ( ( A
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5627_Un__singleton__iff,axiom,
    ! [A: set_nat,B: set_nat,X: nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = ( insert_nat @ X @ bot_bot_set_nat ) )
      = ( ( ( A = bot_bot_set_nat )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5628_Un__singleton__iff,axiom,
    ! [A: set_int,B: set_int,X: int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = ( insert_int @ X @ bot_bot_set_int ) )
      = ( ( ( A = bot_bot_set_int )
          & ( B
            = ( insert_int @ X @ bot_bot_set_int ) ) )
        | ( ( A
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B = bot_bot_set_int ) )
        | ( ( A
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_5629_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat )
        = ( sup_su5525570899277871387at_nat @ A @ B ) )
      = ( ( ( A = bot_bo228742789529271731at_nat )
          & ( B
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B = bot_bo228742789529271731at_nat ) )
        | ( ( A
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5630_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat )
        = ( sup_su6327502436637775413at_nat @ A @ B ) )
      = ( ( ( A = bot_bo2099793752762293965at_nat )
          & ( B
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B = bot_bo2099793752762293965at_nat ) )
        | ( ( A
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5631_singleton__Un__iff,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ( ( insert_o @ X @ bot_bot_set_o )
        = ( sup_sup_set_o @ A @ B ) )
      = ( ( ( A = bot_bot_set_o )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B = bot_bot_set_o ) )
        | ( ( A
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5632_singleton__Un__iff,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( ( insert_nat @ X @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( ( A = bot_bot_set_nat )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5633_singleton__Un__iff,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ( ( insert_int @ X @ bot_bot_set_int )
        = ( sup_sup_set_int @ A @ B ) )
      = ( ( ( A = bot_bot_set_int )
          & ( B
            = ( insert_int @ X @ bot_bot_set_int ) ) )
        | ( ( A
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B = bot_bot_set_int ) )
        | ( ( A
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_5634_Diff__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A @ B ) @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ) ).

% Diff_insert
thf(fact_5635_Diff__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A @ B ) @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% Diff_insert
thf(fact_5636_Diff__insert,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_5637_Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_5638_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_5639_insert__Diff,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A2 @ A )
     => ( ( insert9069300056098147895at_nat @ A2 @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_5640_insert__Diff,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A2 @ A )
     => ( ( insert5175938949040314269_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_5641_insert__Diff,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A )
     => ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_5642_insert__Diff,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_5643_insert__Diff,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_5644_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_5645_Diff__insert2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ B ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_5646_Diff__insert2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ B ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_5647_Diff__insert2,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B ) ) ).

% Diff_insert2
thf(fact_5648_Diff__insert2,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B ) ) ).

% Diff_insert2
thf(fact_5649_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_5650_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_5651_Diff__insert__absorb,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A )
     => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_5652_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_5653_Diff__insert__absorb,axiom,
    ! [X: $o,A: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ X @ bot_bot_set_o ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_5654_Diff__insert__absorb,axiom,
    ! [X: int,A: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_5655_Diff__insert__absorb,axiom,
    ! [X: nat,A: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_5656_set__minus__singleton__eq,axiom,
    ! [X: produc3843707927480180839at_nat,X6: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ X6 )
     => ( ( minus_3314409938677909166at_nat @ X6 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_5657_set__minus__singleton__eq,axiom,
    ! [X: produc3658429121746597890et_nat > $o,X6: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ X6 )
     => ( ( minus_1801376950450012436_nat_o @ X6 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_5658_set__minus__singleton__eq,axiom,
    ! [X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ X6 )
     => ( ( minus_1356011639430497352at_nat @ X6 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_5659_set__minus__singleton__eq,axiom,
    ! [X: $o,X6: set_o] :
      ( ~ ( member_o @ X @ X6 )
     => ( ( minus_minus_set_o @ X6 @ ( insert_o @ X @ bot_bot_set_o ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_5660_set__minus__singleton__eq,axiom,
    ! [X: int,X6: set_int] :
      ( ~ ( member_int @ X @ X6 )
     => ( ( minus_minus_set_int @ X6 @ ( insert_int @ X @ bot_bot_set_int ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_5661_set__minus__singleton__eq,axiom,
    ! [X: nat,X6: set_nat] :
      ( ~ ( member_nat @ X @ X6 )
     => ( ( minus_minus_set_nat @ X6 @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = X6 ) ) ).

% set_minus_singleton_eq
thf(fact_5662_insert__minus__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( X != Y )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A ) @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5663_insert__minus__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( X != Y )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A ) @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5664_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( ( X != Y )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ Y @ bot_bot_set_o ) )
        = ( insert_o @ X @ ( minus_minus_set_o @ A @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5665_insert__minus__eq,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( ( X != Y )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ Y @ bot_bot_set_int ) )
        = ( insert_int @ X @ ( minus_minus_set_int @ A @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5666_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( X != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
        = ( insert_nat @ X @ ( minus_minus_set_nat @ A @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5667_le__mult__floor,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A2 ) @ ( archim3151403230148437115or_rat @ B2 ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ) ).

% le_mult_floor
thf(fact_5668_mult__ceiling__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B2 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_5669_subset__insert__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ X @ B ) )
      = ( ( ( member8757157785044589968at_nat @ X @ A )
         => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ A )
         => ( ord_le1268244103169919719at_nat @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_5670_subset__insert__iff,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B ) )
      = ( ( ( member6576561426505652726_nat_o @ X @ A )
         => ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B ) )
        & ( ~ ( member6576561426505652726_nat_o @ X @ A )
         => ( ord_le2965882846123202637_nat_o @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_5671_subset__insert__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X @ B ) )
      = ( ( ( member8440522571783428010at_nat @ X @ A )
         => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ A )
         => ( ord_le3146513528884898305at_nat @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_5672_subset__insert__iff,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) )
      = ( ( ( member_o @ X @ A )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B ) )
        & ( ~ ( member_o @ X @ A )
         => ( ord_less_eq_set_o @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_5673_subset__insert__iff,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B ) )
      = ( ( ( member_int @ X @ A )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B ) )
        & ( ~ ( member_int @ X @ A )
         => ( ord_less_eq_set_int @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_5674_subset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
      = ( ( ( member_nat @ X @ A )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
        & ( ~ ( member_nat @ X @ A )
         => ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_5675_Diff__single__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B )
     => ( ord_le1268244103169919719at_nat @ A @ ( insert9069300056098147895at_nat @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_5676_Diff__single__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B )
     => ( ord_le3146513528884898305at_nat @ A @ ( insert8211810215607154385at_nat @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_5677_Diff__single__insert,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B )
     => ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_5678_Diff__single__insert,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B )
     => ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_5679_Diff__single__insert,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_5680_card__1__singletonE,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ A )
        = one_one_nat )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( A
           != ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% card_1_singletonE
thf(fact_5681_card__1__singletonE,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A )
        = one_one_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ( A
           != ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% card_1_singletonE
thf(fact_5682_card__1__singletonE,axiom,
    ! [A: set_o] :
      ( ( ( finite_card_o @ A )
        = one_one_nat )
     => ~ ! [X3: $o] :
            ( A
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% card_1_singletonE
thf(fact_5683_card__1__singletonE,axiom,
    ! [A: set_nat] :
      ( ( ( finite_card_nat @ A )
        = one_one_nat )
     => ~ ! [X3: nat] :
            ( A
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_5684_card__1__singletonE,axiom,
    ! [A: set_int] :
      ( ( ( finite_card_int @ A )
        = one_one_nat )
     => ~ ! [X3: int] :
            ( A
           != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% card_1_singletonE
thf(fact_5685_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_5686_remove__subset,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ S )
     => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ S ) ) ).

% remove_subset
thf(fact_5687_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_5688_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_5689_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_5690_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_5691_Compl__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X @ A ) )
      = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).

% Compl_insert
thf(fact_5692_Compl__insert,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X @ A ) )
      = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% Compl_insert
thf(fact_5693_Compl__insert,axiom,
    ! [X: $o,A: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X @ A ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_5694_Compl__insert,axiom,
    ! [X: int,A: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_5695_Compl__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_5696_card__Suc__eq,axiom,
    ! [A: set_Pr4329608150637261639at_nat,K: nat] :
      ( ( ( finite3771342082235030671at_nat @ A )
        = ( suc @ K ) )
      = ( ? [B3: produc3843707927480180839at_nat,B7: set_Pr4329608150637261639at_nat] :
            ( ( A
              = ( insert9069300056098147895at_nat @ B3 @ B7 ) )
            & ~ ( member8757157785044589968at_nat @ B3 @ B7 )
            & ( ( finite3771342082235030671at_nat @ B7 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B7 = bot_bo228742789529271731at_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_5697_card__Suc__eq,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,K: nat] :
      ( ( ( finite1363419556375932405_nat_o @ A )
        = ( suc @ K ) )
      = ( ? [B3: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
            ( ( A
              = ( insert5175938949040314269_nat_o @ B3 @ B7 ) )
            & ~ ( member6576561426505652726_nat_o @ B3 @ B7 )
            & ( ( finite1363419556375932405_nat_o @ B7 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B7 = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).

% card_Suc_eq
thf(fact_5698_card__Suc__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( ( finite711546835091564841at_nat @ A )
        = ( suc @ K ) )
      = ( ? [B3: product_prod_nat_nat,B7: set_Pr1261947904930325089at_nat] :
            ( ( A
              = ( insert8211810215607154385at_nat @ B3 @ B7 ) )
            & ~ ( member8440522571783428010at_nat @ B3 @ B7 )
            & ( ( finite711546835091564841at_nat @ B7 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B7 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_5699_card__Suc__eq,axiom,
    ! [A: set_o,K: nat] :
      ( ( ( finite_card_o @ A )
        = ( suc @ K ) )
      = ( ? [B3: $o,B7: set_o] :
            ( ( A
              = ( insert_o @ B3 @ B7 ) )
            & ~ ( member_o @ B3 @ B7 )
            & ( ( finite_card_o @ B7 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B7 = bot_bot_set_o ) ) ) ) ) ).

% card_Suc_eq
thf(fact_5700_card__Suc__eq,axiom,
    ! [A: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A )
        = ( suc @ K ) )
      = ( ? [B3: nat,B7: set_nat] :
            ( ( A
              = ( insert_nat @ B3 @ B7 ) )
            & ~ ( member_nat @ B3 @ B7 )
            & ( ( finite_card_nat @ B7 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B7 = bot_bot_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_5701_card__Suc__eq,axiom,
    ! [A: set_int,K: nat] :
      ( ( ( finite_card_int @ A )
        = ( suc @ K ) )
      = ( ? [B3: int,B7: set_int] :
            ( ( A
              = ( insert_int @ B3 @ B7 ) )
            & ~ ( member_int @ B3 @ B7 )
            & ( ( finite_card_int @ B7 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B7 = bot_bot_set_int ) ) ) ) ) ).

% card_Suc_eq
thf(fact_5702_card__eq__SucD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,K: nat] :
      ( ( ( finite3771342082235030671at_nat @ A )
        = ( suc @ K ) )
     => ? [B5: produc3843707927480180839at_nat,B8: set_Pr4329608150637261639at_nat] :
          ( ( A
            = ( insert9069300056098147895at_nat @ B5 @ B8 ) )
          & ~ ( member8757157785044589968at_nat @ B5 @ B8 )
          & ( ( finite3771342082235030671at_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bo228742789529271731at_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_5703_card__eq__SucD,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,K: nat] :
      ( ( ( finite1363419556375932405_nat_o @ A )
        = ( suc @ K ) )
     => ? [B5: produc3658429121746597890et_nat > $o,B8: set_Pr4532377907799695533_nat_o] :
          ( ( A
            = ( insert5175938949040314269_nat_o @ B5 @ B8 ) )
          & ~ ( member6576561426505652726_nat_o @ B5 @ B8 )
          & ( ( finite1363419556375932405_nat_o @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bo7824918357723582233_nat_o ) ) ) ) ).

% card_eq_SucD
thf(fact_5704_card__eq__SucD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( ( finite711546835091564841at_nat @ A )
        = ( suc @ K ) )
     => ? [B5: product_prod_nat_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ( A
            = ( insert8211810215607154385at_nat @ B5 @ B8 ) )
          & ~ ( member8440522571783428010at_nat @ B5 @ B8 )
          & ( ( finite711546835091564841at_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_5705_card__eq__SucD,axiom,
    ! [A: set_o,K: nat] :
      ( ( ( finite_card_o @ A )
        = ( suc @ K ) )
     => ? [B5: $o,B8: set_o] :
          ( ( A
            = ( insert_o @ B5 @ B8 ) )
          & ~ ( member_o @ B5 @ B8 )
          & ( ( finite_card_o @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_o ) ) ) ) ).

% card_eq_SucD
thf(fact_5706_card__eq__SucD,axiom,
    ! [A: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A )
        = ( suc @ K ) )
     => ? [B5: nat,B8: set_nat] :
          ( ( A
            = ( insert_nat @ B5 @ B8 ) )
          & ~ ( member_nat @ B5 @ B8 )
          & ( ( finite_card_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_5707_card__eq__SucD,axiom,
    ! [A: set_int,K: nat] :
      ( ( ( finite_card_int @ A )
        = ( suc @ K ) )
     => ? [B5: int,B8: set_int] :
          ( ( A
            = ( insert_int @ B5 @ B8 ) )
          & ~ ( member_int @ B5 @ B8 )
          & ( ( finite_card_int @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
thf(fact_5708_card__1__singleton__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ A )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: produc3843707927480180839at_nat] :
            ( A
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_5709_card__1__singleton__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: product_prod_nat_nat] :
            ( A
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_5710_card__1__singleton__iff,axiom,
    ! [A: set_o] :
      ( ( ( finite_card_o @ A )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: $o] :
            ( A
            = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).

% card_1_singleton_iff
thf(fact_5711_card__1__singleton__iff,axiom,
    ! [A: set_nat] :
      ( ( ( finite_card_nat @ A )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: nat] :
            ( A
            = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_5712_card__1__singleton__iff,axiom,
    ! [A: set_int] :
      ( ( ( finite_card_int @ A )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X2: int] :
            ( A
            = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).

% card_1_singleton_iff
thf(fact_5713_card__Diff1__le,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) @ ( finite3771342082235030671at_nat @ A ) ) ).

% card_Diff1_le
thf(fact_5714_card__Diff1__le,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A ) ) ).

% card_Diff1_le
thf(fact_5715_card__Diff1__le,axiom,
    ! [A: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ).

% card_Diff1_le
thf(fact_5716_card__Diff1__le,axiom,
    ! [A: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ).

% card_Diff1_le
thf(fact_5717_card__Diff1__le,axiom,
    ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ).

% card_Diff1_le
thf(fact_5718_psubset__insert__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A @ ( insert9069300056098147895at_nat @ X @ B ) )
      = ( ( ( member8757157785044589968at_nat @ X @ B )
         => ( ord_le2604355607129572851at_nat @ A @ B ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ B )
         => ( ( ( member8757157785044589968at_nat @ X @ A )
             => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B ) )
            & ( ~ ( member8757157785044589968at_nat @ X @ A )
             => ( ord_le1268244103169919719at_nat @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5719_psubset__insert__iff,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2453136405763929_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B ) )
      = ( ( ( member6576561426505652726_nat_o @ X @ B )
         => ( ord_le2453136405763929_nat_o @ A @ B ) )
        & ( ~ ( member6576561426505652726_nat_o @ X @ B )
         => ( ( ( member6576561426505652726_nat_o @ X @ A )
             => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B ) )
            & ( ~ ( member6576561426505652726_nat_o @ X @ A )
             => ( ord_le2965882846123202637_nat_o @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5720_psubset__insert__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ ( insert8211810215607154385at_nat @ X @ B ) )
      = ( ( ( member8440522571783428010at_nat @ X @ B )
         => ( ord_le7866589430770878221at_nat @ A @ B ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ B )
         => ( ( ( member8440522571783428010at_nat @ X @ A )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B ) )
            & ( ~ ( member8440522571783428010at_nat @ X @ A )
             => ( ord_le3146513528884898305at_nat @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5721_psubset__insert__iff,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_set_o @ A @ ( insert_o @ X @ B ) )
      = ( ( ( member_o @ X @ B )
         => ( ord_less_set_o @ A @ B ) )
        & ( ~ ( member_o @ X @ B )
         => ( ( ( member_o @ X @ A )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B ) )
            & ( ~ ( member_o @ X @ A )
             => ( ord_less_eq_set_o @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5722_psubset__insert__iff,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_set_int @ A @ ( insert_int @ X @ B ) )
      = ( ( ( member_int @ X @ B )
         => ( ord_less_set_int @ A @ B ) )
        & ( ~ ( member_int @ X @ B )
         => ( ( ( member_int @ X @ A )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B ) )
            & ( ~ ( member_int @ X @ A )
             => ( ord_less_eq_set_int @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5723_psubset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B ) )
      = ( ( ( member_nat @ X @ B )
         => ( ord_less_set_nat @ A @ B ) )
        & ( ~ ( member_nat @ X @ B )
         => ( ( ( member_nat @ X @ A )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
            & ( ~ ( member_nat @ X @ A )
             => ( ord_less_eq_set_nat @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5724_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X2: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y2 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_5725_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_5726_card__2__iff,axiom,
    ! [S: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X2: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
            ( ( S
              = ( insert9069300056098147895at_nat @ X2 @ ( insert9069300056098147895at_nat @ Y2 @ bot_bo228742789529271731at_nat ) ) )
            & ( X2 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_5727_card__2__iff,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
            ( ( S
              = ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ Y2 @ bot_bo2099793752762293965at_nat ) ) )
            & ( X2 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_5728_card__2__iff,axiom,
    ! [S: set_o] :
      ( ( ( finite_card_o @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X2: $o,Y2: $o] :
            ( ( S
              = ( insert_o @ X2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
            & ( X2 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_5729_card__2__iff,axiom,
    ! [S: set_nat] :
      ( ( ( finite_card_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X2: nat,Y2: nat] :
            ( ( S
              = ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
            & ( X2 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_5730_card__2__iff,axiom,
    ! [S: set_int] :
      ( ( ( finite_card_int @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X2: int,Y2: int] :
            ( ( S
              = ( insert_int @ X2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
            & ( X2 != Y2 ) ) ) ) ).

% card_2_iff
thf(fact_5731_card__3__iff,axiom,
    ! [S: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X2: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat,Z3: produc3843707927480180839at_nat] :
            ( ( S
              = ( insert9069300056098147895at_nat @ X2 @ ( insert9069300056098147895at_nat @ Y2 @ ( insert9069300056098147895at_nat @ Z3 @ bot_bo228742789529271731at_nat ) ) ) )
            & ( X2 != Y2 )
            & ( Y2 != Z3 )
            & ( X2 != Z3 ) ) ) ) ).

% card_3_iff
thf(fact_5732_card__3__iff,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( ( S
              = ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ Y2 @ ( insert8211810215607154385at_nat @ Z3 @ bot_bo2099793752762293965at_nat ) ) ) )
            & ( X2 != Y2 )
            & ( Y2 != Z3 )
            & ( X2 != Z3 ) ) ) ) ).

% card_3_iff
thf(fact_5733_card__3__iff,axiom,
    ! [S: set_o] :
      ( ( ( finite_card_o @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X2: $o,Y2: $o,Z3: $o] :
            ( ( S
              = ( insert_o @ X2 @ ( insert_o @ Y2 @ ( insert_o @ Z3 @ bot_bot_set_o ) ) ) )
            & ( X2 != Y2 )
            & ( Y2 != Z3 )
            & ( X2 != Z3 ) ) ) ) ).

% card_3_iff
thf(fact_5734_card__3__iff,axiom,
    ! [S: set_nat] :
      ( ( ( finite_card_nat @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X2: nat,Y2: nat,Z3: nat] :
            ( ( S
              = ( insert_nat @ X2 @ ( insert_nat @ Y2 @ ( insert_nat @ Z3 @ bot_bot_set_nat ) ) ) )
            & ( X2 != Y2 )
            & ( Y2 != Z3 )
            & ( X2 != Z3 ) ) ) ) ).

% card_3_iff
thf(fact_5735_card__3__iff,axiom,
    ! [S: set_int] :
      ( ( ( finite_card_int @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X2: int,Y2: int,Z3: int] :
            ( ( S
              = ( insert_int @ X2 @ ( insert_int @ Y2 @ ( insert_int @ Z3 @ bot_bot_set_int ) ) ) )
            & ( X2 != Y2 )
            & ( Y2 != Z3 )
            & ( X2 != Z3 ) ) ) ) ).

% card_3_iff
thf(fact_5736_card__Diff__singleton__if,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A )
       => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
          = ( minus_minus_nat @ ( finite3771342082235030671at_nat @ A ) @ one_one_nat ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A )
       => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
          = ( finite3771342082235030671at_nat @ A ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_5737_card__Diff__singleton__if,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
          = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A ) @ one_one_nat ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ X @ A )
       => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
          = ( finite1363419556375932405_nat_o @ A ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_5738_card__Diff__singleton__if,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A ) @ one_one_nat ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
          = ( finite711546835091564841at_nat @ A ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_5739_card__Diff__singleton__if,axiom,
    ! [X: $o,A: set_o] :
      ( ( ( member_o @ X @ A )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ A ) @ one_one_nat ) ) )
      & ( ~ ( member_o @ X @ A )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( finite_card_o @ A ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_5740_card__Diff__singleton__if,axiom,
    ! [X: int,A: set_int] :
      ( ( ( member_int @ X @ A )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) )
      & ( ~ ( member_int @ X @ A )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( finite_card_int @ A ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_5741_card__Diff__singleton__if,axiom,
    ! [X: nat,A: set_nat] :
      ( ( ( member_nat @ X @ A )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X @ A )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( finite_card_nat @ A ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_5742_card__Diff__singleton,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A )
     => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        = ( minus_minus_nat @ ( finite3771342082235030671at_nat @ A ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_5743_card__Diff__singleton,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ A )
     => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
        = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_5744_card__Diff__singleton,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A )
     => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_5745_card__Diff__singleton,axiom,
    ! [X: $o,A: set_o] :
      ( ( member_o @ X @ A )
     => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
        = ( minus_minus_nat @ ( finite_card_o @ A ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_5746_card__Diff__singleton,axiom,
    ! [X: int,A: set_int] :
      ( ( member_int @ X @ A )
     => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
        = ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_5747_card__Diff__singleton,axiom,
    ! [X: nat,A: set_nat] :
      ( ( member_nat @ X @ A )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_5748_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).

% bezw.pelims
thf(fact_5749_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q5: int,R2: int] :
      ( ( ord_less_eq_int @ B2 @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q5 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q5 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5750_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_5751_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] :
      ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_5752_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_5753_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_5754_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_5755_the__elem__eq,axiom,
    ! [X: int] :
      ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% the_elem_eq
thf(fact_5756_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q5: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q5 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_5757_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).

% is_singletonI
thf(fact_5758_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).

% is_singletonI
thf(fact_5759_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_5760_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_5761_is__singletonI,axiom,
    ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_5762_divmod__BitM__2__eq,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M2 ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_5763_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_5764_mod__int__unique,axiom,
    ! [K: int,L: int,Q5: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q5 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_5765_eucl__rel__int,axiom,
    ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).

% eucl_rel_int
thf(fact_5766_div__int__unique,axiom,
    ! [K: int,L: int,Q5: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q5 @ R2 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q5 ) ) ).

% div_int_unique
thf(fact_5767_unique__quotient,axiom,
    ! [A2: int,B2: int,Q5: int,R2: int,Q7: int,R4: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q5 @ R2 ) )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q7 @ R4 ) )
       => ( Q5 = Q7 ) ) ) ).

% unique_quotient
thf(fact_5768_unique__remainder,axiom,
    ! [A2: int,B2: int,Q5: int,R2: int,Q7: int,R4: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q5 @ R2 ) )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q7 @ R4 ) )
       => ( R2 = R4 ) ) ) ).

% unique_remainder
thf(fact_5769_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_5770_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_5771_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_5772_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_5773_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_5774_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_5775_is__singletonI_H,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( A != bot_bo7824918357723582233_nat_o )
     => ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X3 @ A )
           => ( ( member6576561426505652726_nat_o @ Y3 @ A )
             => ( X3 = Y3 ) ) )
       => ( is_sin5180296473474724033_nat_o @ A ) ) ) ).

% is_singletonI'
thf(fact_5776_is__singletonI_H,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
     => ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A )
           => ( ( member8440522571783428010at_nat @ Y3 @ A )
             => ( X3 = Y3 ) ) )
       => ( is_sin2850979758926227957at_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_5777_is__singletonI_H,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o,Y3: $o] :
            ( ( member_o @ X3 @ A )
           => ( ( member_o @ Y3 @ A )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_o @ A ) ) ) ).

% is_singletonI'
thf(fact_5778_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ( member_nat @ Y3 @ A )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_5779_is__singletonI_H,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
     => ( ! [X3: int,Y3: int] :
            ( ( member_int @ X3 @ A )
           => ( ( member_int @ Y3 @ A )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_int @ A ) ) ) ).

% is_singletonI'
thf(fact_5780_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q5: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q5 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q5 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_5781_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q5: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q5 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q5 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_5782_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
          ( ? [K4: int] :
              ( ( A1 = K4 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K4 ) ) )
          | ? [L2: int,K4: int,Q8: int] :
              ( ( A1 = K4 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q8 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K4
                = ( times_times_int @ Q8 @ L2 ) ) )
          | ? [R5: int,L2: int,K4: int,Q8: int] :
              ( ( A1 = K4 )
              & ( 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 ) )
              & ( K4
                = ( plus_plus_int @ ( times_times_int @ Q8 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_5783_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q6: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q6 @ A23 ) ) ) )
         => ~ ! [R3: int,Q6: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q6 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q6 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_5784_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q5: int,R2: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q5 @ R2 ) )
     => ( ( B2 != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A2 ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q5 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q5 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5785_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bitM @ N ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_5786_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_5787_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bitM @ N ) )
      = ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_BitM
thf(fact_5788_is__singleton__def,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat] :
        ? [X2: produc3843707927480180839at_nat] :
          ( A6
          = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_5789_is__singleton__def,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
        ? [X2: product_prod_nat_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_5790_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
        ? [X2: $o] :
          ( A6
          = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_5791_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
        ? [X2: nat] :
          ( A6
          = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_5792_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
        ? [X2: int] :
          ( A6
          = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_5793_is__singletonE,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( is_sin2937591304547752795at_nat @ A )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( A
           != ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% is_singletonE
thf(fact_5794_is__singletonE,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( is_sin2850979758926227957at_nat @ A )
     => ~ ! [X3: product_prod_nat_nat] :
            ( A
           != ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% is_singletonE
thf(fact_5795_is__singletonE,axiom,
    ! [A: set_o] :
      ( ( is_singleton_o @ A )
     => ~ ! [X3: $o] :
            ( A
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_5796_is__singletonE,axiom,
    ! [A: set_nat] :
      ( ( is_singleton_nat @ A )
     => ~ ! [X3: nat] :
            ( A
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_5797_is__singletonE,axiom,
    ! [A: set_int] :
      ( ( is_singleton_int @ A )
     => ~ ! [X3: int] :
            ( A
           != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_5798_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q5: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q5 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q5 ) @ 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 )
             => ( Q5 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_5799_Divides_Oadjust__div__eq,axiom,
    ! [Q5: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q5 @ R2 ) )
      = ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_5800_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X2 ) ) @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim3151403230148437115or_rat @ X2 ) ) ) ) ).

% round_altdef
thf(fact_5801_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_5802_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_5803_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_5804_normalize__negative,axiom,
    ! [Q5: int,P5: int] :
      ( ( ord_less_int @ Q5 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q5 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q5 ) ) ) ) ) ).

% normalize_negative
thf(fact_5805_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N ) ) ).

% round_unique'
thf(fact_5806_of__int__1,axiom,
    ( ( ring_18347121197199848620nteger @ one_one_int )
    = one_one_Code_integer ) ).

% of_int_1
thf(fact_5807_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_5808_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_5809_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_18347121197199848620nteger @ Z )
        = one_one_Code_integer )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5810_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5811_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5812_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_18347121197199848620nteger @ ( times_times_int @ W @ Z ) )
      = ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_int_mult
thf(fact_5813_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_mult
thf(fact_5814_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_5815_normalize__denom__zero,axiom,
    ! [P5: int] :
      ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_5816_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5817_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5818_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5819_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5820_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5821_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5822_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5823_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5824_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5825_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5826_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5827_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5828_mult__of__int__commute,axiom,
    ! [X: int,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( ring_18347121197199848620nteger @ X ) @ Y )
      = ( times_3573771949741848930nteger @ Y @ ( ring_18347121197199848620nteger @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5829_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5830_mult__of__int__commute,axiom,
    ! [X: int,Y: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5831_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X: rat] :
      ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) @ X )
      = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa ) ) ) ) ).

% mult_inverse_of_int_commute
thf(fact_5832_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_5833_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_5834_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_5835_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_5836_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_5837_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_5838_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_5839_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_5840_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
           => ( P @ I2 ) ) ) ) ).

% floor_split
thf(fact_5841_floor__eq__iff,axiom,
    ! [X: rat,A2: int] :
      ( ( ( archim3151403230148437115or_rat @ X )
        = A2 )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A2 ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_5842_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_5843_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_5844_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_5845_ceiling__eq__iff,axiom,
    ! [X: rat,A2: int] :
      ( ( ( archim2889992004027027881ng_rat @ X )
        = A2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A2 ) @ one_one_rat ) @ X )
        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A2 ) ) ) ) ).

% ceiling_eq_iff
thf(fact_5846_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_5847_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_5848_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_5849_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_5850_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_5851_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P5: int,Q5: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P5 @ Q5 ) )
     => ( ord_less_int @ zero_zero_int @ Q5 ) ) ).

% normalize_denom_pos
thf(fact_5852_normalize__crossproduct,axiom,
    ! [Q5: int,S2: int,P5: int,R2: int] :
      ( ( Q5 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q5 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
         => ( ( times_times_int @ P5 @ S2 )
            = ( times_times_int @ R2 @ Q5 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_5853_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_5854_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_5855_floor__divide__lower,axiom,
    ! [Q5: rat,P5: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q5 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q5 ) ) ) @ Q5 ) @ P5 ) ) ).

% floor_divide_lower
thf(fact_5856_ceiling__divide__upper,axiom,
    ! [Q5: rat,P5: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q5 )
     => ( ord_less_eq_rat @ P5 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q5 ) ) ) @ Q5 ) ) ) ).

% ceiling_divide_upper
thf(fact_5857_floor__divide__upper,axiom,
    ! [Q5: rat,P5: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q5 )
     => ( ord_less_rat @ P5 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P5 @ Q5 ) ) ) @ one_one_rat ) @ Q5 ) ) ) ).

% floor_divide_upper
thf(fact_5858_ceiling__divide__lower,axiom,
    ! [Q5: rat,P5: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q5 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P5 @ Q5 ) ) ) @ one_one_rat ) @ Q5 ) @ P5 ) ) ).

% ceiling_divide_lower
thf(fact_5859_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_5860_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_5861_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_5862_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_5863_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K2: int,L3: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L3 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L3 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_5864_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_5865_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J2 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
             => ( P @ I3 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_5866_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5867_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B2 ) ) ) @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5868_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B2 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5869_le__mult__floor__Ints,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A2 ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5870_le__mult__floor__Ints,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ ( times_times_int @ ( archim3151403230148437115or_rat @ A2 ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_18347121197199848620nteger @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5871_le__mult__floor__Ints,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A2 ) @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_5872_frac__unique__iff,axiom,
    ! [X: rat,A2: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = A2 )
      = ( ( member_rat @ ( minus_minus_rat @ X @ A2 ) @ ring_1_Ints_rat )
        & ( ord_less_eq_rat @ zero_zero_rat @ A2 )
        & ( ord_less_rat @ A2 @ one_one_rat ) ) ) ).

% frac_unique_iff
thf(fact_5873_Gcd__0__iff,axiom,
    ! [A: set_int] :
      ( ( ( gcd_Gcd_int @ A )
        = zero_zero_int )
      = ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).

% Gcd_0_iff
thf(fact_5874_Gcd__0__iff,axiom,
    ! [A: set_nat] :
      ( ( ( gcd_Gcd_nat @ A )
        = zero_zero_nat )
      = ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% Gcd_0_iff
thf(fact_5875_remove__def,axiom,
    ( remove5300788332298244428at_nat
    = ( ^ [X2: produc3843707927480180839at_nat,A6: set_Pr4329608150637261639at_nat] : ( minus_3314409938677909166at_nat @ A6 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% remove_def
thf(fact_5876_remove__def,axiom,
    ( remove6466555014256735590at_nat
    = ( ^ [X2: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] : ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% remove_def
thf(fact_5877_remove__def,axiom,
    ( remove_o
    = ( ^ [X2: $o,A6: set_o] : ( minus_minus_set_o @ A6 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).

% remove_def
thf(fact_5878_remove__def,axiom,
    ( remove_int
    = ( ^ [X2: int,A6: set_int] : ( minus_minus_set_int @ A6 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).

% remove_def
thf(fact_5879_remove__def,axiom,
    ( remove_nat
    = ( ^ [X2: nat,A6: set_nat] : ( minus_minus_set_nat @ A6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% remove_def
thf(fact_5880_quotient__of__number_I5_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_5881_Gcd__empty,axiom,
    ( ( gcd_Gcd_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_empty
thf(fact_5882_Gcd__empty,axiom,
    ( ( gcd_Gcd_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_empty
thf(fact_5883_Gcd__UNIV,axiom,
    ( ( gcd_Gcd_int @ top_top_set_int )
    = one_one_int ) ).

% Gcd_UNIV
thf(fact_5884_Gcd__UNIV,axiom,
    ( ( gcd_Gcd_nat @ top_top_set_nat )
    = one_one_nat ) ).

% Gcd_UNIV
thf(fact_5885_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_5886_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_5887_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).

% quotient_of_number(4)
thf(fact_5888_quotient__of__number_I3_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_5889_Ints__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( member_Code_integer @ A2 @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ B2 @ ring_11222124179247155820nteger )
       => ( member_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ring_11222124179247155820nteger ) ) ) ).

% Ints_mult
thf(fact_5890_Ints__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( member_rat @ A2 @ ring_1_Ints_rat )
     => ( ( member_rat @ B2 @ ring_1_Ints_rat )
       => ( member_rat @ ( times_times_rat @ A2 @ B2 ) @ ring_1_Ints_rat ) ) ) ).

% Ints_mult
thf(fact_5891_Ints__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( member_int @ A2 @ ring_1_Ints_int )
     => ( ( member_int @ B2 @ ring_1_Ints_int )
       => ( member_int @ ( times_times_int @ A2 @ B2 ) @ ring_1_Ints_int ) ) ) ).

% Ints_mult
thf(fact_5892_Ints__1,axiom,
    member_Code_integer @ one_one_Code_integer @ ring_11222124179247155820nteger ).

% Ints_1
thf(fact_5893_Ints__1,axiom,
    member_rat @ one_one_rat @ ring_1_Ints_rat ).

% Ints_1
thf(fact_5894_Ints__1,axiom,
    member_int @ one_one_int @ ring_1_Ints_int ).

% Ints_1
thf(fact_5895_quotient__of__div,axiom,
    ! [R2: rat,N: int,D2: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ N @ D2 ) )
     => ( R2
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D2 ) ) ) ) ).

% quotient_of_div
thf(fact_5896_Gcd__1,axiom,
    ! [A: set_nat] :
      ( ( member_nat @ one_one_nat @ A )
     => ( ( gcd_Gcd_nat @ A )
        = one_one_nat ) ) ).

% Gcd_1
thf(fact_5897_Gcd__1,axiom,
    ! [A: set_int] :
      ( ( member_int @ one_one_int @ A )
     => ( ( gcd_Gcd_int @ A )
        = one_one_int ) ) ).

% Gcd_1
thf(fact_5898_Gcd__eq__1__I,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( member_nat @ A2 @ A )
       => ( ( gcd_Gcd_nat @ A )
          = one_one_nat ) ) ) ).

% Gcd_eq_1_I
thf(fact_5899_Gcd__eq__1__I,axiom,
    ! [A2: int,A: set_int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( member_int @ A2 @ A )
       => ( ( gcd_Gcd_int @ A )
          = one_one_int ) ) ) ).

% Gcd_eq_1_I
thf(fact_5900_Ints__odd__nonzero,axiom,
    ! [A2: code_integer] :
      ( ( member_Code_integer @ A2 @ ring_11222124179247155820nteger )
     => ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ A2 )
       != zero_z3403309356797280102nteger ) ) ).

% Ints_odd_nonzero
thf(fact_5901_Ints__odd__nonzero,axiom,
    ! [A2: rat] :
      ( ( member_rat @ A2 @ ring_1_Ints_rat )
     => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A2 ) @ A2 )
       != zero_zero_rat ) ) ).

% Ints_odd_nonzero
thf(fact_5902_Ints__odd__nonzero,axiom,
    ! [A2: int] :
      ( ( member_int @ A2 @ ring_1_Ints_int )
     => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A2 ) @ A2 )
       != zero_zero_int ) ) ).

% Ints_odd_nonzero
thf(fact_5903_quotient__of__denom__pos,axiom,
    ! [R2: rat,P5: int,Q5: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P5 @ Q5 ) )
     => ( ord_less_int @ zero_zero_int @ Q5 ) ) ).

% quotient_of_denom_pos
thf(fact_5904_Ints__odd__less__0,axiom,
    ! [A2: rat] :
      ( ( member_rat @ A2 @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A2 ) @ A2 ) @ zero_zero_rat )
        = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).

% Ints_odd_less_0
thf(fact_5905_Ints__odd__less__0,axiom,
    ! [A2: int] :
      ( ( member_int @ A2 @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A2 ) @ A2 ) @ zero_zero_int )
        = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).

% Ints_odd_less_0
thf(fact_5906_Ints__odd__less__0,axiom,
    ! [A2: code_integer] :
      ( ( member_Code_integer @ A2 @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ A2 ) @ zero_z3403309356797280102nteger )
        = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ) ).

% Ints_odd_less_0
thf(fact_5907_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_5908_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_5909_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_5910_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_5911_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_5912_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_5913_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_5914_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_5915_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_5916_rat__sgn__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P5 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P5 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_5917_frac__neg,axiom,
    ! [X: rat] :
      ( ( ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = zero_zero_rat ) )
      & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).

% frac_neg
thf(fact_5918_quotient__of__int,axiom,
    ! [A2: int] :
      ( ( quotient_of @ ( of_int @ A2 ) )
      = ( product_Pair_int_int @ A2 @ one_one_int ) ) ).

% quotient_of_int
thf(fact_5919_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_5920_Frct__code__post_I5_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% Frct_code_post(5)
thf(fact_5921_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_5922_Frct__code__post_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( ( frct @ ( product_Pair_int_int @ A2 @ ( uminus_uminus_int @ B2 ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ) ).

% Frct_code_post(8)
thf(fact_5923_Frct__code__post_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A2 ) @ B2 ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ) ).

% Frct_code_post(7)
thf(fact_5924_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
          = ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_5925_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numeral_numeral_rat @ ( num_of_nat @ N ) )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( numeral_numeral_rat @ ( num_of_nat @ N ) )
          = ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_5926_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numera6620942414471956472nteger @ ( num_of_nat @ N ) )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( numera6620942414471956472nteger @ ( num_of_nat @ N ) )
          = ( semiri4939895301339042750nteger @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_5927_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numera5444537566228673987atural @ ( num_of_nat @ N ) )
          = one_one_Code_natural ) )
      & ( ( N != zero_zero_nat )
       => ( ( numera5444537566228673987atural @ ( num_of_nat @ N ) )
          = ( semiri3763490453095760265atural @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_5928_numeral__num__of__nat__unfold,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( numeral_numeral_int @ ( num_of_nat @ N ) )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( numeral_numeral_int @ ( num_of_nat @ N ) )
          = ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_5929_Frct__code__post_I1_J,axiom,
    ! [A2: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A2 ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_5930_Frct__code__post_I2_J,axiom,
    ! [A2: int] :
      ( ( frct @ ( product_Pair_int_int @ A2 @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_5931_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_5932_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_5933_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_5934_of__rat__neg__one,axiom,
    ( ( field_2639924705303425560at_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% of_rat_neg_one
thf(fact_5935_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_5936_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_5937_card__Un__disjoint,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( finite4343798906461161616at_nat @ B )
       => ( ( ( inf_in7913087082777306421at_nat @ A @ B )
            = bot_bo228742789529271731at_nat )
         => ( ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) )
            = ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A ) @ ( finite3771342082235030671at_nat @ B ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_5938_card__Un__disjoint,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( finite6177210948735845034at_nat @ B )
       => ( ( ( inf_in2572325071724192079at_nat @ A @ B )
            = bot_bo2099793752762293965at_nat )
         => ( ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A @ B ) )
            = ( plus_plus_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ B ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_5939_card__Un__disjoint,axiom,
    ! [A: set_o,B: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( finite_finite_o @ B )
       => ( ( ( inf_inf_set_o @ A @ B )
            = bot_bot_set_o )
         => ( ( finite_card_o @ ( sup_sup_set_o @ A @ B ) )
            = ( plus_plus_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ B ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_5940_card__Un__disjoint,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ( ( inf_inf_set_nat @ A @ B )
            = bot_bot_set_nat )
         => ( ( finite_card_nat @ ( sup_sup_set_nat @ A @ B ) )
            = ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_5941_card__Un__disjoint,axiom,
    ! [A: set_int,B: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( finite_finite_int @ B )
       => ( ( ( inf_inf_set_int @ A @ B )
            = bot_bot_set_int )
         => ( ( finite_card_int @ ( sup_sup_set_int @ A @ B ) )
            = ( plus_plus_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_5942_card__Diff1__less,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( member8757157785044589968at_nat @ X @ A )
       => ( ord_less_nat @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) @ ( finite3771342082235030671at_nat @ A ) ) ) ) ).

% card_Diff1_less
thf(fact_5943_card__Diff1__less,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ).

% card_Diff1_less
thf(fact_5944_card__Diff1__less,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ X @ A )
       => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A ) ) ) ) ).

% card_Diff1_less
thf(fact_5945_card__Diff1__less,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ X @ A )
       => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ) ) ).

% card_Diff1_less
thf(fact_5946_card__Diff1__less,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ X @ A )
       => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ).

% card_Diff1_less
thf(fact_5947_card__Diff1__less,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).

% card_Diff1_less
thf(fact_5948_atLeastatMost__empty__iff2,axiom,
    ! [A2: $o,B2: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_o @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5949_atLeastatMost__empty__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or189985376899183464nteger @ A2 @ B2 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5950_atLeastatMost__empty__iff2,axiom,
    ! [A2: assn,B2: assn] :
      ( ( bot_bot_set_assn
        = ( set_or7959216805967363635t_assn @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_assn @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5951_atLeastatMost__empty__iff2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5952_atLeastatMost__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5953_atLeastatMost__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5954_atLeastatMost__empty__iff,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5955_atLeastatMost__empty__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( set_or189985376899183464nteger @ A2 @ B2 )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5956_atLeastatMost__empty__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ( set_or7959216805967363635t_assn @ A2 @ B2 )
        = bot_bot_set_assn )
      = ( ~ ( ord_less_eq_assn @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5957_atLeastatMost__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A2 @ B2 )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5958_atLeastatMost__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( set_or1266510415728281911st_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5959_atLeastatMost__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5960_atLeastatMost__empty,axiom,
    ! [B2: $o,A2: $o] :
      ( ( ord_less_o @ B2 @ A2 )
     => ( ( set_or8904488021354931149Most_o @ A2 @ B2 )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_5961_atLeastatMost__empty,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( set_or7959216805967363635t_assn @ A2 @ B2 )
        = bot_bot_set_assn ) ) ).

% atLeastatMost_empty
thf(fact_5962_atLeastatMost__empty,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( set_or189985376899183464nteger @ A2 @ B2 )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_5963_atLeastatMost__empty,axiom,
    ! [B2: code_natural,A2: code_natural] :
      ( ( ord_le5570908160329646204atural @ B2 @ A2 )
     => ( ( set_or8236952565510676787atural @ A2 @ B2 )
        = bot_bo8448779972843232370atural ) ) ).

% atLeastatMost_empty
thf(fact_5964_atLeastatMost__empty,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( set_or1266510415728281911st_int @ A2 @ B2 )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_5965_atLeastatMost__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_5966_atLeastAtMost__singleton,axiom,
    ! [A2: $o] :
      ( ( set_or8904488021354931149Most_o @ A2 @ A2 )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% atLeastAtMost_singleton
thf(fact_5967_atLeastAtMost__singleton,axiom,
    ! [A2: int] :
      ( ( set_or1266510415728281911st_int @ A2 @ A2 )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_5968_atLeastAtMost__singleton,axiom,
    ! [A2: nat] :
      ( ( set_or1269000886237332187st_nat @ A2 @ A2 )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_5969_atLeastAtMost__singleton__iff,axiom,
    ! [A2: $o,B2: $o,C2: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A2 @ B2 )
        = ( insert_o @ C2 @ bot_bot_set_o ) )
      = ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5970_atLeastAtMost__singleton__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( set_or1266510415728281911st_int @ A2 @ B2 )
        = ( insert_int @ C2 @ bot_bot_set_int ) )
      = ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5971_atLeastAtMost__singleton__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) )
      = ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5972_of__rat__1,axiom,
    ( ( field_2639924705303425560at_rat @ one_one_rat )
    = one_one_rat ) ).

% of_rat_1
thf(fact_5973_of__rat__eq__1__iff,axiom,
    ! [A2: rat] :
      ( ( ( field_2639924705303425560at_rat @ A2 )
        = one_one_rat )
      = ( A2 = one_one_rat ) ) ).

% of_rat_eq_1_iff
thf(fact_5974_one__eq__of__rat__iff,axiom,
    ! [A2: rat] :
      ( ( one_one_rat
        = ( field_2639924705303425560at_rat @ A2 ) )
      = ( one_one_rat = A2 ) ) ).

% one_eq_of_rat_iff
thf(fact_5975_card__0__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( ( finite711546835091564841at_nat @ A )
          = zero_zero_nat )
        = ( A = bot_bo2099793752762293965at_nat ) ) ) ).

% card_0_eq
thf(fact_5976_card__0__eq,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( ( finite_card_o @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_o ) ) ) ).

% card_0_eq
thf(fact_5977_card__0__eq,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( finite_card_nat @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_5978_card__0__eq,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( ( finite_card_int @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_int ) ) ) ).

% card_0_eq
thf(fact_5979_of__rat__le__1__iff,axiom,
    ! [R2: rat] :
      ( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R2 ) @ one_one_rat )
      = ( ord_less_eq_rat @ R2 @ one_one_rat ) ) ).

% of_rat_le_1_iff
thf(fact_5980_one__le__of__rat__iff,axiom,
    ! [R2: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R2 ) )
      = ( ord_less_eq_rat @ one_one_rat @ R2 ) ) ).

% one_le_of_rat_iff
thf(fact_5981_of__rat__less__1__iff,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R2 ) @ one_one_rat )
      = ( ord_less_rat @ R2 @ one_one_rat ) ) ).

% of_rat_less_1_iff
thf(fact_5982_one__less__of__rat__iff,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R2 ) )
      = ( ord_less_rat @ one_one_rat @ R2 ) ) ).

% one_less_of_rat_iff
thf(fact_5983_of__rat__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( field_2639924705303425560at_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( times_times_rat @ ( field_2639924705303425560at_rat @ A2 ) @ ( field_2639924705303425560at_rat @ B2 ) ) ) ).

% of_rat_mult
thf(fact_5984_finite_OemptyI,axiom,
    finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).

% finite.emptyI
thf(fact_5985_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_5986_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_5987_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_5988_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_5989_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ ( finite_finite_o @ S )
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_5990_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_5991_infinite__imp__nonempty,axiom,
    ! [S: set_int] :
      ( ~ ( finite_finite_int @ S )
     => ( S != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_5992_atLeastAtMost__singleton_H,axiom,
    ! [A2: $o,B2: $o] :
      ( ( A2 = B2 )
     => ( ( set_or8904488021354931149Most_o @ A2 @ B2 )
        = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5993_atLeastAtMost__singleton_H,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
     => ( ( set_or1266510415728281911st_int @ A2 @ B2 )
        = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5994_atLeastAtMost__singleton_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
     => ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
        = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5995_finite__has__minimal,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A )
               => ( ( ord_less_eq_o @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5996_finite__has__minimal,axiom,
    ! [A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5997_finite__has__minimal,axiom,
    ! [A: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ? [X3: assn] :
            ( ( member_assn @ X3 @ A )
            & ! [Xa2: assn] :
                ( ( member_assn @ Xa2 @ A )
               => ( ( ord_less_eq_assn @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5998_finite__has__minimal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A )
               => ( ( ord_less_eq_set_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5999_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_6000_finite__has__minimal,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_6001_finite__has__maximal,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A )
               => ( ( ord_less_eq_o @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_6002_finite__has__maximal,axiom,
    ! [A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_6003_finite__has__maximal,axiom,
    ! [A: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ? [X3: assn] :
            ( ( member_assn @ X3 @ A )
            & ! [Xa2: assn] :
                ( ( member_assn @ Xa2 @ A )
               => ( ( ord_less_eq_assn @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_6004_finite__has__maximal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_6005_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_6006_finite__has__maximal,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_6007_finite_Ocases,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( ( A2 != bot_bo228742789529271731at_nat )
       => ~ ! [A7: set_Pr4329608150637261639at_nat] :
              ( ? [A5: produc3843707927480180839at_nat] :
                  ( A2
                  = ( insert9069300056098147895at_nat @ A5 @ A7 ) )
             => ~ ( finite4343798906461161616at_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_6008_finite_Ocases,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( A2 != bot_bo2099793752762293965at_nat )
       => ~ ! [A7: set_Pr1261947904930325089at_nat] :
              ( ? [A5: product_prod_nat_nat] :
                  ( A2
                  = ( insert8211810215607154385at_nat @ A5 @ A7 ) )
             => ~ ( finite6177210948735845034at_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_6009_finite_Ocases,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ~ ! [A7: set_o] :
              ( ? [A5: $o] :
                  ( A2
                  = ( insert_o @ A5 @ A7 ) )
             => ~ ( finite_finite_o @ A7 ) ) ) ) ).

% finite.cases
thf(fact_6010_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A5: nat] :
                  ( A2
                  = ( insert_nat @ A5 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_6011_finite_Ocases,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ~ ! [A7: set_int] :
              ( ? [A5: int] :
                  ( A2
                  = ( insert_int @ A5 @ A7 ) )
             => ~ ( finite_finite_int @ A7 ) ) ) ) ).

% finite.cases
thf(fact_6012_finite_Osimps,axiom,
    ( finite4343798906461161616at_nat
    = ( ^ [A3: set_Pr4329608150637261639at_nat] :
          ( ( A3 = bot_bo228742789529271731at_nat )
          | ? [A6: set_Pr4329608150637261639at_nat,B3: produc3843707927480180839at_nat] :
              ( ( A3
                = ( insert9069300056098147895at_nat @ B3 @ A6 ) )
              & ( finite4343798906461161616at_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_6013_finite_Osimps,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A3: set_Pr1261947904930325089at_nat] :
          ( ( A3 = bot_bo2099793752762293965at_nat )
          | ? [A6: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
              ( ( A3
                = ( insert8211810215607154385at_nat @ B3 @ A6 ) )
              & ( finite6177210948735845034at_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_6014_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A3: set_o] :
          ( ( A3 = bot_bot_set_o )
          | ? [A6: set_o,B3: $o] :
              ( ( A3
                = ( insert_o @ B3 @ A6 ) )
              & ( finite_finite_o @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_6015_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A3: set_nat] :
          ( ( A3 = bot_bot_set_nat )
          | ? [A6: set_nat,B3: nat] :
              ( ( A3
                = ( insert_nat @ B3 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_6016_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A3: set_int] :
          ( ( A3 = bot_bot_set_int )
          | ? [A6: set_int,B3: int] :
              ( ( A3
                = ( insert_int @ B3 @ A6 ) )
              & ( finite_finite_int @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_6017_finite__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X3: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ F5 )
             => ( ~ ( member8757157785044589968at_nat @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert9069300056098147895at_nat @ X3 @ F5 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_6018_finite__induct,axiom,
    ! [F4: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ F4 )
     => ( ( P @ bot_bo7824918357723582233_nat_o )
       => ( ! [X3: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ F5 )
             => ( ~ ( member6576561426505652726_nat_o @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F5 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_6019_finite__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X3: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F5 )
             => ( ~ ( member8440522571783428010at_nat @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert8211810215607154385at_nat @ X3 @ F5 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_6020_finite__induct,axiom,
    ! [F4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F4 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X3: $o,F5: set_o] :
              ( ( finite_finite_o @ F5 )
             => ( ~ ( member_o @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_o @ X3 @ F5 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_6021_finite__induct,axiom,
    ! [F4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,F5: set_nat] :
              ( ( finite_finite_nat @ F5 )
             => ( ~ ( member_nat @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_nat @ X3 @ F5 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_6022_finite__induct,axiom,
    ! [F4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,F5: set_int] :
              ( ( finite_finite_int @ F5 )
             => ( ~ ( member_int @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_int @ X3 @ F5 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_6023_finite__ne__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( F4 != bot_bo228742789529271731at_nat )
       => ( ! [X3: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
         => ( ! [X3: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F5 )
               => ( ( F5 != bot_bo228742789529271731at_nat )
                 => ( ~ ( member8757157785044589968at_nat @ X3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert9069300056098147895at_nat @ X3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_6024_finite__ne__induct,axiom,
    ! [F4: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ F4 )
     => ( ( F4 != bot_bo7824918357723582233_nat_o )
       => ( ! [X3: produc3658429121746597890et_nat > $o] : ( P @ ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) )
         => ( ! [X3: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
                ( ( finite3252695134891459830_nat_o @ F5 )
               => ( ( F5 != bot_bo7824918357723582233_nat_o )
                 => ( ~ ( member6576561426505652726_nat_o @ X3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_6025_finite__ne__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( F4 != bot_bo2099793752762293965at_nat )
       => ( ! [X3: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
         => ( ! [X3: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F5 )
               => ( ( F5 != bot_bo2099793752762293965at_nat )
                 => ( ~ ( member8440522571783428010at_nat @ X3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert8211810215607154385at_nat @ X3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_6026_finite__ne__induct,axiom,
    ! [F4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F4 )
     => ( ( F4 != bot_bot_set_o )
       => ( ! [X3: $o] : ( P @ ( insert_o @ X3 @ bot_bot_set_o ) )
         => ( ! [X3: $o,F5: set_o] :
                ( ( finite_finite_o @ F5 )
               => ( ( F5 != bot_bot_set_o )
                 => ( ~ ( member_o @ X3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_o @ X3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_6027_finite__ne__induct,axiom,
    ! [F4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( F4 != bot_bot_set_nat )
       => ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
         => ( ! [X3: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( F5 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_nat @ X3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_6028_finite__ne__induct,axiom,
    ! [F4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( F4 != bot_bot_set_int )
       => ( ! [X3: int] : ( P @ ( insert_int @ X3 @ bot_bot_set_int ) )
         => ( ! [X3: int,F5: set_int] :
                ( ( finite_finite_int @ F5 )
               => ( ( F5 != bot_bot_set_int )
                 => ( ~ ( member_int @ X3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_int @ X3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_6029_infinite__finite__induct,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat] :
      ( ! [A7: set_Pr4329608150637261639at_nat] :
          ( ~ ( finite4343798906461161616at_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X3: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ F5 )
             => ( ~ ( member8757157785044589968at_nat @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert9069300056098147895at_nat @ X3 @ F5 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_6030_infinite__finite__induct,axiom,
    ! [P: set_Pr4532377907799695533_nat_o > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ! [A7: set_Pr4532377907799695533_nat_o] :
          ( ~ ( finite3252695134891459830_nat_o @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo7824918357723582233_nat_o )
       => ( ! [X3: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ F5 )
             => ( ~ ( member6576561426505652726_nat_o @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F5 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_6031_infinite__finite__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ! [A7: set_Pr1261947904930325089at_nat] :
          ( ~ ( finite6177210948735845034at_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X3: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F5 )
             => ( ~ ( member8440522571783428010at_nat @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert8211810215607154385at_nat @ X3 @ F5 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_6032_infinite__finite__induct,axiom,
    ! [P: set_o > $o,A: set_o] :
      ( ! [A7: set_o] :
          ( ~ ( finite_finite_o @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X3: $o,F5: set_o] :
              ( ( finite_finite_o @ F5 )
             => ( ~ ( member_o @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_o @ X3 @ F5 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_6033_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,F5: set_nat] :
              ( ( finite_finite_nat @ F5 )
             => ( ~ ( member_nat @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_nat @ X3 @ F5 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_6034_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A: set_int] :
      ( ! [A7: set_int] :
          ( ~ ( finite_finite_int @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,F5: set_int] :
              ( ( finite_finite_int @ F5 )
             => ( ~ ( member_int @ X3 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_int @ X3 @ F5 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_6035_Gcd__remove0__nat,axiom,
    ! [M4: set_nat] :
      ( ( finite_finite_nat @ M4 )
     => ( ( gcd_Gcd_nat @ M4 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M4 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_6036_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( set_or2784603332384507286at_nat @ X @ Y )
        = top_to7629004291339433233at_nat )
      = ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_6037_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( set_or370866239135849197et_int @ X @ Y )
        = top_top_set_set_int )
      = ( ( X = bot_bot_set_int )
        & ( Y = top_top_set_int ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_6038_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( set_or7959216805967363635t_assn @ X @ Y )
        = top_top_set_assn )
      = ( ( X = bot_bot_assn )
        & ( Y = top_top_assn ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_6039_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( set_or266616469461849773_set_o @ X @ Y )
        = top_top_set_set_o )
      = ( ( X = bot_bot_set_o )
        & ( Y = top_top_set_o ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_6040_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ X @ Y )
        = top_top_set_set_nat )
      = ( ( X = bot_bot_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_6041_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( set_or269330187037242274t_unit @ X @ Y )
        = top_to1767297665138865437t_unit )
      = ( ( X = bot_bo3957492148770167129t_unit )
        & ( Y = top_to1996260823553986621t_unit ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_6042_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: $o,Y: $o] :
      ( ( ( set_or8904488021354931149Most_o @ X @ Y )
        = top_top_set_o )
      = ( ( X = bot_bot_o )
        & ( Y = top_top_o ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_6043_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ( set_or5692540242118139586t_unit @ X @ Y )
        = top_to1996260823553986621t_unit )
      = ( ( X = bot_bot_Product_unit )
        & ( Y = top_top_Product_unit ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_6044_finite__subset__induct_H,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( ord_le1268244103169919719at_nat @ F4 @ A )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A5: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F5 )
               => ( ( member8757157785044589968at_nat @ A5 @ A )
                 => ( ( ord_le1268244103169919719at_nat @ F5 @ A )
                   => ( ~ ( member8757157785044589968at_nat @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert9069300056098147895at_nat @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_6045_finite__subset__induct_H,axiom,
    ! [F4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ F4 )
     => ( ( ord_le2965882846123202637_nat_o @ F4 @ A )
       => ( ( P @ bot_bo7824918357723582233_nat_o )
         => ( ! [A5: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
                ( ( finite3252695134891459830_nat_o @ F5 )
               => ( ( member6576561426505652726_nat_o @ A5 @ A )
                 => ( ( ord_le2965882846123202637_nat_o @ F5 @ A )
                   => ( ~ ( member6576561426505652726_nat_o @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert5175938949040314269_nat_o @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_6046_finite__subset__induct_H,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( ord_le3146513528884898305at_nat @ F4 @ A )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A5: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F5 )
               => ( ( member8440522571783428010at_nat @ A5 @ A )
                 => ( ( ord_le3146513528884898305at_nat @ F5 @ A )
                   => ( ~ ( member8440522571783428010at_nat @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert8211810215607154385at_nat @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_6047_finite__subset__induct_H,axiom,
    ! [F4: set_o,A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F4 )
     => ( ( ord_less_eq_set_o @ F4 @ A )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A5: $o,F5: set_o] :
                ( ( finite_finite_o @ F5 )
               => ( ( member_o @ A5 @ A )
                 => ( ( ord_less_eq_set_o @ F5 @ A )
                   => ( ~ ( member_o @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_o @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_6048_finite__subset__induct_H,axiom,
    ! [F4: set_int,A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( ord_less_eq_set_int @ F4 @ A )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A5: int,F5: set_int] :
                ( ( finite_finite_int @ F5 )
               => ( ( member_int @ A5 @ A )
                 => ( ( ord_less_eq_set_int @ F5 @ A )
                   => ( ~ ( member_int @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_int @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_6049_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A5 @ A )
                 => ( ( ord_less_eq_set_nat @ F5 @ A )
                   => ( ~ ( member_nat @ A5 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_nat @ A5 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_6050_finite__subset__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( ord_le1268244103169919719at_nat @ F4 @ A )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A5: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F5 )
               => ( ( member8757157785044589968at_nat @ A5 @ A )
                 => ( ~ ( member8757157785044589968at_nat @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert9069300056098147895at_nat @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_6051_finite__subset__induct,axiom,
    ! [F4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ F4 )
     => ( ( ord_le2965882846123202637_nat_o @ F4 @ A )
       => ( ( P @ bot_bo7824918357723582233_nat_o )
         => ( ! [A5: produc3658429121746597890et_nat > $o,F5: set_Pr4532377907799695533_nat_o] :
                ( ( finite3252695134891459830_nat_o @ F5 )
               => ( ( member6576561426505652726_nat_o @ A5 @ A )
                 => ( ~ ( member6576561426505652726_nat_o @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert5175938949040314269_nat_o @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_6052_finite__subset__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( ord_le3146513528884898305at_nat @ F4 @ A )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A5: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F5 )
               => ( ( member8440522571783428010at_nat @ A5 @ A )
                 => ( ~ ( member8440522571783428010at_nat @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert8211810215607154385at_nat @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_6053_finite__subset__induct,axiom,
    ! [F4: set_o,A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F4 )
     => ( ( ord_less_eq_set_o @ F4 @ A )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A5: $o,F5: set_o] :
                ( ( finite_finite_o @ F5 )
               => ( ( member_o @ A5 @ A )
                 => ( ~ ( member_o @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_o @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_6054_finite__subset__induct,axiom,
    ! [F4: set_int,A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( ord_less_eq_set_int @ F4 @ A )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A5: int,F5: set_int] :
                ( ( finite_finite_int @ F5 )
               => ( ( member_int @ A5 @ A )
                 => ( ~ ( member_int @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_int @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_6055_finite__subset__induct,axiom,
    ! [F4: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A5 @ A )
                 => ( ~ ( member_nat @ A5 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_nat @ A5 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_6056_finite__empty__induct,axiom,
    ! [A: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( P @ A )
       => ( ! [A5: produc3843707927480180839at_nat,A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( member8757157785044589968at_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ A5 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
         => ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_6057_finite__empty__induct,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( P @ A )
       => ( ! [A5: produc3658429121746597890et_nat > $o,A7: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ A7 )
             => ( ( member6576561426505652726_nat_o @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_1801376950450012436_nat_o @ A7 @ ( insert5175938949040314269_nat_o @ A5 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
         => ( P @ bot_bo7824918357723582233_nat_o ) ) ) ) ).

% finite_empty_induct
thf(fact_6058_finite__empty__induct,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( P @ A )
       => ( ! [A5: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( member8440522571783428010at_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ A5 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
         => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_6059_finite__empty__induct,axiom,
    ! [A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A )
     => ( ( P @ A )
       => ( ! [A5: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( member_o @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ) )
         => ( P @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_6060_finite__empty__induct,axiom,
    ! [A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A )
     => ( ( P @ A )
       => ( ! [A5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( member_int @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ A5 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_6061_finite__empty__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ A )
       => ( ! [A5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_6062_infinite__coinduct,axiom,
    ! [X6: set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat] :
      ( ( X6 @ A )
     => ( ! [A7: set_Pr4329608150637261639at_nat] :
            ( ( X6 @ A7 )
           => ? [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ A7 )
                & ( ( X6 @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) )
                  | ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
       => ~ ( finite4343798906461161616at_nat @ A ) ) ) ).

% infinite_coinduct
thf(fact_6063_infinite__coinduct,axiom,
    ! [X6: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( X6 @ A )
     => ( ! [A7: set_Pr1261947904930325089at_nat] :
            ( ( X6 @ A7 )
           => ? [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A7 )
                & ( ( X6 @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) )
                  | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
       => ~ ( finite6177210948735845034at_nat @ A ) ) ) ).

% infinite_coinduct
thf(fact_6064_infinite__coinduct,axiom,
    ! [X6: set_o > $o,A: set_o] :
      ( ( X6 @ A )
     => ( ! [A7: set_o] :
            ( ( X6 @ A7 )
           => ? [X4: $o] :
                ( ( member_o @ X4 @ A7 )
                & ( ( X6 @ ( minus_minus_set_o @ A7 @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A7 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A ) ) ) ).

% infinite_coinduct
thf(fact_6065_infinite__coinduct,axiom,
    ! [X6: set_int > $o,A: set_int] :
      ( ( X6 @ A )
     => ( ! [A7: set_int] :
            ( ( X6 @ A7 )
           => ? [X4: int] :
                ( ( member_int @ X4 @ A7 )
                & ( ( X6 @ ( minus_minus_set_int @ A7 @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A7 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A ) ) ) ).

% infinite_coinduct
thf(fact_6066_infinite__coinduct,axiom,
    ! [X6: set_nat > $o,A: set_nat] :
      ( ( X6 @ A )
     => ( ! [A7: set_nat] :
            ( ( X6 @ A7 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ A7 )
                & ( ( X6 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A ) ) ) ).

% infinite_coinduct
thf(fact_6067_infinite__remove,axiom,
    ! [S: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S )
     => ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S @ ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% infinite_remove
thf(fact_6068_infinite__remove,axiom,
    ! [S: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% infinite_remove
thf(fact_6069_infinite__remove,axiom,
    ! [S: set_o,A2: $o] :
      ( ~ ( finite_finite_o @ S )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_6070_infinite__remove,axiom,
    ! [S: set_int,A2: int] :
      ( ~ ( finite_finite_int @ S )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_6071_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_6072_card__eq__0__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A )
        = zero_zero_nat )
      = ( ( A = bot_bo2099793752762293965at_nat )
        | ~ ( finite6177210948735845034at_nat @ A ) ) ) ).

% card_eq_0_iff
thf(fact_6073_card__eq__0__iff,axiom,
    ! [A: set_o] :
      ( ( ( finite_card_o @ A )
        = zero_zero_nat )
      = ( ( A = bot_bot_set_o )
        | ~ ( finite_finite_o @ A ) ) ) ).

% card_eq_0_iff
thf(fact_6074_card__eq__0__iff,axiom,
    ! [A: set_nat] :
      ( ( ( finite_card_nat @ A )
        = zero_zero_nat )
      = ( ( A = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A ) ) ) ).

% card_eq_0_iff
thf(fact_6075_card__eq__0__iff,axiom,
    ! [A: set_int] :
      ( ( ( finite_card_int @ A )
        = zero_zero_nat )
      = ( ( A = bot_bot_set_int )
        | ~ ( finite_finite_int @ A ) ) ) ).

% card_eq_0_iff
thf(fact_6076_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_6077_remove__induct,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,B: set_Pr4329608150637261639at_nat] :
      ( ( P @ bot_bo228742789529271731at_nat )
     => ( ( ~ ( finite4343798906461161616at_nat @ B )
         => ( P @ B ) )
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A7 @ B )
                 => ( ! [X4: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X4 @ A7 )
                       => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_6078_remove__induct,axiom,
    ! [P: set_Pr4532377907799695533_nat_o > $o,B: set_Pr4532377907799695533_nat_o] :
      ( ( P @ bot_bo7824918357723582233_nat_o )
     => ( ( ~ ( finite3252695134891459830_nat_o @ B )
         => ( P @ B ) )
       => ( ! [A7: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ A7 )
             => ( ( A7 != bot_bo7824918357723582233_nat_o )
               => ( ( ord_le2965882846123202637_nat_o @ A7 @ B )
                 => ( ! [X4: produc3658429121746597890et_nat > $o] :
                        ( ( member6576561426505652726_nat_o @ X4 @ A7 )
                       => ( P @ ( minus_1801376950450012436_nat_o @ A7 @ ( insert5175938949040314269_nat_o @ X4 @ bot_bo7824918357723582233_nat_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_6079_remove__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,B: set_Pr1261947904930325089at_nat] :
      ( ( P @ bot_bo2099793752762293965at_nat )
     => ( ( ~ ( finite6177210948735845034at_nat @ B )
         => ( P @ B ) )
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A7 @ B )
                 => ( ! [X4: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X4 @ A7 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_6080_remove__induct,axiom,
    ! [P: set_o > $o,B: set_o] :
      ( ( P @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B )
         => ( P @ B ) )
       => ( ! [A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( A7 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A7 @ B )
                 => ( ! [X4: $o] :
                        ( ( member_o @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_6081_remove__induct,axiom,
    ! [P: set_int > $o,B: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B )
         => ( P @ B ) )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B )
                 => ( ! [X4: int] :
                        ( ( member_int @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_6082_remove__induct,axiom,
    ! [P: set_nat > $o,B: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B )
         => ( P @ B ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B )
                 => ( ! [X4: nat] :
                        ( ( member_nat @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_6083_finite__remove__induct,axiom,
    ! [B: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ B )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A7 @ B )
                 => ( ! [X4: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X4 @ A7 )
                       => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_6084_finite__remove__induct,axiom,
    ! [B: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ B )
     => ( ( P @ bot_bo7824918357723582233_nat_o )
       => ( ! [A7: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ A7 )
             => ( ( A7 != bot_bo7824918357723582233_nat_o )
               => ( ( ord_le2965882846123202637_nat_o @ A7 @ B )
                 => ( ! [X4: produc3658429121746597890et_nat > $o] :
                        ( ( member6576561426505652726_nat_o @ X4 @ A7 )
                       => ( P @ ( minus_1801376950450012436_nat_o @ A7 @ ( insert5175938949040314269_nat_o @ X4 @ bot_bo7824918357723582233_nat_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_6085_finite__remove__induct,axiom,
    ! [B: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A7 @ B )
                 => ( ! [X4: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X4 @ A7 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_6086_finite__remove__induct,axiom,
    ! [B: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ B )
     => ( ( P @ bot_bot_set_o )
       => ( ! [A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( A7 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A7 @ B )
                 => ( ! [X4: $o] :
                        ( ( member_o @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_6087_finite__remove__induct,axiom,
    ! [B: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B )
                 => ( ! [X4: int] :
                        ( ( member_int @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_6088_finite__remove__induct,axiom,
    ! [B: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B )
                 => ( ! [X4: nat] :
                        ( ( member_nat @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_6089_card__gt__0__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A ) )
      = ( ( A != bot_bo2099793752762293965at_nat )
        & ( finite6177210948735845034at_nat @ A ) ) ) ).

% card_gt_0_iff
thf(fact_6090_card__gt__0__iff,axiom,
    ! [A: set_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
      = ( ( A != bot_bot_set_o )
        & ( finite_finite_o @ A ) ) ) ).

% card_gt_0_iff
thf(fact_6091_card__gt__0__iff,axiom,
    ! [A: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
      = ( ( A != bot_bot_set_nat )
        & ( finite_finite_nat @ A ) ) ) ).

% card_gt_0_iff
thf(fact_6092_card__gt__0__iff,axiom,
    ! [A: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
      = ( ( A != bot_bot_set_int )
        & ( finite_finite_int @ A ) ) ) ).

% card_gt_0_iff
thf(fact_6093_card__1__singletonI,axiom,
    ! [S: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( ( finite3771342082235030671at_nat @ S )
          = one_one_nat )
       => ( ( member8757157785044589968at_nat @ X @ S )
         => ( S
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_6094_card__1__singletonI,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ S )
     => ( ( ( finite1363419556375932405_nat_o @ S )
          = one_one_nat )
       => ( ( member6576561426505652726_nat_o @ X @ S )
         => ( S
            = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).

% card_1_singletonI
thf(fact_6095_card__1__singletonI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( ( finite711546835091564841at_nat @ S )
          = one_one_nat )
       => ( ( member8440522571783428010at_nat @ X @ S )
         => ( S
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_6096_card__1__singletonI,axiom,
    ! [S: set_o,X: $o] :
      ( ( finite_finite_o @ S )
     => ( ( ( finite_card_o @ S )
          = one_one_nat )
       => ( ( member_o @ X @ S )
         => ( S
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% card_1_singletonI
thf(fact_6097_card__1__singletonI,axiom,
    ! [S: set_nat,X: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( finite_card_nat @ S )
          = one_one_nat )
       => ( ( member_nat @ X @ S )
         => ( S
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_6098_card__1__singletonI,axiom,
    ! [S: set_int,X: int] :
      ( ( finite_finite_int @ S )
     => ( ( ( finite_card_int @ S )
          = one_one_nat )
       => ( ( member_int @ X @ S )
         => ( S
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% card_1_singletonI
thf(fact_6099_finite__induct__select,axiom,
    ! [S: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [T2: set_Pr4329608150637261639at_nat] :
              ( ( ord_le2604355607129572851at_nat @ T2 @ S )
             => ( ( P @ T2 )
               => ? [X4: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X4 @ ( minus_3314409938677909166at_nat @ S @ T2 ) )
                    & ( P @ ( insert9069300056098147895at_nat @ X4 @ T2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_6100_finite__induct__select,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [T2: set_Pr1261947904930325089at_nat] :
              ( ( ord_le7866589430770878221at_nat @ T2 @ S )
             => ( ( P @ T2 )
               => ? [X4: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X4 @ ( minus_1356011639430497352at_nat @ S @ T2 ) )
                    & ( P @ ( insert8211810215607154385at_nat @ X4 @ T2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_6101_finite__induct__select,axiom,
    ! [S: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [T2: set_o] :
              ( ( ord_less_set_o @ T2 @ S )
             => ( ( P @ T2 )
               => ? [X4: $o] :
                    ( ( member_o @ X4 @ ( minus_minus_set_o @ S @ T2 ) )
                    & ( P @ ( insert_o @ X4 @ T2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_6102_finite__induct__select,axiom,
    ! [S: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T2: set_int] :
              ( ( ord_less_set_int @ T2 @ S )
             => ( ( P @ T2 )
               => ? [X4: int] :
                    ( ( member_int @ X4 @ ( minus_minus_set_int @ S @ T2 ) )
                    & ( P @ ( insert_int @ X4 @ T2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_6103_finite__induct__select,axiom,
    ! [S: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T2: set_nat] :
              ( ( ord_less_set_nat @ T2 @ S )
             => ( ( P @ T2 )
               => ? [X4: nat] :
                    ( ( member_nat @ X4 @ ( minus_minus_set_nat @ S @ T2 ) )
                    & ( P @ ( insert_nat @ X4 @ T2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_6104_one__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).

% one_rat.transfer
thf(fact_6105_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_6106_card__Suc__Diff1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( member8757157785044589968at_nat @ X @ A )
       => ( ( suc @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) )
          = ( finite3771342082235030671at_nat @ A ) ) ) ) ).

% card_Suc_Diff1
thf(fact_6107_card__Suc__Diff1,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) )
          = ( finite1363419556375932405_nat_o @ A ) ) ) ) ).

% card_Suc_Diff1
thf(fact_6108_card__Suc__Diff1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ X @ A )
       => ( ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) )
          = ( finite711546835091564841at_nat @ A ) ) ) ) ).

% card_Suc_Diff1
thf(fact_6109_card__Suc__Diff1,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ X @ A )
       => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
          = ( finite_card_o @ A ) ) ) ) ).

% card_Suc_Diff1
thf(fact_6110_card__Suc__Diff1,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ X @ A )
       => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
          = ( finite_card_int @ A ) ) ) ) ).

% card_Suc_Diff1
thf(fact_6111_card__Suc__Diff1,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
          = ( finite_card_nat @ A ) ) ) ) ).

% card_Suc_Diff1
thf(fact_6112_card_Oinsert__remove,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ X @ A ) )
        = ( suc @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_6113_card_Oinsert__remove,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A ) )
        = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_6114_card_Oinsert__remove,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( finite_card_o @ ( insert_o @ X @ A ) )
        = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_6115_card_Oinsert__remove,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( finite_card_int @ ( insert_int @ X @ A ) )
        = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_6116_card_Oinsert__remove,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
        = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_6117_card_Oremove,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( member8757157785044589968at_nat @ X @ A )
       => ( ( finite3771342082235030671at_nat @ A )
          = ( suc @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_6118_card_Oremove,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ( finite1363419556375932405_nat_o @ A )
          = ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ).

% card.remove
thf(fact_6119_card_Oremove,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ X @ A )
       => ( ( finite711546835091564841at_nat @ A )
          = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_6120_card_Oremove,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ X @ A )
       => ( ( finite_card_o @ A )
          = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% card.remove
thf(fact_6121_card_Oremove,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ X @ A )
       => ( ( finite_card_int @ A )
          = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% card.remove
thf(fact_6122_card_Oremove,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( finite_card_nat @ A )
          = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_6123_card__Diff1__less__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_less_nat @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) @ ( finite3771342082235030671at_nat @ A ) )
      = ( ( finite4343798906461161616at_nat @ A )
        & ( member8757157785044589968at_nat @ X @ A ) ) ) ).

% card_Diff1_less_iff
thf(fact_6124_card__Diff1__less__iff,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) )
      = ( ( finite3252695134891459830_nat_o @ A )
        & ( member6576561426505652726_nat_o @ X @ A ) ) ) ).

% card_Diff1_less_iff
thf(fact_6125_card__Diff1__less__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A ) )
      = ( ( finite6177210948735845034at_nat @ A )
        & ( member8440522571783428010at_nat @ X @ A ) ) ) ).

% card_Diff1_less_iff
thf(fact_6126_card__Diff1__less__iff,axiom,
    ! [A: set_o,X: $o] :
      ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) )
      = ( ( finite_finite_o @ A )
        & ( member_o @ X @ A ) ) ) ).

% card_Diff1_less_iff
thf(fact_6127_card__Diff1__less__iff,axiom,
    ! [A: set_int,X: int] :
      ( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) )
      = ( ( finite_finite_int @ A )
        & ( member_int @ X @ A ) ) ) ).

% card_Diff1_less_iff
thf(fact_6128_card__Diff1__less__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
      = ( ( finite_finite_nat @ A )
        & ( member_nat @ X @ A ) ) ) ).

% card_Diff1_less_iff
thf(fact_6129_card__Diff2__less,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( member8757157785044589968at_nat @ X @ A )
       => ( ( member8757157785044589968at_nat @ Y @ A )
         => ( ord_less_nat @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) ) @ ( finite3771342082235030671at_nat @ A ) ) ) ) ) ).

% card_Diff2_less
thf(fact_6130_card__Diff2__less,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ( member6576561426505652726_nat_o @ Y @ A )
         => ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ).

% card_Diff2_less
thf(fact_6131_card__Diff2__less,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ X @ A )
       => ( ( member8440522571783428010at_nat @ Y @ A )
         => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A ) ) ) ) ) ).

% card_Diff2_less
thf(fact_6132_card__Diff2__less,axiom,
    ! [A: set_o,X: $o,Y: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ X @ A )
       => ( ( member_o @ Y @ A )
         => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( insert_o @ Y @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ) ) ) ).

% card_Diff2_less
thf(fact_6133_card__Diff2__less,axiom,
    ! [A: set_int,X: int,Y: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ X @ A )
       => ( ( member_int @ Y @ A )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( insert_int @ Y @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ) ).

% card_Diff2_less
thf(fact_6134_card__Diff2__less,axiom,
    ! [A: set_nat,X: nat,Y: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( member_nat @ Y @ A )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).

% card_Diff2_less
thf(fact_6135_finite__linorder__max__induct,axiom,
    ! [A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B5: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ! [X4: $o] :
                    ( ( member_o @ X4 @ A7 )
                   => ( ord_less_o @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_o @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_6136_finite__linorder__max__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A7 )
                   => ( ord_less_nat @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_6137_finite__linorder__max__induct,axiom,
    ! [A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A7 )
                   => ( ord_less_int @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_6138_finite__linorder__max__induct,axiom,
    ! [A: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B5: code_integer,A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ! [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ A7 )
                   => ( ord_le6747313008572928689nteger @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_Code_integer @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_6139_finite__linorder__max__induct,axiom,
    ! [A: set_Code_natural,P: set_Code_natural > $o] :
      ( ( finite4840673202314680255atural @ A )
     => ( ( P @ bot_bo8448779972843232370atural )
       => ( ! [B5: code_natural,A7: set_Code_natural] :
              ( ( finite4840673202314680255atural @ A7 )
             => ( ! [X4: code_natural] :
                    ( ( member_Code_natural @ X4 @ A7 )
                   => ( ord_le5570908160329646204atural @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_Code_natural @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_6140_finite__linorder__min__induct,axiom,
    ! [A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B5: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ! [X4: $o] :
                    ( ( member_o @ X4 @ A7 )
                   => ( ord_less_o @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_o @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_6141_finite__linorder__min__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A7 )
                   => ( ord_less_nat @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_6142_finite__linorder__min__induct,axiom,
    ! [A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A7 )
                   => ( ord_less_int @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_6143_finite__linorder__min__induct,axiom,
    ! [A: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B5: code_integer,A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ! [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ A7 )
                   => ( ord_le6747313008572928689nteger @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_Code_integer @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_6144_finite__linorder__min__induct,axiom,
    ! [A: set_Code_natural,P: set_Code_natural > $o] :
      ( ( finite4840673202314680255atural @ A )
     => ( ( P @ bot_bo8448779972843232370atural )
       => ( ! [B5: code_natural,A7: set_Code_natural] :
              ( ( finite4840673202314680255atural @ A7 )
             => ( ! [X4: code_natural] :
                    ( ( member_Code_natural @ X4 @ A7 )
                   => ( ord_le5570908160329646204atural @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_Code_natural @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_6145_finite__ranking__induct,axiom,
    ! [S: set_o,P: set_o > $o,F: $o > code_integer] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X3: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S4 )
                   => ( ord_le3102999989581377725nteger @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6146_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > code_integer] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S4 )
                   => ( ord_le3102999989581377725nteger @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6147_finite__ranking__induct,axiom,
    ! [S: set_int,P: set_int > $o,F: int > code_integer] :
      ( ( finite_finite_int @ S )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S4 )
                   => ( ord_le3102999989581377725nteger @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6148_finite__ranking__induct,axiom,
    ! [S: set_o,P: set_o > $o,F: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X3: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6149_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6150_finite__ranking__induct,axiom,
    ! [S: set_int,P: set_int > $o,F: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S4 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6151_finite__ranking__induct,axiom,
    ! [S: set_o,P: set_o > $o,F: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X3: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S4 )
                   => ( ord_less_eq_int @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6152_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S4 )
                   => ( ord_less_eq_int @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6153_finite__ranking__induct,axiom,
    ! [S: set_int,P: set_int > $o,F: int > int] :
      ( ( finite_finite_int @ S )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S4 )
                   => ( ord_less_eq_int @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6154_finite__ranking__induct,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o,F: product_prod_nat_nat > code_integer] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X3: product_prod_nat_nat,S4: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ S4 )
             => ( ! [Y5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y5 @ S4 )
                   => ( ord_le3102999989581377725nteger @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert8211810215607154385at_nat @ X3 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_6155_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ S )
            & ~ ? [Xa2: $o] :
                  ( ( member_o @ Xa2 @ S )
                  & ( ord_less_o @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6156_ex__min__if__finite,axiom,
    ! [S: set_assn] :
      ( ( finite_finite_assn @ S )
     => ( ( S != bot_bot_set_assn )
       => ? [X3: assn] :
            ( ( member_assn @ X3 @ S )
            & ~ ? [Xa2: assn] :
                  ( ( member_assn @ Xa2 @ S )
                  & ( ord_less_assn @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6157_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S )
                  & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6158_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S )
            & ~ ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ S )
                  & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6159_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S )
            & ~ ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ S )
                  & ( ord_le6747313008572928689nteger @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6160_ex__min__if__finite,axiom,
    ! [S: set_Code_natural] :
      ( ( finite4840673202314680255atural @ S )
     => ( ( S != bot_bo8448779972843232370atural )
       => ? [X3: code_natural] :
            ( ( member_Code_natural @ X3 @ S )
            & ~ ? [Xa2: code_natural] :
                  ( ( member_Code_natural @ Xa2 @ S )
                  & ( ord_le5570908160329646204atural @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_6161_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ X6 )
           => ? [Xa2: $o] :
                ( ( member_o @ Xa2 @ X6 )
                & ( ord_less_o @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_o @ X6 ) ) ) ).

% infinite_growing
thf(fact_6162_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X6 )
                & ( ord_less_nat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X6 ) ) ) ).

% infinite_growing
thf(fact_6163_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X6 )
           => ? [Xa2: int] :
                ( ( member_int @ Xa2 @ X6 )
                & ( ord_less_int @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_int @ X6 ) ) ) ).

% infinite_growing
thf(fact_6164_infinite__growing,axiom,
    ! [X6: set_Code_integer] :
      ( ( X6 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X6 )
           => ? [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ X6 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa2 ) ) )
       => ~ ( finite6017078050557962740nteger @ X6 ) ) ) ).

% infinite_growing
thf(fact_6165_infinite__growing,axiom,
    ! [X6: set_Code_natural] :
      ( ( X6 != bot_bo8448779972843232370atural )
     => ( ! [X3: code_natural] :
            ( ( member_Code_natural @ X3 @ X6 )
           => ? [Xa2: code_natural] :
                ( ( member_Code_natural @ Xa2 @ X6 )
                & ( ord_le5570908160329646204atural @ X3 @ Xa2 ) ) )
       => ~ ( finite4840673202314680255atural @ X6 ) ) ) ).

% infinite_growing
thf(fact_6166_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% uminus_rat.abs_eq
thf(fact_6167_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_6168_one__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).

% one_rat.rsp
thf(fact_6169_zero__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).

% zero_rat.rsp
thf(fact_6170_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa @ Xa )
     => ( ( ratrel @ X @ X )
       => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% times_rat.abs_eq
thf(fact_6171_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_fst_int_int @ X )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).

% inverse_rat.abs_eq
thf(fact_6172_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > assn] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( ( member_o @ X4 @ S )
              & ( ord_less_assn @ ( F @ X4 ) @ ( F @ ( lattic7708394068118247271o_assn @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6173_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > assn] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ S )
              & ( ord_less_assn @ ( F @ X4 ) @ ( F @ ( lattic2187264146484958483t_assn @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6174_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: int > assn] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( ( member_int @ X4 @ S )
              & ( ord_less_assn @ ( F @ X4 ) @ ( F @ ( lattic3162799611691135855t_assn @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6175_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( ( member_o @ X4 @ S )
              & ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6176_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ S )
              & ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6177_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( ( member_int @ X4 @ S )
              & ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6178_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( ( member_o @ X4 @ S )
              & ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic2773365557947402859_o_int @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6179_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ S )
              & ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic7444442490073309207at_int @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6180_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: int > int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( ( member_int @ X4 @ S )
              & ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic8443796201974363763nt_int @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6181_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( ( member_o @ X4 @ S )
              & ( ord_le6747313008572928689nteger @ ( F @ X4 ) @ ( F @ ( lattic8357703018590397596nteger @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_6182_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: $o > code_integer] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( member_o @ Y @ S )
         => ( ord_le3102999989581377725nteger @ ( F @ ( lattic8357703018590397596nteger @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6183_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S )
         => ( ord_le3102999989581377725nteger @ ( F @ ( lattic6339492004331817544nteger @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6184_arg__min__least,axiom,
    ! [S: set_int,Y: int,F: int > code_integer] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ( ( member_int @ Y @ S )
         => ( ord_le3102999989581377725nteger @ ( F @ ( lattic6711145564568662180nteger @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6185_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( member_o @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6186_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6187_arg__min__least,axiom,
    ! [S: set_int,Y: int,F: int > nat] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ( ( member_int @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6188_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( member_o @ Y @ S )
         => ( ord_less_eq_int @ ( F @ ( lattic2773365557947402859_o_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6189_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S )
         => ( ord_less_eq_int @ ( F @ ( lattic7444442490073309207at_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6190_arg__min__least,axiom,
    ! [S: set_int,Y: int,F: int > int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ( ( member_int @ Y @ S )
         => ( ord_less_eq_int @ ( F @ ( lattic8443796201974363763nt_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6191_arg__min__least,axiom,
    ! [S: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,F: product_prod_nat_nat > code_integer] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( S != bot_bo2099793752762293965at_nat )
       => ( ( member8440522571783428010at_nat @ Y @ S )
         => ( ord_le3102999989581377725nteger @ ( F @ ( lattic1349466628417373189nteger @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_6192_finite__transitivity__chain,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,R: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ! [X3: produc3658429121746597890et_nat > $o] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z2: produc3658429121746597890et_nat > $o] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: produc3658429121746597890et_nat > $o] :
                ( ( member6576561426505652726_nat_o @ X3 @ A )
               => ? [Y5: produc3658429121746597890et_nat > $o] :
                    ( ( member6576561426505652726_nat_o @ Y5 @ A )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_6193_finite__transitivity__chain,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ! [X3: product_prod_nat_nat] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ A )
               => ? [Y5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y5 @ A )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_6194_finite__transitivity__chain,axiom,
    ! [A: set_o,R: $o > $o > $o] :
      ( ( finite_finite_o @ A )
     => ( ! [X3: $o] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: $o,Y3: $o,Z2: $o] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ A )
               => ? [Y5: $o] :
                    ( ( member_o @ Y5 @ A )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_6195_finite__transitivity__chain,axiom,
    ! [A: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [X3: nat] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A )
               => ? [Y5: nat] :
                    ( ( member_nat @ Y5 @ A )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_6196_finite__transitivity__chain,axiom,
    ! [A: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A )
     => ( ! [X3: int] :
            ~ ( R @ X3 @ X3 )
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ A )
               => ? [Y5: int] :
                    ( ( member_int @ Y5 @ A )
                    & ( R @ X3 @ Y5 ) ) )
           => ( A = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_6197_Gcd__fin__0__iff,axiom,
    ! [A: set_int] :
      ( ( ( semiri4256215615220890538in_int @ A )
        = zero_zero_int )
      = ( ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
        & ( finite_finite_int @ A ) ) ) ).

% Gcd_fin_0_iff
thf(fact_6198_Gcd__fin__0__iff,axiom,
    ! [A: set_nat] :
      ( ( ( semiri4258706085729940814in_nat @ A )
        = zero_zero_nat )
      = ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
        & ( finite_finite_nat @ A ) ) ) ).

% Gcd_fin_0_iff
thf(fact_6199_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_6200_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
            = bot_bot_set_assn )
         => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
            = X ) )
        & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
           != bot_bot_set_assn )
         => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
            = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_6201_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
            = bot_bot_set_set_nat )
         => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
            = X ) )
        & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
           != bot_bot_set_set_nat )
         => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
            = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_6202_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ( ( ( minus_5090130588299000206at_nat @ A @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
            = bot_bo1534825418459847827at_nat )
         => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A ) )
            = X ) )
        & ( ( ( minus_5090130588299000206at_nat @ A @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
           != bot_bo1534825418459847827at_nat )
         => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A ) )
            = ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ ( minus_5090130588299000206at_nat @ A @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_6203_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
        = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X )
          & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_6204_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
            = bot_bot_set_int )
         => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
            = X ) )
        & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
           != bot_bot_set_int )
         => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
            = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_6205_Sup__fin_Oinsert__remove,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
            = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_6206_Sup__fin_Oremove,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
              = bot_bo7824918357723582233_nat_o )
           => ( ( lattic7320199455484906628_nat_o @ A )
              = X ) )
          & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
             != bot_bo7824918357723582233_nat_o )
           => ( ( lattic7320199455484906628_nat_o @ A )
              = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_6207_Sup__fin_Oremove,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( member_assn @ X @ A )
       => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
              = bot_bot_set_assn )
           => ( ( lattic2150320897289308081n_assn @ A )
              = X ) )
          & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
             != bot_bot_set_assn )
           => ( ( lattic2150320897289308081n_assn @ A )
              = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_6208_Sup__fin_Oremove,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ X @ A )
       => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
              = bot_bot_set_set_nat )
           => ( ( lattic3835124923745554447et_nat @ A )
              = X ) )
          & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
             != bot_bot_set_set_nat )
           => ( ( lattic3835124923745554447et_nat @ A )
              = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_6209_Sup__fin_Oremove,axiom,
    ! [A: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ( member6273874620674182256at_nat @ X @ A )
       => ( ( ( ( minus_5090130588299000206at_nat @ A @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
              = bot_bo1534825418459847827at_nat )
           => ( ( lattic8394993312974875902at_nat @ A )
              = X ) )
          & ( ( ( minus_5090130588299000206at_nat @ A @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
             != bot_bo1534825418459847827at_nat )
           => ( ( lattic8394993312974875902at_nat @ A )
              = ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ ( minus_5090130588299000206at_nat @ A @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_6210_Sup__fin_Oremove,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ X @ A )
       => ( ( lattic1508158080041050831_fin_o @ A )
          = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_6211_Sup__fin_Oremove,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ X @ A )
       => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
              = bot_bot_set_int )
           => ( ( lattic1091506334969745077in_int @ A )
              = X ) )
          & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
             != bot_bot_set_int )
           => ( ( lattic1091506334969745077in_int @ A )
              = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_6212_Sup__fin_Oremove,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ A )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ A )
              = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_6213_Gcd__fin_Oempty,axiom,
    ( ( semiri4258706085729940814in_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_fin.empty
thf(fact_6214_Gcd__fin_Oempty,axiom,
    ( ( semiri4256215615220890538in_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_fin.empty
thf(fact_6215_Sup__fin_Osingleton,axiom,
    ! [X: $o] :
      ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Sup_fin.singleton
thf(fact_6216_Sup__fin_Osingleton,axiom,
    ! [X: nat] :
      ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% Sup_fin.singleton
thf(fact_6217_Sup__fin_Osingleton,axiom,
    ! [X: int] :
      ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% Sup_fin.singleton
thf(fact_6218_Gcd__fin_Oinfinite,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( semiri4258706085729940814in_nat @ A )
        = one_one_nat ) ) ).

% Gcd_fin.infinite
thf(fact_6219_Gcd__fin_Oinfinite,axiom,
    ! [A: set_int] :
      ( ~ ( finite_finite_int @ A )
     => ( ( semiri4256215615220890538in_int @ A )
        = one_one_int ) ) ).

% Gcd_fin.infinite
thf(fact_6220_is__unit__Gcd__fin__iff,axiom,
    ! [A: set_nat] :
      ( ( dvd_dvd_nat @ ( semiri4258706085729940814in_nat @ A ) @ one_one_nat )
      = ( ( semiri4258706085729940814in_nat @ A )
        = one_one_nat ) ) ).

% is_unit_Gcd_fin_iff
thf(fact_6221_is__unit__Gcd__fin__iff,axiom,
    ! [A: set_int] :
      ( ( dvd_dvd_int @ ( semiri4256215615220890538in_int @ A ) @ one_one_int )
      = ( ( semiri4256215615220890538in_int @ A )
        = one_one_int ) ) ).

% is_unit_Gcd_fin_iff
thf(fact_6222_Sup__fin_Oinsert,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
          = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_6223_Sup__fin_Oinsert,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
          = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_6224_Sup__fin_Oinsert,axiom,
    ! [A: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ( A != bot_bo1534825418459847827at_nat )
       => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A ) )
          = ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_6225_Sup__fin_Oinsert,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
          = ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_6226_Sup__fin_Oinsert,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
          = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_6227_Sup__fin_Oinsert,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
          = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_6228_greaterThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_or1207661135979820486an_int @ X )
     != bot_bot_set_int ) ).

% greaterThan_non_empty
thf(fact_6229_greaterThan__non__empty,axiom,
    ! [X: nat] :
      ( ( set_or1210151606488870762an_nat @ X )
     != bot_bot_set_nat ) ).

% greaterThan_non_empty
thf(fact_6230_Sup__fin_Oin__idem,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ X @ A )
       => ( ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) )
          = ( lattic1508158080041050831_fin_o @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_6231_Sup__fin_Oin__idem,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ X @ A )
       => ( ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) )
          = ( lattic1091506334969745077in_int @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_6232_Sup__fin_Oin__idem,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A ) )
          = ( lattic7320199455484906628_nat_o @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_6233_Sup__fin_Oin__idem,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) )
          = ( lattic1093996805478795353in_nat @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_6234_Sup__fin_Oin__idem,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( member_assn @ X @ A )
       => ( ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) )
          = ( lattic2150320897289308081n_assn @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_6235_Sup__fin_Oin__idem,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ X @ A )
       => ( ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) )
          = ( lattic3835124923745554447et_nat @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_6236_Sup__fin_Oin__idem,axiom,
    ! [A: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ( member6273874620674182256at_nat @ X @ A )
       => ( ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ A ) )
          = ( lattic8394993312974875902at_nat @ A ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_6237_Sup__fin_OboundedE,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( A != bot_bo7824918357723582233_nat_o )
       => ( ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X )
         => ! [A8: produc3658429121746597890et_nat > $o] :
              ( ( member6576561426505652726_nat_o @ A8 @ A )
             => ( ord_le729326519192465773_nat_o @ A8 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_6238_Sup__fin_OboundedE,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
         => ! [A8: $o] :
              ( ( member_o @ A8 @ A )
             => ( ord_less_eq_o @ A8 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_6239_Sup__fin_OboundedE,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ( ord_less_eq_assn @ ( lattic2150320897289308081n_assn @ A ) @ X )
         => ! [A8: assn] :
              ( ( member_assn @ A8 @ A )
             => ( ord_less_eq_assn @ A8 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_6240_Sup__fin_OboundedE,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
         => ! [A8: set_nat] :
              ( ( member_set_nat @ A8 @ A )
             => ( ord_less_eq_set_nat @ A8 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_6241_Sup__fin_OboundedE,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
         => ! [A8: nat] :
              ( ( member_nat @ A8 @ A )
             => ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_6242_Sup__fin_OboundedE,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
         => ! [A8: int] :
              ( ( member_int @ A8 @ A )
             => ( ord_less_eq_int @ A8 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_6243_Sup__fin_OboundedI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( A != bot_bo7824918357723582233_nat_o )
       => ( ! [A5: produc3658429121746597890et_nat > $o] :
              ( ( member6576561426505652726_nat_o @ A5 @ A )
             => ( ord_le729326519192465773_nat_o @ A5 @ X ) )
         => ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_6244_Sup__fin_OboundedI,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ A )
             => ( ord_less_eq_o @ A5 @ X ) )
         => ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_6245_Sup__fin_OboundedI,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ! [A5: assn] :
              ( ( member_assn @ A5 @ A )
             => ( ord_less_eq_assn @ A5 @ X ) )
         => ( ord_less_eq_assn @ ( lattic2150320897289308081n_assn @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_6246_Sup__fin_OboundedI,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [A5: set_nat] :
              ( ( member_set_nat @ A5 @ A )
             => ( ord_less_eq_set_nat @ A5 @ X ) )
         => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_6247_Sup__fin_OboundedI,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A5: nat] :
              ( ( member_nat @ A5 @ A )
             => ( ord_less_eq_nat @ A5 @ X ) )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_6248_Sup__fin_OboundedI,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ A )
             => ( ord_less_eq_int @ A5 @ X ) )
         => ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_6249_Sup__fin_Obounded__iff,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
          = ( ! [X2: $o] :
                ( ( member_o @ X2 @ A )
               => ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_6250_Sup__fin_Obounded__iff,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ( ord_less_eq_assn @ ( lattic2150320897289308081n_assn @ A ) @ X )
          = ( ! [X2: assn] :
                ( ( member_assn @ X2 @ A )
               => ( ord_less_eq_assn @ X2 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_6251_Sup__fin_Obounded__iff,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
          = ( ! [X2: set_nat] :
                ( ( member_set_nat @ X2 @ A )
               => ( ord_less_eq_set_nat @ X2 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_6252_Sup__fin_Obounded__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A )
               => ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_6253_Sup__fin_Obounded__iff,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A )
               => ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_6254_ivl__disj__int__one_I7_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or8904488021354931149Most_o @ L @ U ) @ ( set_or6416164934427428222Than_o @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(7)
thf(fact_6255_ivl__disj__int__one_I7_J,axiom,
    ! [L: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or1266510415728281911st_int @ L @ U ) @ ( set_or1207661135979820486an_int @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_one(7)
thf(fact_6256_ivl__disj__int__one_I7_J,axiom,
    ! [L: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L @ U ) @ ( set_or1210151606488870762an_nat @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_one(7)
thf(fact_6257_Sup__fin_Osubset__imp,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ( A != bot_bot_set_o )
       => ( ( finite_finite_o @ B )
         => ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_6258_Sup__fin_Osubset__imp,axiom,
    ! [A: set_assn,B: set_assn] :
      ( ( ord_less_eq_set_assn @ A @ B )
     => ( ( A != bot_bot_set_assn )
       => ( ( finite_finite_assn @ B )
         => ( ord_less_eq_assn @ ( lattic2150320897289308081n_assn @ A ) @ ( lattic2150320897289308081n_assn @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_6259_Sup__fin_Osubset__imp,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B )
         => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_6260_Sup__fin_Osubset__imp,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( A != bot_bot_set_int )
       => ( ( finite_finite_int @ B )
         => ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ ( lattic1091506334969745077in_int @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_6261_Sup__fin_Osubset__imp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_6262_Sup__fin_Osubset,axiom,
    ! [A: set_assn,B: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( B != bot_bot_set_assn )
       => ( ( ord_less_eq_set_assn @ B @ A )
         => ( ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ B ) @ ( lattic2150320897289308081n_assn @ A ) )
            = ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_6263_Sup__fin_Osubset,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( B != bot_bot_set_set_nat )
       => ( ( ord_le6893508408891458716et_nat @ B @ A )
         => ( ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ B ) @ ( lattic3835124923745554447et_nat @ A ) )
            = ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_6264_Sup__fin_Osubset,axiom,
    ! [A: set_se5479201105039950375at_nat,B: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ( B != bot_bo1534825418459847827at_nat )
       => ( ( ord_le7966958417815640519at_nat @ B @ A )
         => ( ( sup_su5525570899277871387at_nat @ ( lattic8394993312974875902at_nat @ B ) @ ( lattic8394993312974875902at_nat @ A ) )
            = ( lattic8394993312974875902at_nat @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_6265_Sup__fin_Osubset,axiom,
    ! [A: set_o,B: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( B != bot_bot_set_o )
       => ( ( ord_less_eq_set_o @ B @ A )
         => ( ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ B ) @ ( lattic1508158080041050831_fin_o @ A ) )
            = ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_6266_Sup__fin_Osubset,axiom,
    ! [A: set_int,B: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( B != bot_bot_set_int )
       => ( ( ord_less_eq_set_int @ B @ A )
         => ( ( sup_sup_int @ ( lattic1091506334969745077in_int @ B ) @ ( lattic1091506334969745077in_int @ A ) )
            = ( lattic1091506334969745077in_int @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_6267_Sup__fin_Osubset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( B != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B @ A )
         => ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
            = ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_6268_Sup__fin_Oclosed,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( A != bot_bo7824918357723582233_nat_o )
       => ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( sup_su5453871518329203617_nat_o @ X3 @ Y3 ) @ ( insert5175938949040314269_nat_o @ X3 @ ( insert5175938949040314269_nat_o @ Y3 @ bot_bo7824918357723582233_nat_o ) ) )
         => ( member6576561426505652726_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ A ) ) ) ) ).

% Sup_fin.closed
thf(fact_6269_Sup__fin_Oclosed,axiom,
    ! [A: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ! [X3: assn,Y3: assn] : ( member_assn @ ( sup_sup_assn @ X3 @ Y3 ) @ ( insert_assn @ X3 @ ( insert_assn @ Y3 @ bot_bot_set_assn ) ) )
         => ( member_assn @ ( lattic2150320897289308081n_assn @ A ) @ A ) ) ) ) ).

% Sup_fin.closed
thf(fact_6270_Sup__fin_Oclosed,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [X3: set_nat,Y3: set_nat] : ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y3 @ bot_bot_set_set_nat ) ) )
         => ( member_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ A ) ) ) ) ).

% Sup_fin.closed
thf(fact_6271_Sup__fin_Oclosed,axiom,
    ! [A: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ( A != bot_bo1534825418459847827at_nat )
       => ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( member6273874620674182256at_nat @ ( sup_su5525570899277871387at_nat @ X3 @ Y3 ) @ ( insert2290795007606087959at_nat @ X3 @ ( insert2290795007606087959at_nat @ Y3 @ bot_bo1534825418459847827at_nat ) ) )
         => ( member6273874620674182256at_nat @ ( lattic8394993312974875902at_nat @ A ) @ A ) ) ) ) ).

% Sup_fin.closed
thf(fact_6272_Sup__fin_Oclosed,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o,Y3: $o] : ( member_o @ ( sup_sup_o @ X3 @ Y3 ) @ ( insert_o @ X3 @ ( insert_o @ Y3 @ bot_bot_set_o ) ) )
         => ( member_o @ ( lattic1508158080041050831_fin_o @ A ) @ A ) ) ) ) ).

% Sup_fin.closed
thf(fact_6273_Sup__fin_Oclosed,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [X3: nat,Y3: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
         => ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).

% Sup_fin.closed
thf(fact_6274_Sup__fin_Oclosed,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ! [X3: int,Y3: int] : ( member_int @ ( sup_sup_int @ X3 @ Y3 ) @ ( insert_int @ X3 @ ( insert_int @ Y3 @ bot_bot_set_int ) ) )
         => ( member_int @ ( lattic1091506334969745077in_int @ A ) @ A ) ) ) ) ).

% Sup_fin.closed
thf(fact_6275_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ~ ( member6576561426505652726_nat_o @ X @ A )
       => ( ( A != bot_bo7824918357723582233_nat_o )
         => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
            = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_6276_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ~ ( member_assn @ X @ A )
       => ( ( A != bot_bot_set_assn )
         => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
            = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_6277_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ~ ( member_set_nat @ X @ A )
       => ( ( A != bot_bot_set_set_nat )
         => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
            = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_6278_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ~ ( member6273874620674182256at_nat @ X @ A )
       => ( ( A != bot_bo1534825418459847827at_nat )
         => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A ) )
            = ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ A ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_6279_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ~ ( member_o @ X @ A )
       => ( ( A != bot_bot_set_o )
         => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
            = ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_6280_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ~ ( member_nat @ X @ A )
       => ( ( A != bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
            = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_6281_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ~ ( member_int @ X @ A )
       => ( ( A != bot_bot_set_int )
         => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
            = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_6282_Sup__fin_Ounion,axiom,
    ! [A: set_assn,B: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ( finite_finite_assn @ B )
         => ( ( B != bot_bot_set_assn )
           => ( ( lattic2150320897289308081n_assn @ ( sup_sup_set_assn @ A @ B ) )
              = ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ A ) @ ( lattic2150320897289308081n_assn @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_6283_Sup__fin_Ounion,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B )
         => ( ( B != bot_bot_set_set_nat )
           => ( ( lattic3835124923745554447et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
              = ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_6284_Sup__fin_Ounion,axiom,
    ! [A: set_se5479201105039950375at_nat,B: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ( A != bot_bo1534825418459847827at_nat )
       => ( ( finite4768004253697144688at_nat @ B )
         => ( ( B != bot_bo1534825418459847827at_nat )
           => ( ( lattic8394993312974875902at_nat @ ( sup_su474284620922681339at_nat @ A @ B ) )
              = ( sup_su5525570899277871387at_nat @ ( lattic8394993312974875902at_nat @ A ) @ ( lattic8394993312974875902at_nat @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_6285_Sup__fin_Ounion,axiom,
    ! [A: set_o,B: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( finite_finite_o @ B )
         => ( ( B != bot_bot_set_o )
           => ( ( lattic1508158080041050831_fin_o @ ( sup_sup_set_o @ A @ B ) )
              = ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_6286_Sup__fin_Ounion,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ( B != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
              = ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_6287_Sup__fin_Ounion,axiom,
    ! [A: set_int,B: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( finite_finite_int @ B )
         => ( ( B != bot_bot_set_int )
           => ( ( lattic1091506334969745077in_int @ ( sup_sup_set_int @ A @ B ) )
              = ( sup_sup_int @ ( lattic1091506334969745077in_int @ A ) @ ( lattic1091506334969745077in_int @ B ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_6288_enumerate__Suc_H,axiom,
    ! [S: set_nat,N: nat] :
      ( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
      = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ ( infini8530281810654367211te_nat @ S @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N ) ) ).

% enumerate_Suc'
thf(fact_6289_Inf__fin_Oremove,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( member6576561426505652726_nat_o @ X @ A )
       => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
              = bot_bo7824918357723582233_nat_o )
           => ( ( lattic956194824204696298_nat_o @ A )
              = X ) )
          & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
             != bot_bo7824918357723582233_nat_o )
           => ( ( lattic956194824204696298_nat_o @ A )
              = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_6290_Inf__fin_Oremove,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ X @ A )
       => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
              = bot_bot_set_set_nat )
           => ( ( lattic3014633134055518761et_nat @ A )
              = X ) )
          & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
             != bot_bot_set_set_nat )
           => ( ( lattic3014633134055518761et_nat @ A )
              = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_6291_Inf__fin_Oremove,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( member_assn @ X @ A )
       => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
              = bot_bot_set_assn )
           => ( ( lattic47131356835913163n_assn @ A )
              = X ) )
          & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
             != bot_bot_set_assn )
           => ( ( lattic47131356835913163n_assn @ A )
              = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_6292_Inf__fin_Oremove,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ X @ A )
       => ( ( lattic4107685809792843317_fin_o @ A )
          = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_6293_Inf__fin_Oremove,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ X @ A )
       => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
              = bot_bot_set_int )
           => ( ( lattic5235898064620869839in_int @ A )
              = X ) )
          & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
             != bot_bot_set_int )
           => ( ( lattic5235898064620869839in_int @ A )
              = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_6294_Inf__fin_Oremove,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ A )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ A )
              = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_6295_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
            = bot_bot_set_set_nat )
         => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
            = X ) )
        & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
           != bot_bot_set_set_nat )
         => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
            = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_6296_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
            = bot_bot_set_assn )
         => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
            = X ) )
        & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
           != bot_bot_set_assn )
         => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
            = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_6297_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
        = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X )
          & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_6298_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
            = bot_bot_set_int )
         => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
            = X ) )
        & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
           != bot_bot_set_int )
         => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
            = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_6299_Inf__fin_Oinsert__remove,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
            = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_6300_Inf__fin_Oinsert,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
          = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_6301_Inf__fin_Oinsert,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
          = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_6302_Inf__fin_Oinsert,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
          = ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_6303_Inf__fin_Oinsert,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
          = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_6304_Inf__fin_Oinsert,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
          = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_6305_image__mult__atLeastAtMost__if,axiom,
    ! [C2: rat,X: rat,Y: rat] :
      ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ( image_rat_rat @ ( times_times_rat @ C2 ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
          = ( set_or633870826150836451st_rat @ ( times_times_rat @ C2 @ X ) @ ( times_times_rat @ C2 @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ( ( ord_less_eq_rat @ X @ Y )
           => ( ( image_rat_rat @ ( times_times_rat @ C2 ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = ( set_or633870826150836451st_rat @ ( times_times_rat @ C2 @ Y ) @ ( times_times_rat @ C2 @ X ) ) ) )
          & ( ~ ( ord_less_eq_rat @ X @ Y )
           => ( ( image_rat_rat @ ( times_times_rat @ C2 ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = bot_bot_set_rat ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
thf(fact_6306_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_6307_ivl__disj__un__singleton_I5_J,axiom,
    ! [L: $o,U: $o] :
      ( ( ord_less_eq_o @ L @ U )
     => ( ( sup_sup_set_o @ ( insert_o @ L @ bot_bot_set_o ) @ ( set_or8254209520273421544Most_o @ L @ U ) )
        = ( set_or8904488021354931149Most_o @ L @ U ) ) ) ).

% ivl_disj_un_singleton(5)
thf(fact_6308_ivl__disj__un__singleton_I5_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( insert_Code_integer @ L @ bot_bo3990330152332043303nteger ) @ ( set_or2715278749043346189nteger @ L @ U ) )
        = ( set_or189985376899183464nteger @ L @ U ) ) ) ).

% ivl_disj_un_singleton(5)
thf(fact_6309_ivl__disj__un__singleton_I5_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_eq_int @ L @ U )
     => ( ( sup_sup_set_int @ ( insert_int @ L @ bot_bot_set_int ) @ ( set_or6656581121297822940st_int @ L @ U ) )
        = ( set_or1266510415728281911st_int @ L @ U ) ) ) ).

% ivl_disj_un_singleton(5)
thf(fact_6310_ivl__disj__un__singleton_I5_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or6659071591806873216st_nat @ L @ U ) )
        = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(5)
thf(fact_6311_image__is__empty,axiom,
    ! [F: $o > $o,A: set_o] :
      ( ( ( image_o_o @ F @ A )
        = bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_6312_image__is__empty,axiom,
    ! [F: nat > $o,A: set_nat] :
      ( ( ( image_nat_o @ F @ A )
        = bot_bot_set_o )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_6313_image__is__empty,axiom,
    ! [F: int > $o,A: set_int] :
      ( ( ( image_int_o @ F @ A )
        = bot_bot_set_o )
      = ( A = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_6314_image__is__empty,axiom,
    ! [F: $o > nat,A: set_o] :
      ( ( ( image_o_nat @ F @ A )
        = bot_bot_set_nat )
      = ( A = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_6315_image__is__empty,axiom,
    ! [F: nat > nat,A: set_nat] :
      ( ( ( image_nat_nat @ F @ A )
        = bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_6316_image__is__empty,axiom,
    ! [F: int > nat,A: set_int] :
      ( ( ( image_int_nat @ F @ A )
        = bot_bot_set_nat )
      = ( A = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_6317_image__is__empty,axiom,
    ! [F: $o > int,A: set_o] :
      ( ( ( image_o_int @ F @ A )
        = bot_bot_set_int )
      = ( A = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_6318_image__is__empty,axiom,
    ! [F: nat > int,A: set_nat] :
      ( ( ( image_nat_int @ F @ A )
        = bot_bot_set_int )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_6319_image__is__empty,axiom,
    ! [F: int > int,A: set_int] :
      ( ( ( image_int_int @ F @ A )
        = bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% image_is_empty
thf(fact_6320_image__is__empty,axiom,
    ! [F: nat > set_nat,A: set_nat] :
      ( ( ( image_nat_set_nat @ F @ A )
        = bot_bot_set_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_6321_empty__is__image,axiom,
    ! [F: $o > $o,A: set_o] :
      ( ( bot_bot_set_o
        = ( image_o_o @ F @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_6322_empty__is__image,axiom,
    ! [F: nat > $o,A: set_nat] :
      ( ( bot_bot_set_o
        = ( image_nat_o @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_6323_empty__is__image,axiom,
    ! [F: int > $o,A: set_int] :
      ( ( bot_bot_set_o
        = ( image_int_o @ F @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_6324_empty__is__image,axiom,
    ! [F: $o > nat,A: set_o] :
      ( ( bot_bot_set_nat
        = ( image_o_nat @ F @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_6325_empty__is__image,axiom,
    ! [F: nat > nat,A: set_nat] :
      ( ( bot_bot_set_nat
        = ( image_nat_nat @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_6326_empty__is__image,axiom,
    ! [F: int > nat,A: set_int] :
      ( ( bot_bot_set_nat
        = ( image_int_nat @ F @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_6327_empty__is__image,axiom,
    ! [F: $o > int,A: set_o] :
      ( ( bot_bot_set_int
        = ( image_o_int @ F @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_6328_empty__is__image,axiom,
    ! [F: nat > int,A: set_nat] :
      ( ( bot_bot_set_int
        = ( image_nat_int @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_6329_empty__is__image,axiom,
    ! [F: int > int,A: set_int] :
      ( ( bot_bot_set_int
        = ( image_int_int @ F @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% empty_is_image
thf(fact_6330_empty__is__image,axiom,
    ! [F: nat > set_nat,A: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( image_nat_set_nat @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_6331_image__empty,axiom,
    ! [F: $o > $o] :
      ( ( image_o_o @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_6332_image__empty,axiom,
    ! [F: $o > nat] :
      ( ( image_o_nat @ F @ bot_bot_set_o )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_6333_image__empty,axiom,
    ! [F: $o > int] :
      ( ( image_o_int @ F @ bot_bot_set_o )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_6334_image__empty,axiom,
    ! [F: nat > $o] :
      ( ( image_nat_o @ F @ bot_bot_set_nat )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_6335_image__empty,axiom,
    ! [F: nat > nat] :
      ( ( image_nat_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_6336_image__empty,axiom,
    ! [F: nat > int] :
      ( ( image_nat_int @ F @ bot_bot_set_nat )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_6337_image__empty,axiom,
    ! [F: int > $o] :
      ( ( image_int_o @ F @ bot_bot_set_int )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_6338_image__empty,axiom,
    ! [F: int > nat] :
      ( ( image_int_nat @ F @ bot_bot_set_int )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_6339_image__empty,axiom,
    ! [F: int > int] :
      ( ( image_int_int @ F @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% image_empty
thf(fact_6340_image__empty,axiom,
    ! [F: nat > set_nat] :
      ( ( image_nat_set_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_set_nat ) ).

% image_empty
thf(fact_6341_img__fst,axiom,
    ! [A2: nat,B2: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ S )
     => ( member_nat @ A2 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) ) ) ).

% img_fst
thf(fact_6342_img__fst,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,S: set_Pr1281608226676607948nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A2 @ B2 ) @ S )
     => ( member4242434998011752849e_term @ A2 @ ( image_4436781500471753239e_term @ produc7822682618958472924nteger @ S ) ) ) ).

% img_fst
thf(fact_6343_img__fst,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ S )
     => ( member6576561426505652726_nat_o @ A2 @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S ) ) ) ).

% img_fst
thf(fact_6344_img__fst,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ S )
     => ( member6576561426505652726_nat_o @ A2 @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S ) ) ) ).

% img_fst
thf(fact_6345_img__fst,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,S: set_Pr9222295170931077689nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A2 @ B2 ) @ S )
     => ( member3222579708246209666e_term @ A2 @ ( image_142517975987106933e_term @ produc6816164490631068361nt_int @ S ) ) ) ).

% img_fst
thf(fact_6346_img__fst,axiom,
    ! [A2: int > option6357759511663192854e_term,B2: product_prod_int_int,S: set_Pr1872883991513573699nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A2 @ B2 ) @ S )
     => ( member8845023287901829240e_term @ A2 @ ( image_4217645046469565813e_term @ produc6230002227079971283nt_int @ S ) ) ) ).

% img_fst
thf(fact_6347_img__fst,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B2: set_nat,S: set_Pr3948176798113811640et_nat] :
      ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ S )
     => ( member242389114213480694t_unit @ A2 @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ S ) ) ) ).

% img_fst
thf(fact_6348_img__fst,axiom,
    ! [A2: int,B2: int,S: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ S )
     => ( member_int @ A2 @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S ) ) ) ).

% img_fst
thf(fact_6349_img__snd,axiom,
    ! [A2: nat,B2: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ S )
     => ( member_nat @ B2 @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S ) ) ) ).

% img_snd
thf(fact_6350_img__snd,axiom,
    ! [A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,S: set_Pr1281608226676607948nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A2 @ B2 ) @ S )
     => ( member157494554546826820nteger @ B2 @ ( image_3141030398089242942nteger @ produc7856867400915047194nteger @ S ) ) ) ).

% img_snd
thf(fact_6351_img__snd,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ S )
     => ( member6260224972018164377et_nat @ B2 @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S ) ) ) ).

% img_snd
thf(fact_6352_img__snd,axiom,
    ! [A2: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ S )
     => ( member1996754912294343701et_nat @ B2 @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S ) ) ) ).

% img_snd
thf(fact_6353_img__snd,axiom,
    ! [A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,S: set_Pr9222295170931077689nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A2 @ B2 ) @ S )
     => ( member5262025264175285858nt_int @ B2 @ ( image_3915291023682402351nt_int @ produc7328097813583171335nt_int @ S ) ) ) ).

% img_snd
thf(fact_6354_img__snd,axiom,
    ! [A2: int > option6357759511663192854e_term,B2: product_prod_int_int,S: set_Pr1872883991513573699nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A2 @ B2 ) @ S )
     => ( member5262025264175285858nt_int @ B2 @ ( image_3108077152341812517nt_int @ produc3162348030201620241nt_int @ S ) ) ) ).

% img_snd
thf(fact_6355_img__snd,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B2: set_nat,S: set_Pr3948176798113811640et_nat] :
      ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ S )
     => ( member_set_nat @ B2 @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ S ) ) ) ).

% img_snd
thf(fact_6356_img__snd,axiom,
    ! [A2: int,B2: int,S: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ S )
     => ( member_int @ B2 @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S ) ) ) ).

% img_snd
thf(fact_6357_img__snd,axiom,
    ! [A2: code_natural,B2: produc7822875418678951345atural,S: set_Pr2352657477744078772atural] :
      ( ( member2108296798382401941atural @ ( produc6639722614265839536atural @ A2 @ B2 ) @ S )
     => ( member8280416476884352090atural @ B2 @ ( image_3837677093542026988atural @ produc6591912806276919810atural @ S ) ) ) ).

% img_snd
thf(fact_6358_pair__in__swap__image,axiom,
    ! [Y: nat,X: nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6359_pair__in__swap__image,axiom,
    ! [Y: product_prod_int_int,X: int > option6357759511663192854e_term,A: set_Pr1872883991513573699nt_int] :
      ( ( member3093737243930835296e_term @ ( produc7516973260301520963e_term @ Y @ X ) @ ( image_463623704579739357e_term @ produc8399074013560602399nt_int @ A ) )
      = ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6360_pair__in__swap__image,axiom,
    ! [Y: int > option6357759511663192854e_term,X: product_prod_int_int,A: set_Pr9020418513037773311e_term] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y @ X ) @ ( image_1292712855239066845nt_int @ produc2386993194027891363e_term @ A ) )
      = ( member3093737243930835296e_term @ ( produc7516973260301520963e_term @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6361_pair__in__swap__image,axiom,
    ! [Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,A: set_Pr3286484037609594932et_nat] :
      ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ Y @ X ) @ ( image_6229134628169569989_nat_o @ produc3251101204053596376et_nat @ A ) )
      = ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6362_pair__in__swap__image,axiom,
    ! [Y: produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat,A: set_Pr4208978187861016310_nat_o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y @ X ) @ ( image_4501728140019976901et_nat @ produc1078497009909708626_nat_o @ A ) )
      = ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6363_pair__in__swap__image,axiom,
    ! [Y: produc8923325533196201883nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term,A: set_Pr1281608226676607948nteger] :
      ( ( member3254845646535258459e_term @ ( produc1898379890548282302e_term @ Y @ X ) @ ( image_610307673866561761e_term @ produc736171624797870632nteger @ A ) )
      = ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6364_pair__in__swap__image,axiom,
    ! [Y: product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,A: set_Pr9222295170931077689nt_int] :
      ( ( member7154775377335690986e_term @ ( produc5219906170222623181e_term @ Y @ X ) @ ( image_3490000813295636061e_term @ produc1841954058824580629nt_int @ A ) )
      = ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6365_pair__in__swap__image,axiom,
    ! [Y: produc6241069584506657477e_term > option6357759511663192854e_term,X: produc8923325533196201883nteger,A: set_Pr1628540301243981434e_term] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ Y @ X ) @ ( image_8159590125932948193nteger @ produc3254817899252985374e_term @ A ) )
      = ( member3254845646535258459e_term @ ( produc1898379890548282302e_term @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6366_pair__in__swap__image,axiom,
    ! [Y: produc8551481072490612790e_term > option6357759511663192854e_term,X: product_prod_int_int,A: set_Pr7309386651551657097e_term] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ Y @ X ) @ ( image_1640129903684976221nt_int @ produc1360913580328244269e_term @ A ) )
      = ( member7154775377335690986e_term @ ( produc5219906170222623181e_term @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6367_pair__in__swap__image,axiom,
    ! [Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,A: set_Pr8536935166611901872et_nat] :
      ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ Y @ X ) @ ( image_4071802983491761093_nat_o @ produc1151471490888544852et_nat @ A ) )
      = ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A ) ) ).

% pair_in_swap_image
thf(fact_6368_greaterThanAtMost__empty,axiom,
    ! [L: $o,K: $o] :
      ( ( ord_less_eq_o @ L @ K )
     => ( ( set_or8254209520273421544Most_o @ K @ L )
        = bot_bot_set_o ) ) ).

% greaterThanAtMost_empty
thf(fact_6369_greaterThanAtMost__empty,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ K )
     => ( ( set_or2715278749043346189nteger @ K @ L )
        = bot_bo3990330152332043303nteger ) ) ).

% greaterThanAtMost_empty
thf(fact_6370_greaterThanAtMost__empty,axiom,
    ! [L: assn,K: assn] :
      ( ( ord_less_eq_assn @ L @ K )
     => ( ( set_or2869495477957074904t_assn @ K @ L )
        = bot_bot_set_assn ) ) ).

% greaterThanAtMost_empty
thf(fact_6371_greaterThanAtMost__empty,axiom,
    ! [L: set_nat,K: set_nat] :
      ( ( ord_less_eq_set_nat @ L @ K )
     => ( ( set_or7074010630789208630et_nat @ K @ L )
        = bot_bot_set_set_nat ) ) ).

% greaterThanAtMost_empty
thf(fact_6372_greaterThanAtMost__empty,axiom,
    ! [L: nat,K: nat] :
      ( ( ord_less_eq_nat @ L @ K )
     => ( ( set_or6659071591806873216st_nat @ K @ L )
        = bot_bot_set_nat ) ) ).

% greaterThanAtMost_empty
thf(fact_6373_greaterThanAtMost__empty,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( set_or6656581121297822940st_int @ K @ L )
        = bot_bot_set_int ) ) ).

% greaterThanAtMost_empty
thf(fact_6374_greaterThanAtMost__empty__iff,axiom,
    ! [K: $o,L: $o] :
      ( ( ( set_or8254209520273421544Most_o @ K @ L )
        = bot_bot_set_o )
      = ( ~ ( ord_less_o @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff
thf(fact_6375_greaterThanAtMost__empty__iff,axiom,
    ! [K: assn,L: assn] :
      ( ( ( set_or2869495477957074904t_assn @ K @ L )
        = bot_bot_set_assn )
      = ( ~ ( ord_less_assn @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff
thf(fact_6376_greaterThanAtMost__empty__iff,axiom,
    ! [K: nat,L: nat] :
      ( ( ( set_or6659071591806873216st_nat @ K @ L )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff
thf(fact_6377_greaterThanAtMost__empty__iff,axiom,
    ! [K: int,L: int] :
      ( ( ( set_or6656581121297822940st_int @ K @ L )
        = bot_bot_set_int )
      = ( ~ ( ord_less_int @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff
thf(fact_6378_greaterThanAtMost__empty__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( set_or2715278749043346189nteger @ K @ L )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le6747313008572928689nteger @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff
thf(fact_6379_greaterThanAtMost__empty__iff,axiom,
    ! [K: code_natural,L: code_natural] :
      ( ( ( set_or1538873900800063704atural @ K @ L )
        = bot_bo8448779972843232370atural )
      = ( ~ ( ord_le5570908160329646204atural @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff
thf(fact_6380_greaterThanAtMost__empty__iff2,axiom,
    ! [K: $o,L: $o] :
      ( ( bot_bot_set_o
        = ( set_or8254209520273421544Most_o @ K @ L ) )
      = ( ~ ( ord_less_o @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff2
thf(fact_6381_greaterThanAtMost__empty__iff2,axiom,
    ! [K: assn,L: assn] :
      ( ( bot_bot_set_assn
        = ( set_or2869495477957074904t_assn @ K @ L ) )
      = ( ~ ( ord_less_assn @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff2
thf(fact_6382_greaterThanAtMost__empty__iff2,axiom,
    ! [K: nat,L: nat] :
      ( ( bot_bot_set_nat
        = ( set_or6659071591806873216st_nat @ K @ L ) )
      = ( ~ ( ord_less_nat @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff2
thf(fact_6383_greaterThanAtMost__empty__iff2,axiom,
    ! [K: int,L: int] :
      ( ( bot_bot_set_int
        = ( set_or6656581121297822940st_int @ K @ L ) )
      = ( ~ ( ord_less_int @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff2
thf(fact_6384_greaterThanAtMost__empty__iff2,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or2715278749043346189nteger @ K @ L ) )
      = ( ~ ( ord_le6747313008572928689nteger @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff2
thf(fact_6385_greaterThanAtMost__empty__iff2,axiom,
    ! [K: code_natural,L: code_natural] :
      ( ( bot_bo8448779972843232370atural
        = ( set_or1538873900800063704atural @ K @ L ) )
      = ( ~ ( ord_le5570908160329646204atural @ K @ L ) ) ) ).

% greaterThanAtMost_empty_iff2
thf(fact_6386_Inf__fin_Osingleton,axiom,
    ! [X: $o] :
      ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Inf_fin.singleton
thf(fact_6387_Inf__fin_Osingleton,axiom,
    ! [X: nat] :
      ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% Inf_fin.singleton
thf(fact_6388_Inf__fin_Osingleton,axiom,
    ! [X: int] :
      ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% Inf_fin.singleton
thf(fact_6389_sup__Inf__absorb,axiom,
    ! [A: set_o,A2: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ A2 @ A )
       => ( ( sup_sup_o @ ( lattic4107685809792843317_fin_o @ A ) @ A2 )
          = A2 ) ) ) ).

% sup_Inf_absorb
thf(fact_6390_sup__Inf__absorb,axiom,
    ! [A: set_int,A2: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ A2 @ A )
       => ( ( sup_sup_int @ ( lattic5235898064620869839in_int @ A ) @ A2 )
          = A2 ) ) ) ).

% sup_Inf_absorb
thf(fact_6391_sup__Inf__absorb,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( member6576561426505652726_nat_o @ A2 @ A )
       => ( ( sup_su5453871518329203617_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ A2 )
          = A2 ) ) ) ).

% sup_Inf_absorb
thf(fact_6392_sup__Inf__absorb,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
          = A2 ) ) ) ).

% sup_Inf_absorb
thf(fact_6393_sup__Inf__absorb,axiom,
    ! [A: set_assn,A2: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( member_assn @ A2 @ A )
       => ( ( sup_sup_assn @ ( lattic47131356835913163n_assn @ A ) @ A2 )
          = A2 ) ) ) ).

% sup_Inf_absorb
thf(fact_6394_sup__Inf__absorb,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A2 @ A )
       => ( ( sup_sup_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ A2 )
          = A2 ) ) ) ).

% sup_Inf_absorb
thf(fact_6395_sup__Inf__absorb,axiom,
    ! [A: set_se5479201105039950375at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A )
     => ( ( member6273874620674182256at_nat @ A2 @ A )
       => ( ( sup_su5525570899277871387at_nat @ ( lattic6378378157708517732at_nat @ A ) @ A2 )
          = A2 ) ) ) ).

% sup_Inf_absorb
thf(fact_6396_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_6397_image__mult__atLeastAtMost,axiom,
    ! [D2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D2 )
     => ( ( image_rat_rat @ ( times_times_rat @ D2 ) @ ( set_or633870826150836451st_rat @ A2 @ B2 ) )
        = ( set_or633870826150836451st_rat @ ( times_times_rat @ D2 @ A2 ) @ ( times_times_rat @ D2 @ B2 ) ) ) ) ).

% image_mult_atLeastAtMost
thf(fact_6398_ivl__disj__int__one_I3_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_ord_atMost_o @ L ) @ ( set_or8254209520273421544Most_o @ L @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(3)
thf(fact_6399_ivl__disj__int__one_I3_J,axiom,
    ! [L: int,U: int] :
      ( ( inf_inf_set_int @ ( set_ord_atMost_int @ L ) @ ( set_or6656581121297822940st_int @ L @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_one(3)
thf(fact_6400_ivl__disj__int__one_I3_J,axiom,
    ! [L: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_one(3)
thf(fact_6401_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: $o] :
      ( bot_bot_set_o
     != ( set_ord_atMost_o @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6402_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: int] :
      ( bot_bot_set_int
     != ( set_ord_atMost_int @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6403_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: nat] :
      ( bot_bot_set_nat
     != ( set_ord_atMost_nat @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6404_Inf__fin_Ohom__commute,axiom,
    ! [H2: set_nat > set_nat,N3: set_set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] :
          ( ( H2 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
          = ( inf_inf_set_nat @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite1152437895449049373et_nat @ N3 )
       => ( ( N3 != bot_bot_set_set_nat )
         => ( ( H2 @ ( lattic3014633134055518761et_nat @ N3 ) )
            = ( lattic3014633134055518761et_nat @ ( image_7916887816326733075et_nat @ H2 @ N3 ) ) ) ) ) ) ).

% Inf_fin.hom_commute
thf(fact_6405_Inf__fin_Ohom__commute,axiom,
    ! [H2: assn > assn,N3: set_assn] :
      ( ! [X3: assn,Y3: assn] :
          ( ( H2 @ ( inf_inf_assn @ X3 @ Y3 ) )
          = ( inf_inf_assn @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_assn @ N3 )
       => ( ( N3 != bot_bot_set_assn )
         => ( ( H2 @ ( lattic47131356835913163n_assn @ N3 ) )
            = ( lattic47131356835913163n_assn @ ( image_assn_assn @ H2 @ N3 ) ) ) ) ) ) ).

% Inf_fin.hom_commute
thf(fact_6406_Inf__fin_Ohom__commute,axiom,
    ! [H2: $o > $o,N3: set_o] :
      ( ! [X3: $o,Y3: $o] :
          ( ( H2 @ ( inf_inf_o @ X3 @ Y3 ) )
          = ( inf_inf_o @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_o @ N3 )
       => ( ( N3 != bot_bot_set_o )
         => ( ( H2 @ ( lattic4107685809792843317_fin_o @ N3 ) )
            = ( lattic4107685809792843317_fin_o @ ( image_o_o @ H2 @ N3 ) ) ) ) ) ) ).

% Inf_fin.hom_commute
thf(fact_6407_Inf__fin_Ohom__commute,axiom,
    ! [H2: nat > nat,N3: set_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( H2 @ ( inf_inf_nat @ X3 @ Y3 ) )
          = ( inf_inf_nat @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_nat @ N3 )
       => ( ( N3 != bot_bot_set_nat )
         => ( ( H2 @ ( lattic5238388535129920115in_nat @ N3 ) )
            = ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H2 @ N3 ) ) ) ) ) ) ).

% Inf_fin.hom_commute
thf(fact_6408_Inf__fin_Ohom__commute,axiom,
    ! [H2: int > int,N3: set_int] :
      ( ! [X3: int,Y3: int] :
          ( ( H2 @ ( inf_inf_int @ X3 @ Y3 ) )
          = ( inf_inf_int @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_int @ N3 )
       => ( ( N3 != bot_bot_set_int )
         => ( ( H2 @ ( lattic5235898064620869839in_int @ N3 ) )
            = ( lattic5235898064620869839in_int @ ( image_int_int @ H2 @ N3 ) ) ) ) ) ) ).

% Inf_fin.hom_commute
thf(fact_6409_in__fst__imageE,axiom,
    ! [X: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ X @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) )
     => ~ ! [Y3: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S ) ) ).

% in_fst_imageE
thf(fact_6410_in__fst__imageE,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,S: set_Pr1281608226676607948nteger] :
      ( ( member4242434998011752849e_term @ X @ ( image_4436781500471753239e_term @ produc7822682618958472924nteger @ S ) )
     => ~ ! [Y3: produc8923325533196201883nteger] :
            ~ ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y3 ) @ S ) ) ).

% in_fst_imageE
thf(fact_6411_in__fst__imageE,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S: set_Pr3286484037609594932et_nat] :
      ( ( member6576561426505652726_nat_o @ X @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S ) )
     => ~ ! [Y3: produc3658429121746597890et_nat] :
            ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y3 ) @ S ) ) ).

% in_fst_imageE
thf(fact_6412_in__fst__imageE,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S: set_Pr8536935166611901872et_nat] :
      ( ( member6576561426505652726_nat_o @ X @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S ) )
     => ~ ! [Y3: produc3925858234332021118et_nat] :
            ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y3 ) @ S ) ) ).

% in_fst_imageE
thf(fact_6413_in__fst__imageE,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,S: set_Pr9222295170931077689nt_int] :
      ( ( member3222579708246209666e_term @ X @ ( image_142517975987106933e_term @ produc6816164490631068361nt_int @ S ) )
     => ~ ! [Y3: product_prod_int_int] :
            ~ ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y3 ) @ S ) ) ).

% in_fst_imageE
thf(fact_6414_in__fst__imageE,axiom,
    ! [X: int > option6357759511663192854e_term,S: set_Pr1872883991513573699nt_int] :
      ( ( member8845023287901829240e_term @ X @ ( image_4217645046469565813e_term @ produc6230002227079971283nt_int @ S ) )
     => ~ ! [Y3: product_prod_int_int] :
            ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y3 ) @ S ) ) ).

% in_fst_imageE
thf(fact_6415_in__fst__imageE,axiom,
    ! [X: heap_e7401611519738050253t_unit,S: set_Pr3948176798113811640et_nat] :
      ( ( member242389114213480694t_unit @ X @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ S ) )
     => ~ ! [Y3: set_nat] :
            ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y3 ) @ S ) ) ).

% in_fst_imageE
thf(fact_6416_in__fst__imageE,axiom,
    ! [X: int,S: set_Pr958786334691620121nt_int] :
      ( ( member_int @ X @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S ) )
     => ~ ! [Y3: int] :
            ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ S ) ) ).

% in_fst_imageE
thf(fact_6417_in__snd__imageE,axiom,
    ! [Y: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ Y @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S ) )
     => ~ ! [X3: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6418_in__snd__imageE,axiom,
    ! [Y: produc8923325533196201883nteger,S: set_Pr1281608226676607948nteger] :
      ( ( member157494554546826820nteger @ Y @ ( image_3141030398089242942nteger @ produc7856867400915047194nteger @ S ) )
     => ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ~ ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6419_in__snd__imageE,axiom,
    ! [Y: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( member6260224972018164377et_nat @ Y @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o] :
            ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6420_in__snd__imageE,axiom,
    ! [Y: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( member1996754912294343701et_nat @ Y @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o] :
            ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6421_in__snd__imageE,axiom,
    ! [Y: product_prod_int_int,S: set_Pr9222295170931077689nt_int] :
      ( ( member5262025264175285858nt_int @ Y @ ( image_3915291023682402351nt_int @ produc7328097813583171335nt_int @ S ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ~ ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6422_in__snd__imageE,axiom,
    ! [Y: product_prod_int_int,S: set_Pr1872883991513573699nt_int] :
      ( ( member5262025264175285858nt_int @ Y @ ( image_3108077152341812517nt_int @ produc3162348030201620241nt_int @ S ) )
     => ~ ! [X3: int > option6357759511663192854e_term] :
            ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6423_in__snd__imageE,axiom,
    ! [Y: set_nat,S: set_Pr3948176798113811640et_nat] :
      ( ( member_set_nat @ Y @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ S ) )
     => ~ ! [X3: heap_e7401611519738050253t_unit] :
            ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6424_in__snd__imageE,axiom,
    ! [Y: int,S: set_Pr958786334691620121nt_int] :
      ( ( member_int @ Y @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S ) )
     => ~ ! [X3: int] :
            ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6425_in__snd__imageE,axiom,
    ! [Y: produc7822875418678951345atural,S: set_Pr2352657477744078772atural] :
      ( ( member8280416476884352090atural @ Y @ ( image_3837677093542026988atural @ produc6591912806276919810atural @ S ) )
     => ~ ! [X3: code_natural] :
            ~ ( member2108296798382401941atural @ ( produc6639722614265839536atural @ X3 @ Y ) @ S ) ) ).

% in_snd_imageE
thf(fact_6426_ivl__disj__int__two_I6_J,axiom,
    ! [L: $o,M2: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or8254209520273421544Most_o @ L @ M2 ) @ ( set_or8254209520273421544Most_o @ M2 @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_two(6)
thf(fact_6427_ivl__disj__int__two_I6_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or6659071591806873216st_nat @ M2 @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_two(6)
thf(fact_6428_ivl__disj__int__two_I6_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or6656581121297822940st_int @ L @ M2 ) @ ( set_or6656581121297822940st_int @ M2 @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_two(6)
thf(fact_6429_atMost__eq__UNIV__iff,axiom,
    ! [X: assn] :
      ( ( ( set_ord_atMost_assn @ X )
        = top_top_set_assn )
      = ( X = top_top_assn ) ) ).

% atMost_eq_UNIV_iff
thf(fact_6430_atMost__eq__UNIV__iff,axiom,
    ! [X: set_o] :
      ( ( ( set_ord_atMost_set_o @ X )
        = top_top_set_set_o )
      = ( X = top_top_set_o ) ) ).

% atMost_eq_UNIV_iff
thf(fact_6431_atMost__eq__UNIV__iff,axiom,
    ! [X: set_nat] :
      ( ( ( set_or4236626031148496127et_nat @ X )
        = top_top_set_set_nat )
      = ( X = top_top_set_nat ) ) ).

% atMost_eq_UNIV_iff
thf(fact_6432_atMost__eq__UNIV__iff,axiom,
    ! [X: set_Product_unit] :
      ( ( ( set_or2827140217781692084t_unit @ X )
        = top_to1767297665138865437t_unit )
      = ( X = top_to1996260823553986621t_unit ) ) ).

% atMost_eq_UNIV_iff
thf(fact_6433_atMost__eq__UNIV__iff,axiom,
    ! [X: $o] :
      ( ( ( set_ord_atMost_o @ X )
        = top_top_set_o )
      = ( X = top_top_o ) ) ).

% atMost_eq_UNIV_iff
thf(fact_6434_atMost__eq__UNIV__iff,axiom,
    ! [X: product_unit] :
      ( ( ( set_or8621721900777396692t_unit @ X )
        = top_to1996260823553986621t_unit )
      = ( X = top_top_Product_unit ) ) ).

% atMost_eq_UNIV_iff
thf(fact_6435_the__elem__image__unique,axiom,
    ! [A: set_nat,F: nat > nat,X: nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A )
           => ( ( F @ Y3 )
              = ( F @ X ) ) )
       => ( ( the_elem_nat @ ( image_nat_nat @ F @ A ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_6436_the__elem__image__unique,axiom,
    ! [A: set_nat,F: nat > set_nat,X: nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ A )
           => ( ( F @ Y3 )
              = ( F @ X ) ) )
       => ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_6437_range__eq__singletonD,axiom,
    ! [F: $o > $o,A2: $o,X: $o] :
      ( ( ( image_o_o @ F @ top_top_set_o )
        = ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6438_range__eq__singletonD,axiom,
    ! [F: $o > nat,A2: nat,X: $o] :
      ( ( ( image_o_nat @ F @ top_top_set_o )
        = ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6439_range__eq__singletonD,axiom,
    ! [F: $o > int,A2: int,X: $o] :
      ( ( ( image_o_int @ F @ top_top_set_o )
        = ( insert_int @ A2 @ bot_bot_set_int ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6440_range__eq__singletonD,axiom,
    ! [F: nat > $o,A2: $o,X: nat] :
      ( ( ( image_nat_o @ F @ top_top_set_nat )
        = ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6441_range__eq__singletonD,axiom,
    ! [F: nat > nat,A2: nat,X: nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6442_range__eq__singletonD,axiom,
    ! [F: nat > int,A2: int,X: nat] :
      ( ( ( image_nat_int @ F @ top_top_set_nat )
        = ( insert_int @ A2 @ bot_bot_set_int ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6443_range__eq__singletonD,axiom,
    ! [F: product_unit > $o,A2: $o,X: product_unit] :
      ( ( ( image_Product_unit_o @ F @ top_to1996260823553986621t_unit )
        = ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6444_range__eq__singletonD,axiom,
    ! [F: product_unit > nat,A2: nat,X: product_unit] :
      ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
        = ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6445_range__eq__singletonD,axiom,
    ! [F: product_unit > int,A2: int,X: product_unit] :
      ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
        = ( insert_int @ A2 @ bot_bot_set_int ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6446_range__eq__singletonD,axiom,
    ! [F: nat > set_nat,A2: set_nat,X: nat] :
      ( ( ( image_nat_set_nat @ F @ top_top_set_nat )
        = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_6447_fst__image__mp,axiom,
    ! [A: set_Pr1281608226676607948nteger,B: set_Pr7604974323444597168e_term,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( ord_le7106919388332126736e_term @ ( image_4436781500471753239e_term @ produc7822682618958472924nteger @ A ) @ B )
     => ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ A )
       => ( member4242434998011752849e_term @ X @ B ) ) ) ).

% fst_image_mp
thf(fact_6448_fst__image__mp,axiom,
    ! [A: set_Pr3286484037609594932et_nat,B: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( ord_le2965882846123202637_nat_o @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ A ) @ B )
     => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A )
       => ( member6576561426505652726_nat_o @ X @ B ) ) ) ).

% fst_image_mp
thf(fact_6449_fst__image__mp,axiom,
    ! [A: set_Pr8536935166611901872et_nat,B: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( ord_le2965882846123202637_nat_o @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ A ) @ B )
     => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A )
       => ( member6576561426505652726_nat_o @ X @ B ) ) ) ).

% fst_image_mp
thf(fact_6450_fst__image__mp,axiom,
    ! [A: set_Pr9222295170931077689nt_int,B: set_Pr3642885161833720865e_term,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( ord_le4129093240937979009e_term @ ( image_142517975987106933e_term @ produc6816164490631068361nt_int @ A ) @ B )
     => ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ A )
       => ( member3222579708246209666e_term @ X @ B ) ) ) ).

% fst_image_mp
thf(fact_6451_fst__image__mp,axiom,
    ! [A: set_Pr1872883991513573699nt_int,B: set_in3461395444621081367e_term,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( ord_le567748327753481079e_term @ ( image_4217645046469565813e_term @ produc6230002227079971283nt_int @ A ) @ B )
     => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ A )
       => ( member8845023287901829240e_term @ X @ B ) ) ) ).

% fst_image_mp
thf(fact_6452_fst__image__mp,axiom,
    ! [A: set_Pr3948176798113811640et_nat,B: set_he8644174446132254637t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
      ( ( ord_le4903435234951335757t_unit @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ A ) @ B )
     => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A )
       => ( member242389114213480694t_unit @ X @ B ) ) ) ).

% fst_image_mp
thf(fact_6453_fst__image__mp,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_int,X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ A ) @ B )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A )
       => ( member_int @ X @ B ) ) ) ).

% fst_image_mp
thf(fact_6454_fst__image__mp,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_nat,X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A ) @ B )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A )
       => ( member_nat @ X @ B ) ) ) ).

% fst_image_mp
thf(fact_6455_snd__image__mp,axiom,
    ! [A: set_Pr1281608226676607948nteger,B: set_Pr4811707699266497531nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( ord_le3725938330318615451nteger @ ( image_3141030398089242942nteger @ produc7856867400915047194nteger @ A ) @ B )
     => ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ A )
       => ( member157494554546826820nteger @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6456_snd__image__mp,axiom,
    ! [A: set_Pr3286484037609594932et_nat,B: set_Pr3948176798113811640et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( ord_le8090042543978450200et_nat @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ A ) @ B )
     => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A )
       => ( member6260224972018164377et_nat @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6457_snd__image__mp,axiom,
    ! [A: set_Pr8536935166611901872et_nat,B: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( ord_le5966269811547037844et_nat @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ A ) @ B )
     => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A )
       => ( member1996754912294343701et_nat @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6458_snd__image__mp,axiom,
    ! [A: set_Pr9222295170931077689nt_int,B: set_Pr958786334691620121nt_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ ( image_3915291023682402351nt_int @ produc7328097813583171335nt_int @ A ) @ B )
     => ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ A )
       => ( member5262025264175285858nt_int @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6459_snd__image__mp,axiom,
    ! [A: set_Pr1872883991513573699nt_int,B: set_Pr958786334691620121nt_int,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ ( image_3108077152341812517nt_int @ produc3162348030201620241nt_int @ A ) @ B )
     => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ A )
       => ( member5262025264175285858nt_int @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6460_snd__image__mp,axiom,
    ! [A: set_Pr3948176798113811640et_nat,B: set_set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ A ) @ B )
     => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A )
       => ( member_set_nat @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6461_snd__image__mp,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_int,X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ A ) @ B )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A )
       => ( member_int @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6462_snd__image__mp,axiom,
    ! [A: set_Pr2352657477744078772atural,B: set_Pr7564037697178849041atural,X: code_natural,Y: produc7822875418678951345atural] :
      ( ( ord_le6478268328230966961atural @ ( image_3837677093542026988atural @ produc6591912806276919810atural @ A ) @ B )
     => ( ( member2108296798382401941atural @ ( produc6639722614265839536atural @ X @ Y ) @ A )
       => ( member8280416476884352090atural @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6463_snd__image__mp,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_nat,X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A ) @ B )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A )
       => ( member_nat @ Y @ B ) ) ) ).

% snd_image_mp
thf(fact_6464_in__image__insert__iff,axiom,
    ! [B: set_se5479201105039950375at_nat,X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ! [C6: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ C6 @ B )
         => ~ ( member8757157785044589968at_nat @ X @ C6 ) )
     => ( ( member6273874620674182256at_nat @ A @ ( image_4303274092540115909at_nat @ ( insert9069300056098147895at_nat @ X ) @ B ) )
        = ( ( member8757157785044589968at_nat @ X @ A )
          & ( member6273874620674182256at_nat @ ( minus_3314409938677909166at_nat @ A @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_6465_in__image__insert__iff,axiom,
    ! [B: set_se1353816476878259085_nat_o,X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
      ( ! [C6: set_Pr4532377907799695533_nat_o] :
          ( ( member1452989533377600726_nat_o @ C6 @ B )
         => ~ ( member6576561426505652726_nat_o @ X @ C6 ) )
     => ( ( member1452989533377600726_nat_o @ A @ ( image_8925030434418351941_nat_o @ ( insert5175938949040314269_nat_o @ X ) @ B ) )
        = ( ( member6576561426505652726_nat_o @ X @ A )
          & ( member1452989533377600726_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_6466_in__image__insert__iff,axiom,
    ! [B: set_se7855581050983116737at_nat,X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ! [C6: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ C6 @ B )
         => ~ ( member8440522571783428010at_nat @ X @ C6 ) )
     => ( ( member2643936169264416010at_nat @ A @ ( image_3684629450409544005at_nat @ ( insert8211810215607154385at_nat @ X ) @ B ) )
        = ( ( member8440522571783428010at_nat @ X @ A )
          & ( member2643936169264416010at_nat @ ( minus_1356011639430497352at_nat @ A @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_6467_in__image__insert__iff,axiom,
    ! [B: set_set_o,X: $o,A: set_o] :
      ( ! [C6: set_o] :
          ( ( member_set_o @ C6 @ B )
         => ~ ( member_o @ X @ C6 ) )
     => ( ( member_set_o @ A @ ( image_set_o_set_o @ ( insert_o @ X ) @ B ) )
        = ( ( member_o @ X @ A )
          & ( member_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_6468_in__image__insert__iff,axiom,
    ! [B: set_set_int,X: int,A: set_int] :
      ( ! [C6: set_int] :
          ( ( member_set_int @ C6 @ B )
         => ~ ( member_int @ X @ C6 ) )
     => ( ( member_set_int @ A @ ( image_524474410958335435et_int @ ( insert_int @ X ) @ B ) )
        = ( ( member_int @ X @ A )
          & ( member_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_6469_in__image__insert__iff,axiom,
    ! [B: set_set_nat,X: nat,A: set_nat] :
      ( ! [C6: set_nat] :
          ( ( member_set_nat @ C6 @ B )
         => ~ ( member_nat @ X @ C6 ) )
     => ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B ) )
        = ( ( member_nat @ X @ A )
          & ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) ) ) ) ).

% in_image_insert_iff
thf(fact_6470_Ioc__disjoint,axiom,
    ! [A2: $o,B2: $o,C2: $o,D2: $o] :
      ( ( ( inf_inf_set_o @ ( set_or8254209520273421544Most_o @ A2 @ B2 ) @ ( set_or8254209520273421544Most_o @ C2 @ D2 ) )
        = bot_bot_set_o )
      = ( ( ord_less_eq_o @ B2 @ A2 )
        | ( ord_less_eq_o @ D2 @ C2 )
        | ( ord_less_eq_o @ B2 @ C2 )
        | ( ord_less_eq_o @ D2 @ A2 ) ) ) ).

% Ioc_disjoint
thf(fact_6471_Ioc__disjoint,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ( inf_in1364745209274528805nteger @ ( set_or2715278749043346189nteger @ A2 @ B2 ) @ ( set_or2715278749043346189nteger @ C2 @ D2 ) )
        = bot_bo3990330152332043303nteger )
      = ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
        | ( ord_le3102999989581377725nteger @ D2 @ C2 )
        | ( ord_le3102999989581377725nteger @ B2 @ C2 )
        | ( ord_le3102999989581377725nteger @ D2 @ A2 ) ) ) ).

% Ioc_disjoint
thf(fact_6472_Ioc__disjoint,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ A2 @ B2 ) @ ( set_or6659071591806873216st_nat @ C2 @ D2 ) )
        = bot_bot_set_nat )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        | ( ord_less_eq_nat @ D2 @ C2 )
        | ( ord_less_eq_nat @ B2 @ C2 )
        | ( ord_less_eq_nat @ D2 @ A2 ) ) ) ).

% Ioc_disjoint
thf(fact_6473_Ioc__disjoint,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( inf_inf_set_int @ ( set_or6656581121297822940st_int @ A2 @ B2 ) @ ( set_or6656581121297822940st_int @ C2 @ D2 ) )
        = bot_bot_set_int )
      = ( ( ord_less_eq_int @ B2 @ A2 )
        | ( ord_less_eq_int @ D2 @ C2 )
        | ( ord_less_eq_int @ B2 @ C2 )
        | ( ord_less_eq_int @ D2 @ A2 ) ) ) ).

% Ioc_disjoint
thf(fact_6474_ivl__disj__int__two_I8_J,axiom,
    ! [L: $o,M2: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or8904488021354931149Most_o @ L @ M2 ) @ ( set_or8254209520273421544Most_o @ M2 @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_two(8)
thf(fact_6475_ivl__disj__int__two_I8_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or6656581121297822940st_int @ M2 @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_two(8)
thf(fact_6476_ivl__disj__int__two_I8_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or6659071591806873216st_nat @ M2 @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_two(8)
thf(fact_6477_Inf__fin_Obounded__iff,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
          = ( ! [X2: $o] :
                ( ( member_o @ X2 @ A )
               => ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_6478_Inf__fin_Obounded__iff,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ( ord_less_eq_assn @ X @ ( lattic47131356835913163n_assn @ A ) )
          = ( ! [X2: assn] :
                ( ( member_assn @ X2 @ A )
               => ( ord_less_eq_assn @ X @ X2 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_6479_Inf__fin_Obounded__iff,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
          = ( ! [X2: set_nat] :
                ( ( member_set_nat @ X2 @ A )
               => ( ord_less_eq_set_nat @ X @ X2 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_6480_Inf__fin_Obounded__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A )
               => ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_6481_Inf__fin_Obounded__iff,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A )
               => ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_6482_Inf__fin_OboundedI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( A != bot_bo7824918357723582233_nat_o )
       => ( ! [A5: produc3658429121746597890et_nat > $o] :
              ( ( member6576561426505652726_nat_o @ A5 @ A )
             => ( ord_le729326519192465773_nat_o @ X @ A5 ) )
         => ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_6483_Inf__fin_OboundedI,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ! [A5: $o] :
              ( ( member_o @ A5 @ A )
             => ( ord_less_eq_o @ X @ A5 ) )
         => ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_6484_Inf__fin_OboundedI,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ! [A5: assn] :
              ( ( member_assn @ A5 @ A )
             => ( ord_less_eq_assn @ X @ A5 ) )
         => ( ord_less_eq_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_6485_Inf__fin_OboundedI,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [A5: set_nat] :
              ( ( member_set_nat @ A5 @ A )
             => ( ord_less_eq_set_nat @ X @ A5 ) )
         => ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_6486_Inf__fin_OboundedI,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A5: nat] :
              ( ( member_nat @ A5 @ A )
             => ( ord_less_eq_nat @ X @ A5 ) )
         => ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_6487_Inf__fin_OboundedI,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ A )
             => ( ord_less_eq_int @ X @ A5 ) )
         => ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_6488_Inf__fin_OboundedE,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( A != bot_bo7824918357723582233_nat_o )
       => ( ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) )
         => ! [A8: produc3658429121746597890et_nat > $o] :
              ( ( member6576561426505652726_nat_o @ A8 @ A )
             => ( ord_le729326519192465773_nat_o @ X @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_6489_Inf__fin_OboundedE,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
         => ! [A8: $o] :
              ( ( member_o @ A8 @ A )
             => ( ord_less_eq_o @ X @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_6490_Inf__fin_OboundedE,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ( ord_less_eq_assn @ X @ ( lattic47131356835913163n_assn @ A ) )
         => ! [A8: assn] :
              ( ( member_assn @ A8 @ A )
             => ( ord_less_eq_assn @ X @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_6491_Inf__fin_OboundedE,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
         => ! [A8: set_nat] :
              ( ( member_set_nat @ A8 @ A )
             => ( ord_less_eq_set_nat @ X @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_6492_Inf__fin_OboundedE,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
         => ! [A8: nat] :
              ( ( member_nat @ A8 @ A )
             => ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_6493_Inf__fin_OboundedE,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
         => ! [A8: int] :
              ( ( member_int @ A8 @ A )
             => ( ord_less_eq_int @ X @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_6494_ivl__disj__int__one_I5_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or8254209520273421544Most_o @ L @ U ) @ ( set_or6416164934427428222Than_o @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(5)
thf(fact_6495_ivl__disj__int__one_I5_J,axiom,
    ! [L: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or6656581121297822940st_int @ L @ U ) @ ( set_or1207661135979820486an_int @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_one(5)
thf(fact_6496_ivl__disj__int__one_I5_J,axiom,
    ! [L: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L @ U ) @ ( set_or1210151606488870762an_nat @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_one(5)
thf(fact_6497_Sup__fin_Ohom__commute,axiom,
    ! [H2: assn > assn,N3: set_assn] :
      ( ! [X3: assn,Y3: assn] :
          ( ( H2 @ ( sup_sup_assn @ X3 @ Y3 ) )
          = ( sup_sup_assn @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_assn @ N3 )
       => ( ( N3 != bot_bot_set_assn )
         => ( ( H2 @ ( lattic2150320897289308081n_assn @ N3 ) )
            = ( lattic2150320897289308081n_assn @ ( image_assn_assn @ H2 @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_6498_Sup__fin_Ohom__commute,axiom,
    ! [H2: set_nat > set_nat,N3: set_set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] :
          ( ( H2 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
          = ( sup_sup_set_nat @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite1152437895449049373et_nat @ N3 )
       => ( ( N3 != bot_bot_set_set_nat )
         => ( ( H2 @ ( lattic3835124923745554447et_nat @ N3 ) )
            = ( lattic3835124923745554447et_nat @ ( image_7916887816326733075et_nat @ H2 @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_6499_Sup__fin_Ohom__commute,axiom,
    ! [H2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,N3: set_se5479201105039950375at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( H2 @ ( sup_su5525570899277871387at_nat @ X3 @ Y3 ) )
          = ( sup_su5525570899277871387at_nat @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite4768004253697144688at_nat @ N3 )
       => ( ( N3 != bot_bo1534825418459847827at_nat )
         => ( ( H2 @ ( lattic8394993312974875902at_nat @ N3 ) )
            = ( lattic8394993312974875902at_nat @ ( image_4303274092540115909at_nat @ H2 @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_6500_Sup__fin_Ohom__commute,axiom,
    ! [H2: $o > $o,N3: set_o] :
      ( ! [X3: $o,Y3: $o] :
          ( ( H2 @ ( sup_sup_o @ X3 @ Y3 ) )
          = ( sup_sup_o @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_o @ N3 )
       => ( ( N3 != bot_bot_set_o )
         => ( ( H2 @ ( lattic1508158080041050831_fin_o @ N3 ) )
            = ( lattic1508158080041050831_fin_o @ ( image_o_o @ H2 @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_6501_Sup__fin_Ohom__commute,axiom,
    ! [H2: nat > nat,N3: set_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( H2 @ ( sup_sup_nat @ X3 @ Y3 ) )
          = ( sup_sup_nat @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_nat @ N3 )
       => ( ( N3 != bot_bot_set_nat )
         => ( ( H2 @ ( lattic1093996805478795353in_nat @ N3 ) )
            = ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H2 @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_6502_Sup__fin_Ohom__commute,axiom,
    ! [H2: int > int,N3: set_int] :
      ( ! [X3: int,Y3: int] :
          ( ( H2 @ ( sup_sup_int @ X3 @ Y3 ) )
          = ( sup_sup_int @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
     => ( ( finite_finite_int @ N3 )
       => ( ( N3 != bot_bot_set_int )
         => ( ( H2 @ ( lattic1091506334969745077in_int @ N3 ) )
            = ( lattic1091506334969745077in_int @ ( image_int_int @ H2 @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_6503_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ( set_or8254209520273421544Most_o
    = ( ^ [A3: $o,B3: $o] : ( minus_minus_set_o @ ( set_or8904488021354931149Most_o @ A3 @ B3 ) @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_6504_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [A3: int,B3: int] : ( minus_minus_set_int @ ( set_or1266510415728281911st_int @ A3 @ B3 ) @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_6505_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [A3: nat,B3: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_6506_Inf__fin_Osubset__imp,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ( A != bot_bot_set_o )
       => ( ( finite_finite_o @ B )
         => ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ B ) @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_6507_Inf__fin_Osubset__imp,axiom,
    ! [A: set_assn,B: set_assn] :
      ( ( ord_less_eq_set_assn @ A @ B )
     => ( ( A != bot_bot_set_assn )
       => ( ( finite_finite_assn @ B )
         => ( ord_less_eq_assn @ ( lattic47131356835913163n_assn @ B ) @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_6508_Inf__fin_Osubset__imp,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B )
         => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B ) @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_6509_Inf__fin_Osubset__imp,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( A != bot_bot_set_int )
       => ( ( finite_finite_int @ B )
         => ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ B ) @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_6510_Inf__fin_Osubset__imp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_6511_Inf__fin_Osubset,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( B != bot_bot_set_set_nat )
       => ( ( ord_le6893508408891458716et_nat @ B @ A )
         => ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B ) @ ( lattic3014633134055518761et_nat @ A ) )
            = ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_6512_Inf__fin_Osubset,axiom,
    ! [A: set_assn,B: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( B != bot_bot_set_assn )
       => ( ( ord_less_eq_set_assn @ B @ A )
         => ( ( inf_inf_assn @ ( lattic47131356835913163n_assn @ B ) @ ( lattic47131356835913163n_assn @ A ) )
            = ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_6513_Inf__fin_Osubset,axiom,
    ! [A: set_o,B: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( B != bot_bot_set_o )
       => ( ( ord_less_eq_set_o @ B @ A )
         => ( ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ B ) @ ( lattic4107685809792843317_fin_o @ A ) )
            = ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_6514_Inf__fin_Osubset,axiom,
    ! [A: set_int,B: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( B != bot_bot_set_int )
       => ( ( ord_less_eq_set_int @ B @ A )
         => ( ( inf_inf_int @ ( lattic5235898064620869839in_int @ B ) @ ( lattic5235898064620869839in_int @ A ) )
            = ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_6515_Inf__fin_Osubset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( B != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B @ A )
         => ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
            = ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_6516_Inf__fin_Oclosed,axiom,
    ! [A: set_Pr4532377907799695533_nat_o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ( A != bot_bo7824918357723582233_nat_o )
       => ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( inf_in1318976480646536635_nat_o @ X3 @ Y3 ) @ ( insert5175938949040314269_nat_o @ X3 @ ( insert5175938949040314269_nat_o @ Y3 @ bot_bo7824918357723582233_nat_o ) ) )
         => ( member6576561426505652726_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ A ) ) ) ) ).

% Inf_fin.closed
thf(fact_6517_Inf__fin_Oclosed,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ! [X3: set_nat,Y3: set_nat] : ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y3 @ bot_bot_set_set_nat ) ) )
         => ( member_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ A ) ) ) ) ).

% Inf_fin.closed
thf(fact_6518_Inf__fin_Oclosed,axiom,
    ! [A: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ! [X3: assn,Y3: assn] : ( member_assn @ ( inf_inf_assn @ X3 @ Y3 ) @ ( insert_assn @ X3 @ ( insert_assn @ Y3 @ bot_bot_set_assn ) ) )
         => ( member_assn @ ( lattic47131356835913163n_assn @ A ) @ A ) ) ) ) ).

% Inf_fin.closed
thf(fact_6519_Inf__fin_Oclosed,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ! [X3: $o,Y3: $o] : ( member_o @ ( inf_inf_o @ X3 @ Y3 ) @ ( insert_o @ X3 @ ( insert_o @ Y3 @ bot_bot_set_o ) ) )
         => ( member_o @ ( lattic4107685809792843317_fin_o @ A ) @ A ) ) ) ) ).

% Inf_fin.closed
thf(fact_6520_Inf__fin_Oclosed,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [X3: nat,Y3: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
         => ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).

% Inf_fin.closed
thf(fact_6521_Inf__fin_Oclosed,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ! [X3: int,Y3: int] : ( member_int @ ( inf_inf_int @ X3 @ Y3 ) @ ( insert_int @ X3 @ ( insert_int @ Y3 @ bot_bot_set_int ) ) )
         => ( member_int @ ( lattic5235898064620869839in_int @ A ) @ A ) ) ) ) ).

% Inf_fin.closed
thf(fact_6522_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A )
     => ( ~ ( member6576561426505652726_nat_o @ X @ A )
       => ( ( A != bot_bo7824918357723582233_nat_o )
         => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
            = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_6523_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ~ ( member_set_nat @ X @ A )
       => ( ( A != bot_bot_set_set_nat )
         => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
            = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_6524_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_assn,X: assn] :
      ( ( finite_finite_assn @ A )
     => ( ~ ( member_assn @ X @ A )
       => ( ( A != bot_bot_set_assn )
         => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
            = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_6525_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ~ ( member_o @ X @ A )
       => ( ( A != bot_bot_set_o )
         => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
            = ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_6526_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ~ ( member_nat @ X @ A )
       => ( ( A != bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
            = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_6527_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ~ ( member_int @ X @ A )
       => ( ( A != bot_bot_set_int )
         => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
            = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_6528_Inf__fin_Ounion,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B )
         => ( ( B != bot_bot_set_set_nat )
           => ( ( lattic3014633134055518761et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
              = ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3014633134055518761et_nat @ B ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_6529_Inf__fin_Ounion,axiom,
    ! [A: set_assn,B: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ( finite_finite_assn @ B )
         => ( ( B != bot_bot_set_assn )
           => ( ( lattic47131356835913163n_assn @ ( sup_sup_set_assn @ A @ B ) )
              = ( inf_inf_assn @ ( lattic47131356835913163n_assn @ A ) @ ( lattic47131356835913163n_assn @ B ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_6530_Inf__fin_Ounion,axiom,
    ! [A: set_o,B: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( finite_finite_o @ B )
         => ( ( B != bot_bot_set_o )
           => ( ( lattic4107685809792843317_fin_o @ ( sup_sup_set_o @ A @ B ) )
              = ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic4107685809792843317_fin_o @ B ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_6531_Inf__fin_Ounion,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ( B != bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
              = ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_6532_Inf__fin_Ounion,axiom,
    ! [A: set_int,B: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( finite_finite_int @ B )
         => ( ( B != bot_bot_set_int )
           => ( ( lattic5235898064620869839in_int @ ( sup_sup_set_int @ A @ B ) )
              = ( inf_inf_int @ ( lattic5235898064620869839in_int @ A ) @ ( lattic5235898064620869839in_int @ B ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_6533_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_6534_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_assn] :
      ( ( finite_finite_assn @ A )
     => ( ( A != bot_bot_set_assn )
       => ( ord_less_eq_assn @ ( lattic47131356835913163n_assn @ A ) @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_6535_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_6536_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_6537_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ A ) @ ( lattic1091506334969745077in_int @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_6538_ivl__disj__un__singleton_I4_J,axiom,
    ! [L: $o,U: $o] :
      ( ( ord_less_o @ L @ U )
     => ( ( sup_sup_set_o @ ( set_or1716231572884733764Than_o @ L @ U ) @ ( insert_o @ U @ bot_bot_set_o ) )
        = ( set_or8254209520273421544Most_o @ L @ U ) ) ) ).

% ivl_disj_un_singleton(4)
thf(fact_6539_ivl__disj__un__singleton_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
        = ( set_or6659071591806873216st_nat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(4)
thf(fact_6540_ivl__disj__un__singleton_I4_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_or5832277885323065728an_int @ L @ U ) @ ( insert_int @ U @ bot_bot_set_int ) )
        = ( set_or6656581121297822940st_int @ L @ U ) ) ) ).

% ivl_disj_un_singleton(4)
thf(fact_6541_ivl__disj__un__singleton_I4_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le6747313008572928689nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( set_or4266950643985792945nteger @ L @ U ) @ ( insert_Code_integer @ U @ bot_bo3990330152332043303nteger ) )
        = ( set_or2715278749043346189nteger @ L @ U ) ) ) ).

% ivl_disj_un_singleton(4)
thf(fact_6542_ivl__disj__un__singleton_I4_J,axiom,
    ! [L: code_natural,U: code_natural] :
      ( ( ord_le5570908160329646204atural @ L @ U )
     => ( ( sup_su5306851075354978058atural @ ( set_or3090545795742510460atural @ L @ U ) @ ( insert_Code_natural @ U @ bot_bo8448779972843232370atural ) )
        = ( set_or1538873900800063704atural @ L @ U ) ) ) ).

% ivl_disj_un_singleton(4)
thf(fact_6543_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6544_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6545_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6546_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6547_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6548_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M2 @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_6549_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_6550_Min_Oremove,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( member_o @ X @ A )
       => ( ( lattic1973801136483472281_Min_o @ A )
          = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( ord_min_o @ X @ ( lattic1973801136483472281_Min_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Min.remove
thf(fact_6551_Min_Oremove,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ X @ A )
       => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
              = bot_bot_set_int )
           => ( ( lattic8718645017227715691in_int @ A )
              = X ) )
          & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
             != bot_bot_set_int )
           => ( ( lattic8718645017227715691in_int @ A )
              = ( ord_min_int @ X @ ( lattic8718645017227715691in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).

% Min.remove
thf(fact_6552_Min_Oremove,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic8721135487736765967in_nat @ A )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic8721135487736765967in_nat @ A )
              = ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Min.remove
thf(fact_6553_Min_Oinsert__remove,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ A ) )
        = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X )
          & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( ord_min_o @ X @ ( lattic1973801136483472281_Min_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Min.insert_remove
thf(fact_6554_Min_Oinsert__remove,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
            = bot_bot_set_int )
         => ( ( lattic8718645017227715691in_int @ ( insert_int @ X @ A ) )
            = X ) )
        & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
           != bot_bot_set_int )
         => ( ( lattic8718645017227715691in_int @ ( insert_int @ X @ A ) )
            = ( ord_min_int @ X @ ( lattic8718645017227715691in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).

% Min.insert_remove
thf(fact_6555_Min_Oinsert__remove,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ A ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ A ) )
            = ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Min.insert_remove
thf(fact_6556_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A3: rat,K4: nat] :
          ( if_rat @ ( K4 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K4 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ) ).

% gbinomial_code
thf(fact_6557_bool__assn__proper_I2_J,axiom,
    ( proper
    @ ^ [Uu: produc3658429121746597890et_nat] : $false ) ).

% bool_assn_proper(2)
thf(fact_6558_bool__assn__proper_I3_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( P @ H )
              | ( Q @ H ) ) ) ) ) ).

% bool_assn_proper(3)
thf(fact_6559_bool__assn__proper_I4_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( P @ H )
              & ( Q @ H ) ) ) ) ) ).

% bool_assn_proper(4)
thf(fact_6560_singleton__conv2,axiom,
    ! [A2: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ( ^ [Y4: produc3843707927480180839at_nat,Z4: produc3843707927480180839at_nat] : Y4 = Z4
          @ A2 ) )
      = ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv2
thf(fact_6561_singleton__conv2,axiom,
    ! [A2: produc3658429121746597890et_nat > $o] :
      ( ( collec939566748876313656_nat_o
        @ ( ^ [Y4: produc3658429121746597890et_nat > $o,Z4: produc3658429121746597890et_nat > $o] : Y4 = Z4
          @ A2 ) )
      = ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ).

% singleton_conv2
thf(fact_6562_singleton__conv2,axiom,
    ! [A2: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y4: product_prod_nat_nat,Z4: product_prod_nat_nat] : Y4 = Z4
          @ A2 ) )
      = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_6563_singleton__conv2,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ( ^ [Y4: $o,Z4: $o] : Y4 = Z4
          @ A2 ) )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_6564_singleton__conv2,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ( ^ [Y4: nat,Z4: nat] : Y4 = Z4
          @ A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_6565_singleton__conv2,axiom,
    ! [A2: int] :
      ( ( collect_int
        @ ( ^ [Y4: int,Z4: int] : Y4 = Z4
          @ A2 ) )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_6566_singleton__conv,axiom,
    ! [A2: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X2: produc3843707927480180839at_nat] : X2 = A2 )
      = ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv
thf(fact_6567_singleton__conv,axiom,
    ! [A2: produc3658429121746597890et_nat > $o] :
      ( ( collec939566748876313656_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o] : X2 = A2 )
      = ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ).

% singleton_conv
thf(fact_6568_singleton__conv,axiom,
    ! [A2: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : X2 = A2 )
      = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_6569_singleton__conv,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ^ [X2: $o] : X2 = A2 )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_6570_singleton__conv,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : X2 = A2 )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_6571_singleton__conv,axiom,
    ! [A2: int] :
      ( ( collect_int
        @ ^ [X2: int] : X2 = A2 )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_6572_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec939566748876313656_nat_o
            @ ^ [S5: produc3658429121746597890et_nat > $o] : P )
          = top_to4869887016220417533_nat_o ) )
      & ( ~ P
       => ( ( collec939566748876313656_nat_o
            @ ^ [S5: produc3658429121746597890et_nat > $o] : P )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% Collect_const
thf(fact_6573_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec3392354462482085612at_nat
            @ ^ [S5: product_prod_nat_nat] : P )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P
       => ( ( collec3392354462482085612at_nat
            @ ^ [S5: product_prod_nat_nat] : P )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const
thf(fact_6574_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_int
            @ ^ [S5: int] : P )
          = top_top_set_int ) )
      & ( ~ P
       => ( ( collect_int
            @ ^ [S5: int] : P )
          = bot_bot_set_int ) ) ) ).

% Collect_const
thf(fact_6575_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_o
            @ ^ [S5: $o] : P )
          = top_top_set_o ) )
      & ( ~ P
       => ( ( collect_o
            @ ^ [S5: $o] : P )
          = bot_bot_set_o ) ) ) ).

% Collect_const
thf(fact_6576_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S5: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S5: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_6577_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_Product_unit
            @ ^ [S5: product_unit] : P )
          = top_to1996260823553986621t_unit ) )
      & ( ~ P
       => ( ( collect_Product_unit
            @ ^ [S5: product_unit] : P )
          = bot_bo3957492148770167129t_unit ) ) ) ).

% Collect_const
thf(fact_6578_bool__assn__proper_I5_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( proper
        @ ^ [H: produc3658429121746597890et_nat] :
            ( ( in_range @ H )
            & ~ ( P @ H ) ) ) ) ).

% bool_assn_proper(5)
thf(fact_6579_atLeastLessThan__empty,axiom,
    ! [B2: $o,A2: $o] :
      ( ( ord_less_eq_o @ B2 @ A2 )
     => ( ( set_or7139685690850216873Than_o @ A2 @ B2 )
        = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
thf(fact_6580_atLeastLessThan__empty,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( set_or8404916559141939852nteger @ A2 @ B2 )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
thf(fact_6581_atLeastLessThan__empty,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( set_or5523502641901747543n_assn @ A2 @ B2 )
        = bot_bot_set_assn ) ) ).

% atLeastLessThan_empty
thf(fact_6582_atLeastLessThan__empty,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( set_or3540276404033026485et_nat @ A2 @ B2 )
        = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_6583_atLeastLessThan__empty,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( set_or4662586982721622107an_int @ A2 @ B2 )
        = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_6584_atLeastLessThan__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
        = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_6585_atLeastLessThan__empty__iff2,axiom,
    ! [A2: $o,B2: $o] :
      ( ( bot_bot_set_o
        = ( set_or7139685690850216873Than_o @ A2 @ B2 ) )
      = ( ~ ( ord_less_o @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6586_atLeastLessThan__empty__iff2,axiom,
    ! [A2: assn,B2: assn] :
      ( ( bot_bot_set_assn
        = ( set_or5523502641901747543n_assn @ A2 @ B2 ) )
      = ( ~ ( ord_less_assn @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6587_atLeastLessThan__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int
        = ( set_or4662586982721622107an_int @ A2 @ B2 ) )
      = ( ~ ( ord_less_int @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6588_atLeastLessThan__empty__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or8404916559141939852nteger @ A2 @ B2 ) )
      = ( ~ ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6589_atLeastLessThan__empty__iff2,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( bot_bo8448779972843232370atural
        = ( set_or7228511710898657367atural @ A2 @ B2 ) )
      = ( ~ ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6590_atLeastLessThan__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat
        = ( set_or4665077453230672383an_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_nat @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_6591_atLeastLessThan__empty__iff,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( set_or7139685690850216873Than_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ~ ( ord_less_o @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6592_atLeastLessThan__empty__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ( set_or5523502641901747543n_assn @ A2 @ B2 )
        = bot_bot_set_assn )
      = ( ~ ( ord_less_assn @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6593_atLeastLessThan__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_int @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6594_atLeastLessThan__empty__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A2 @ B2 )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6595_atLeastLessThan__empty__iff,axiom,
    ! [A2: code_natural,B2: code_natural] :
      ( ( ( set_or7228511710898657367atural @ A2 @ B2 )
        = bot_bo8448779972843232370atural )
      = ( ~ ( ord_le5570908160329646204atural @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6596_atLeastLessThan__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_6597_greaterThanLessThan__empty,axiom,
    ! [L: $o,K: $o] :
      ( ( ord_less_eq_o @ L @ K )
     => ( ( set_or1716231572884733764Than_o @ K @ L )
        = bot_bot_set_o ) ) ).

% greaterThanLessThan_empty
thf(fact_6598_greaterThanLessThan__empty,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ K )
     => ( ( set_or4266950643985792945nteger @ K @ L )
        = bot_bo3990330152332043303nteger ) ) ).

% greaterThanLessThan_empty
thf(fact_6599_greaterThanLessThan__empty,axiom,
    ! [L: assn,K: assn] :
      ( ( ord_less_eq_assn @ L @ K )
     => ( ( set_or4250685708352838780n_assn @ K @ L )
        = bot_bot_set_assn ) ) ).

% greaterThanLessThan_empty
thf(fact_6600_greaterThanLessThan__empty,axiom,
    ! [L: set_nat,K: set_nat] :
      ( ( ord_less_eq_set_nat @ L @ K )
     => ( ( set_or8625682525731655386et_nat @ K @ L )
        = bot_bot_set_set_nat ) ) ).

% greaterThanLessThan_empty
thf(fact_6601_greaterThanLessThan__empty,axiom,
    ! [L: nat,K: nat] :
      ( ( ord_less_eq_nat @ L @ K )
     => ( ( set_or5834768355832116004an_nat @ K @ L )
        = bot_bot_set_nat ) ) ).

% greaterThanLessThan_empty
thf(fact_6602_greaterThanLessThan__empty,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( set_or5832277885323065728an_int @ K @ L )
        = bot_bot_set_int ) ) ).

% greaterThanLessThan_empty
thf(fact_6603_Min__singleton,axiom,
    ! [X: $o] :
      ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Min_singleton
thf(fact_6604_Min__singleton,axiom,
    ! [X: nat] :
      ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% Min_singleton
thf(fact_6605_Min__singleton,axiom,
    ! [X: int] :
      ( ( lattic8718645017227715691in_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% Min_singleton
thf(fact_6606_Min_Obounded__iff,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
          = ( ! [X2: $o] :
                ( ( member_o @ X2 @ A )
               => ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_6607_Min_Obounded__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
          = ( ! [X2: code_integer] :
                ( ( member_Code_integer @ X2 @ A )
               => ( ord_le3102999989581377725nteger @ X @ X2 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_6608_Min_Obounded__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A )
               => ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_6609_Min_Obounded__iff,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A )
               => ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_6610_Min__gr__iff,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( ord_less_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
          = ( ! [X2: $o] :
                ( ( member_o @ X2 @ A )
               => ( ord_less_o @ X @ X2 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_6611_Min__gr__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A )
               => ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_6612_Min__gr__iff,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A ) )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A )
               => ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_6613_Min__gr__iff,axiom,
    ! [A: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( A != bot_bo3990330152332043303nteger )
       => ( ( ord_le6747313008572928689nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
          = ( ! [X2: code_integer] :
                ( ( member_Code_integer @ X2 @ A )
               => ( ord_le6747313008572928689nteger @ X @ X2 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_6614_Min__gr__iff,axiom,
    ! [A: set_Code_natural,X: code_natural] :
      ( ( finite4840673202314680255atural @ A )
     => ( ( A != bot_bo8448779972843232370atural )
       => ( ( ord_le5570908160329646204atural @ X @ ( lattic9110812603455646823atural @ A ) )
          = ( ! [X2: code_natural] :
                ( ( member_Code_natural @ X2 @ A )
               => ( ord_le5570908160329646204atural @ X @ X2 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_6615_prod_OatMost__Suc,axiom,
    ! [G: nat > code_integer,N: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6616_prod_OatMost__Suc,axiom,
    ! [G: nat > assn,N: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6617_prod_OatMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6618_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6619_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6620_range__constant,axiom,
    ! [X: $o] :
      ( ( image_o_o
        @ ^ [Uu: $o] : X
        @ top_top_set_o )
      = ( insert_o @ X @ bot_bot_set_o ) ) ).

% range_constant
thf(fact_6621_range__constant,axiom,
    ! [X: nat] :
      ( ( image_o_nat
        @ ^ [Uu: $o] : X
        @ top_top_set_o )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% range_constant
thf(fact_6622_range__constant,axiom,
    ! [X: int] :
      ( ( image_o_int
        @ ^ [Uu: $o] : X
        @ top_top_set_o )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% range_constant
thf(fact_6623_range__constant,axiom,
    ! [X: $o] :
      ( ( image_nat_o
        @ ^ [Uu: nat] : X
        @ top_top_set_nat )
      = ( insert_o @ X @ bot_bot_set_o ) ) ).

% range_constant
thf(fact_6624_range__constant,axiom,
    ! [X: nat] :
      ( ( image_nat_nat
        @ ^ [Uu: nat] : X
        @ top_top_set_nat )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% range_constant
thf(fact_6625_range__constant,axiom,
    ! [X: int] :
      ( ( image_nat_int
        @ ^ [Uu: nat] : X
        @ top_top_set_nat )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% range_constant
thf(fact_6626_range__constant,axiom,
    ! [X: $o] :
      ( ( image_Product_unit_o
        @ ^ [Uu: product_unit] : X
        @ top_to1996260823553986621t_unit )
      = ( insert_o @ X @ bot_bot_set_o ) ) ).

% range_constant
thf(fact_6627_range__constant,axiom,
    ! [X: nat] :
      ( ( image_875570014554754200it_nat
        @ ^ [Uu: product_unit] : X
        @ top_to1996260823553986621t_unit )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% range_constant
thf(fact_6628_range__constant,axiom,
    ! [X: int] :
      ( ( image_873079544045703924it_int
        @ ^ [Uu: product_unit] : X
        @ top_to1996260823553986621t_unit )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% range_constant
thf(fact_6629_range__constant,axiom,
    ! [X: set_nat] :
      ( ( image_nat_set_nat
        @ ^ [Uu: nat] : X
        @ top_top_set_nat )
      = ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ).

% range_constant
thf(fact_6630_atLeastLessThan__singleton,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
      = ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_6631_Min__const,axiom,
    ! [A: set_nat,C2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic8721135487736765967in_nat
            @ ( image_nat_nat
              @ ^ [Uu: nat] : C2
              @ A ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_6632_Min__insert,axiom,
    ! [A: set_o,X: $o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ A ) )
          = ( ord_min_o @ X @ ( lattic1973801136483472281_Min_o @ A ) ) ) ) ) ).

% Min_insert
thf(fact_6633_Min__insert,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ A ) )
          = ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).

% Min_insert
thf(fact_6634_Min__insert,axiom,
    ! [A: set_int,X: int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ( ( lattic8718645017227715691in_int @ ( insert_int @ X @ A ) )
          = ( ord_min_int @ X @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).

% Min_insert
thf(fact_6635_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6636_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6637_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6638_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6639_prod_Oop__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_6640_lambda__zero,axiom,
    ( ( ^ [H: code_integer] : zero_z3403309356797280102nteger )
    = ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger ) ) ).

% lambda_zero
thf(fact_6641_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_6642_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_6643_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_6644_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_6645_top__empty__eq2,axiom,
    ( top_to8112647782992986859eger_o
    = ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ top_to7512759353274530428nteger ) ) ) ).

% top_empty_eq2
thf(fact_6646_top__empty__eq2,axiom,
    ( top_to2428096842796733269_nat_o
    = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ top_to8753217654552796900et_nat ) ) ) ).

% top_empty_eq2
thf(fact_6647_top__empty__eq2,axiom,
    ( top_to7190503160269336793_nat_o
    = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ top_to1599102959340997728et_nat ) ) ) ).

% top_empty_eq2
thf(fact_6648_top__empty__eq2,axiom,
    ( top_to6261655714344447806_int_o
    = ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ top_to3181862456159035625nt_int ) ) ) ).

% top_empty_eq2
thf(fact_6649_top__empty__eq2,axiom,
    ( top_to6513043852502318900_int_o
    = ( ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ top_to2069137843433766899nt_int ) ) ) ).

% top_empty_eq2
thf(fact_6650_inf__Int__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_inf_nat_nat_o
        @ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( inf_in2572325071724192079at_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_6651_inf__Int__eq2,axiom,
    ! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
      ( ( inf_in3563340378267265229eger_o
        @ ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ ( inf_in3408396081536963678nteger @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_6652_inf__Int__eq2,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( inf_in3295504058751909687_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ ( inf_in1768905781608824518et_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_6653_inf__Int__eq2,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( inf_in2641120393918057659_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ ( inf_in3088352823822785602et_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_6654_inf__Int__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( inf_in8045883074377559328_int_o
        @ ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ ( inf_in2576786181675586251nt_int @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_6655_inf__Int__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( inf_in8201999774870019094_int_o
        @ ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ R )
        @ ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ ( inf_in8806012763027673301nt_int @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_6656_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_6657_pred__equals__eq2,axiom,
    ! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
      ( ( ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_6658_pred__equals__eq2,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_6659_pred__equals__eq2,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_6660_pred__equals__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_6661_pred__equals__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( ( ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_6662_sup__Un__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( sup_su6327502436637775413at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_6663_sup__Un__eq2,axiom,
    ! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
      ( ( sup_su234547053653886311eger_o
        @ ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ ( sup_su4591284015454442744nteger @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_6664_sup__Un__eq2,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( sup_su1630790145277462993_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ ( sup_su7128418612487073120et_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_6665_sup__Un__eq2,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( sup_su6535292691877529429_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ ( sup_su8975264963432250076et_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_6666_sup__Un__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( sup_su4182031696650224058_int_o
        @ ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ ( sup_su3382966977382714213nt_int @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_6667_sup__Un__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( sup_su600977994968626096_int_o
        @ ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ R )
        @ ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ ( sup_su3298353300217089135nt_int @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_6668_sup__Un__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su7519161239522478338_nat_o
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y2 ) @ S ) )
      = ( ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y2 ) @ ( sup_su5525570899277871387at_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_6669_Set_Oempty__def,axiom,
    ( bot_bo7824918357723582233_nat_o
    = ( collec939566748876313656_nat_o
      @ ^ [X2: produc3658429121746597890et_nat > $o] : $false ) ) ).

% Set.empty_def
thf(fact_6670_Set_Oempty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X2: product_prod_nat_nat] : $false ) ) ).

% Set.empty_def
thf(fact_6671_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X2: $o] : $false ) ) ).

% Set.empty_def
thf(fact_6672_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X2: nat] : $false ) ) ).

% Set.empty_def
thf(fact_6673_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X2: int] : $false ) ) ).

% Set.empty_def
thf(fact_6674_bot__empty__eq2,axiom,
    ( bot_bo3000040243691356879eger_o
    = ( ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ bot_bo5443222936135328352nteger ) ) ) ).

% bot_empty_eq2
thf(fact_6675_bot__empty__eq2,axiom,
    ( bot_bo5580076615179976505_nat_o
    = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ bot_bo1481135142794719944et_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6676_bot__empty__eq2,axiom,
    ( bot_bo3790638025767943357_nat_o
    = ( ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ bot_bo5635537948650799172et_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6677_bot__empty__eq2,axiom,
    ( bot_bo8662317086119403298_int_o
    = ( ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ bot_bo572930865798478029nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_6678_bot__empty__eq2,axiom,
    ( bot_bo1403522918969695512_int_o
    = ( ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ bot_bo4508923176915781079nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_6679_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6680_lambda__one,axiom,
    ( ( ^ [X2: code_integer] : X2 )
    = ( times_3573771949741848930nteger @ one_one_Code_integer ) ) ).

% lambda_one
thf(fact_6681_lambda__one,axiom,
    ( ( ^ [X2: assn] : X2 )
    = ( times_times_assn @ one_one_assn ) ) ).

% lambda_one
thf(fact_6682_lambda__one,axiom,
    ( ( ^ [X2: rat] : X2 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_6683_lambda__one,axiom,
    ( ( ^ [X2: nat] : X2 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_6684_lambda__one,axiom,
    ( ( ^ [X2: int] : X2 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_6685_Collect__conv__if2,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A2: produc3843707927480180839at_nat] :
      ( ( ( P @ A2 )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X2: produc3843707927480180839at_nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X2: produc3843707927480180839at_nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6686_Collect__conv__if2,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,A2: produc3658429121746597890et_nat > $o] :
      ( ( ( P @ A2 )
       => ( ( collec939566748876313656_nat_o
            @ ^ [X2: produc3658429121746597890et_nat > $o] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec939566748876313656_nat_o
            @ ^ [X2: produc3658429121746597890et_nat > $o] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% Collect_conv_if2
thf(fact_6687_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
      ( ( ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6688_Collect__conv__if2,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_6689_Collect__conv__if2,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6690_Collect__conv__if2,axiom,
    ! [P: int > $o,A2: int] :
      ( ( ( P @ A2 )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_int @ A2 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_6691_Collect__conv__if,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A2: produc3843707927480180839at_nat] :
      ( ( ( P @ A2 )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X2: produc3843707927480180839at_nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert9069300056098147895at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X2: produc3843707927480180839at_nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if
thf(fact_6692_Collect__conv__if,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,A2: produc3658429121746597890et_nat > $o] :
      ( ( ( P @ A2 )
       => ( ( collec939566748876313656_nat_o
            @ ^ [X2: produc3658429121746597890et_nat > $o] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec939566748876313656_nat_o
            @ ^ [X2: produc3658429121746597890et_nat > $o] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% Collect_conv_if
thf(fact_6693_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
      ( ( ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_6694_Collect__conv__if,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_6695_Collect__conv__if,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_6696_Collect__conv__if,axiom,
    ! [P: int > $o,A2: int] :
      ( ( ( P @ A2 )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_int @ A2 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_6697_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_6698_pred__subset__eq2,axiom,
    ! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
      ( ( ord_le4340812435750786203eger_o
        @ ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) @ S ) )
      = ( ord_le653643898420964396nteger @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_6699_pred__subset__eq2,axiom,
    ! [R: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ( ord_le8000401564054156549_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) @ S ) )
      = ( ord_le5966269811547037844et_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_6700_pred__subset__eq2,axiom,
    ! [R: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ( ord_le6753239538765779593_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) @ S ) )
      = ( ord_le4763372923235995152et_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_6701_pred__subset__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ( ord_le5643404153117327598_int_o
        @ ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ R )
        @ ^ [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) @ S ) )
      = ( ord_le8725513860283290265nt_int @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_6702_pred__subset__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ( ord_le2124322318746777828_int_o
        @ ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ R )
        @ ^ [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) @ S ) )
      = ( ord_le135402666524580259nt_int @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_6703_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N: nat,P5: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ P5 )
       => ( ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ N @ P5 ) ) )
          = ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ P5 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6704_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N: nat,P5: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ P5 )
       => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ N @ P5 ) ) )
          = ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ P5 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6705_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N: nat,P5: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ P5 )
       => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ N @ P5 ) ) )
          = ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ P5 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6706_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N: nat,P5: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ P5 )
       => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ P5 ) ) )
          = ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ P5 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6707_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N: nat,P5: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ P5 )
       => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N @ P5 ) ) )
          = ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ P5 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_6708_ivl__disj__int__one_I8_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or7139685690850216873Than_o @ L @ U ) @ ( set_ord_atLeast_o @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(8)
thf(fact_6709_ivl__disj__int__one_I8_J,axiom,
    ! [L: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or4662586982721622107an_int @ L @ U ) @ ( set_ord_atLeast_int @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_one(8)
thf(fact_6710_ivl__disj__int__one_I8_J,axiom,
    ! [L: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( set_ord_atLeast_nat @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_one(8)
thf(fact_6711_ivl__disj__int__two_I1_J,axiom,
    ! [L: $o,M2: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or1716231572884733764Than_o @ L @ M2 ) @ ( set_or7139685690850216873Than_o @ M2 @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_two(1)
thf(fact_6712_ivl__disj__int__two_I1_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or5832277885323065728an_int @ L @ M2 ) @ ( set_or4662586982721622107an_int @ M2 @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_two(1)
thf(fact_6713_ivl__disj__int__two_I1_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_two(1)
thf(fact_6714_ivl__disj__int__one_I6_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_or1716231572884733764Than_o @ L @ U ) @ ( set_ord_atLeast_o @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(6)
thf(fact_6715_ivl__disj__int__one_I6_J,axiom,
    ! [L: int,U: int] :
      ( ( inf_inf_set_int @ ( set_or5832277885323065728an_int @ L @ U ) @ ( set_ord_atLeast_int @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_one(6)
thf(fact_6716_ivl__disj__int__one_I6_J,axiom,
    ! [L: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L @ U ) @ ( set_ord_atLeast_nat @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_one(6)
thf(fact_6717_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_3573771949741848930nteger @ ( G @ M2 )
          @ ( groups3455450783089532116nteger
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6718_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_assn @ ( G @ M2 )
          @ ( groups6906906614972039071t_assn
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6719_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ M2 )
          @ ( groups73079841787564623at_rat
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6720_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ M2 )
          @ ( groups708209901874060359at_nat
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6721_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ M2 )
          @ ( groups705719431365010083at_int
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6722_Min__add__commute,axiom,
    ! [S: set_o,F: $o > code_integer,K: code_integer] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( lattic1063845414844153500nteger
            @ ( image_o_Code_integer
              @ ^ [X2: $o] : ( plus_p5714425477246183910nteger @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_p5714425477246183910nteger @ ( lattic1063845414844153500nteger @ ( image_o_Code_integer @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6723_Min__add__commute,axiom,
    ! [S: set_o,F: $o > rat,K: rat] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( lattic8086005427650270231in_rat
            @ ( image_o_rat
              @ ^ [X2: $o] : ( plus_plus_rat @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_plus_rat @ ( lattic8086005427650270231in_rat @ ( image_o_rat @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6724_Min__add__commute,axiom,
    ! [S: set_o,F: $o > nat,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( lattic8721135487736765967in_nat
            @ ( image_o_nat
              @ ^ [X2: $o] : ( plus_plus_nat @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_plus_nat @ ( lattic8721135487736765967in_nat @ ( image_o_nat @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6725_Min__add__commute,axiom,
    ! [S: set_o,F: $o > int,K: int] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( lattic8718645017227715691in_int
            @ ( image_o_int
              @ ^ [X2: $o] : ( plus_plus_int @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_plus_int @ ( lattic8718645017227715691in_int @ ( image_o_int @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6726_Min__add__commute,axiom,
    ! [S: set_nat,F: nat > code_integer,K: code_integer] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( lattic1063845414844153500nteger
            @ ( image_1215581382706833972nteger
              @ ^ [X2: nat] : ( plus_p5714425477246183910nteger @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_p5714425477246183910nteger @ ( lattic1063845414844153500nteger @ ( image_1215581382706833972nteger @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6727_Min__add__commute,axiom,
    ! [S: set_nat,F: nat > rat,K: rat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( lattic8086005427650270231in_rat
            @ ( image_nat_rat
              @ ^ [X2: nat] : ( plus_plus_rat @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_plus_rat @ ( lattic8086005427650270231in_rat @ ( image_nat_rat @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6728_Min__add__commute,axiom,
    ! [S: set_nat,F: nat > nat,K: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( lattic8721135487736765967in_nat
            @ ( image_nat_nat
              @ ^ [X2: nat] : ( plus_plus_nat @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_plus_nat @ ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6729_Min__add__commute,axiom,
    ! [S: set_nat,F: nat > int,K: int] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( lattic8718645017227715691in_int
            @ ( image_nat_int
              @ ^ [X2: nat] : ( plus_plus_int @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_plus_int @ ( lattic8718645017227715691in_int @ ( image_nat_int @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6730_Min__add__commute,axiom,
    ! [S: set_int,F: int > code_integer,K: code_integer] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ( ( lattic1063845414844153500nteger
            @ ( image_1587234942943678608nteger
              @ ^ [X2: int] : ( plus_p5714425477246183910nteger @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_p5714425477246183910nteger @ ( lattic1063845414844153500nteger @ ( image_1587234942943678608nteger @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6731_Min__add__commute,axiom,
    ! [S: set_int,F: int > rat,K: rat] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ( ( lattic8086005427650270231in_rat
            @ ( image_int_rat
              @ ^ [X2: int] : ( plus_plus_rat @ ( F @ X2 ) @ K )
              @ S ) )
          = ( plus_plus_rat @ ( lattic8086005427650270231in_rat @ ( image_int_rat @ F @ S ) ) @ K ) ) ) ) ).

% Min_add_commute
thf(fact_6732_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > code_integer,N: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_3573771949741848930nteger @ ( G @ zero_zero_nat )
        @ ( groups3455450783089532116nteger
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6733_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > assn,N: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_assn @ ( G @ zero_zero_nat )
        @ ( groups6906906614972039071t_assn
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6734_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6735_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6736_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6737_bot__assn__def,axiom,
    ( bot_bot_assn
    = ( abs_assn
      @ ^ [Uu: produc3658429121746597890et_nat] : $false ) ) ).

% bot_assn_def
thf(fact_6738_prod__atLeastAtMost__code,axiom,
    ! [F: nat > code_integer,A2: nat,B2: nat] :
      ( ( groups3455450783089532116nteger @ F @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
      = ( set_fo1084959871951514735nteger
        @ ^ [A3: nat] : ( times_3573771949741848930nteger @ ( F @ A3 ) )
        @ A2
        @ B2
        @ one_one_Code_integer ) ) ).

% prod_atLeastAtMost_code
thf(fact_6739_prod__atLeastAtMost__code,axiom,
    ! [F: nat > assn,A2: nat,B2: nat] :
      ( ( groups6906906614972039071t_assn @ F @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
      = ( set_fo1959793692361082170t_assn
        @ ^ [A3: nat] : ( times_times_assn @ ( F @ A3 ) )
        @ A2
        @ B2
        @ one_one_assn ) ) ).

% prod_atLeastAtMost_code
thf(fact_6740_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A2: nat,B2: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A3: nat] : ( times_times_rat @ ( F @ A3 ) )
        @ A2
        @ B2
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6741_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A2: nat,B2: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A3: nat] : ( times_times_nat @ ( F @ A3 ) )
        @ A2
        @ B2
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6742_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A2: nat,B2: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A3: nat] : ( times_times_int @ ( F @ A3 ) )
        @ A2
        @ B2
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_6743_not__empty__eq__Ici__eq__empty,axiom,
    ! [L: $o] :
      ( bot_bot_set_o
     != ( set_ord_atLeast_o @ L ) ) ).

% not_empty_eq_Ici_eq_empty
thf(fact_6744_not__empty__eq__Ici__eq__empty,axiom,
    ! [L: int] :
      ( bot_bot_set_int
     != ( set_ord_atLeast_int @ L ) ) ).

% not_empty_eq_Ici_eq_empty
thf(fact_6745_not__empty__eq__Ici__eq__empty,axiom,
    ! [L: nat] :
      ( bot_bot_set_nat
     != ( set_ord_atLeast_nat @ L ) ) ).

% not_empty_eq_Ici_eq_empty
thf(fact_6746_image__constant,axiom,
    ! [X: $o,A: set_o,C2: $o] :
      ( ( member_o @ X @ A )
     => ( ( image_o_o
          @ ^ [X2: $o] : C2
          @ A )
        = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_6747_image__constant,axiom,
    ! [X: nat,A: set_nat,C2: $o] :
      ( ( member_nat @ X @ A )
     => ( ( image_nat_o
          @ ^ [X2: nat] : C2
          @ A )
        = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_6748_image__constant,axiom,
    ! [X: int,A: set_int,C2: $o] :
      ( ( member_int @ X @ A )
     => ( ( image_int_o
          @ ^ [X2: int] : C2
          @ A )
        = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_6749_image__constant,axiom,
    ! [X: $o,A: set_o,C2: nat] :
      ( ( member_o @ X @ A )
     => ( ( image_o_nat
          @ ^ [X2: $o] : C2
          @ A )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_6750_image__constant,axiom,
    ! [X: nat,A: set_nat,C2: nat] :
      ( ( member_nat @ X @ A )
     => ( ( image_nat_nat
          @ ^ [X2: nat] : C2
          @ A )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_6751_image__constant,axiom,
    ! [X: int,A: set_int,C2: nat] :
      ( ( member_int @ X @ A )
     => ( ( image_int_nat
          @ ^ [X2: int] : C2
          @ A )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_6752_image__constant,axiom,
    ! [X: $o,A: set_o,C2: int] :
      ( ( member_o @ X @ A )
     => ( ( image_o_int
          @ ^ [X2: $o] : C2
          @ A )
        = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ).

% image_constant
thf(fact_6753_image__constant,axiom,
    ! [X: nat,A: set_nat,C2: int] :
      ( ( member_nat @ X @ A )
     => ( ( image_nat_int
          @ ^ [X2: nat] : C2
          @ A )
        = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ).

% image_constant
thf(fact_6754_image__constant,axiom,
    ! [X: int,A: set_int,C2: int] :
      ( ( member_int @ X @ A )
     => ( ( image_int_int
          @ ^ [X2: int] : C2
          @ A )
        = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ).

% image_constant
thf(fact_6755_image__constant,axiom,
    ! [X: nat,A: set_nat,C2: set_nat] :
      ( ( member_nat @ X @ A )
     => ( ( image_nat_set_nat
          @ ^ [X2: nat] : C2
          @ A )
        = ( insert_set_nat @ C2 @ bot_bot_set_set_nat ) ) ) ).

% image_constant
thf(fact_6756_image__constant__conv,axiom,
    ! [A: set_o,C2: $o] :
      ( ( ( A = bot_bot_set_o )
       => ( ( image_o_o
            @ ^ [X2: $o] : C2
            @ A )
          = bot_bot_set_o ) )
      & ( ( A != bot_bot_set_o )
       => ( ( image_o_o
            @ ^ [X2: $o] : C2
            @ A )
          = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_6757_image__constant__conv,axiom,
    ! [A: set_o,C2: nat] :
      ( ( ( A = bot_bot_set_o )
       => ( ( image_o_nat
            @ ^ [X2: $o] : C2
            @ A )
          = bot_bot_set_nat ) )
      & ( ( A != bot_bot_set_o )
       => ( ( image_o_nat
            @ ^ [X2: $o] : C2
            @ A )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_6758_image__constant__conv,axiom,
    ! [A: set_o,C2: int] :
      ( ( ( A = bot_bot_set_o )
       => ( ( image_o_int
            @ ^ [X2: $o] : C2
            @ A )
          = bot_bot_set_int ) )
      & ( ( A != bot_bot_set_o )
       => ( ( image_o_int
            @ ^ [X2: $o] : C2
            @ A )
          = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ) ).

% image_constant_conv
thf(fact_6759_image__constant__conv,axiom,
    ! [A: set_nat,C2: $o] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( image_nat_o
            @ ^ [X2: nat] : C2
            @ A )
          = bot_bot_set_o ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( image_nat_o
            @ ^ [X2: nat] : C2
            @ A )
          = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_6760_image__constant__conv,axiom,
    ! [A: set_nat,C2: nat] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( image_nat_nat
            @ ^ [X2: nat] : C2
            @ A )
          = bot_bot_set_nat ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( image_nat_nat
            @ ^ [X2: nat] : C2
            @ A )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_6761_image__constant__conv,axiom,
    ! [A: set_nat,C2: int] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( image_nat_int
            @ ^ [X2: nat] : C2
            @ A )
          = bot_bot_set_int ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( image_nat_int
            @ ^ [X2: nat] : C2
            @ A )
          = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ) ).

% image_constant_conv
thf(fact_6762_image__constant__conv,axiom,
    ! [A: set_int,C2: $o] :
      ( ( ( A = bot_bot_set_int )
       => ( ( image_int_o
            @ ^ [X2: int] : C2
            @ A )
          = bot_bot_set_o ) )
      & ( ( A != bot_bot_set_int )
       => ( ( image_int_o
            @ ^ [X2: int] : C2
            @ A )
          = ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_6763_image__constant__conv,axiom,
    ! [A: set_int,C2: nat] :
      ( ( ( A = bot_bot_set_int )
       => ( ( image_int_nat
            @ ^ [X2: int] : C2
            @ A )
          = bot_bot_set_nat ) )
      & ( ( A != bot_bot_set_int )
       => ( ( image_int_nat
            @ ^ [X2: int] : C2
            @ A )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_6764_image__constant__conv,axiom,
    ! [A: set_int,C2: int] :
      ( ( ( A = bot_bot_set_int )
       => ( ( image_int_int
            @ ^ [X2: int] : C2
            @ A )
          = bot_bot_set_int ) )
      & ( ( A != bot_bot_set_int )
       => ( ( image_int_int
            @ ^ [X2: int] : C2
            @ A )
          = ( insert_int @ C2 @ bot_bot_set_int ) ) ) ) ).

% image_constant_conv
thf(fact_6765_image__constant__conv,axiom,
    ! [A: set_nat,C2: set_nat] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( image_nat_set_nat
            @ ^ [X2: nat] : C2
            @ A )
          = bot_bot_set_set_nat ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( image_nat_set_nat
            @ ^ [X2: nat] : C2
            @ A )
          = ( insert_set_nat @ C2 @ bot_bot_set_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_6766_gbinomial__mult__fact,axiom,
    ! [K: nat,A2: rat] :
      ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( gbinomial_rat @ A2 @ K ) )
      = ( groups73079841787564623at_rat
        @ ^ [I2: nat] : ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ I2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).

% gbinomial_mult_fact
thf(fact_6767_gbinomial__mult__fact_H,axiom,
    ! [A2: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A2 @ K ) @ ( semiri773545260158071498ct_rat @ K ) )
      = ( groups73079841787564623at_rat
        @ ^ [I2: nat] : ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ I2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).

% gbinomial_mult_fact'
thf(fact_6768_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > code_integer,N: nat] :
      ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_6769_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > assn,N: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_6770_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_6771_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_6772_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_6773_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N: nat,G: nat > code_integer] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
        = ( times_3573771949741848930nteger @ ( G @ M2 ) @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_6774_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N: nat,G: nat > assn] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
        = ( times_times_assn @ ( G @ M2 ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_6775_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
        = ( times_times_rat @ ( G @ M2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_6776_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
        = ( times_times_nat @ ( G @ M2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_6777_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
        = ( times_times_int @ ( G @ M2 ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_6778_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ A2 @ ( suc @ B2 ) ) )
        = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_6779_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ A2 @ ( suc @ B2 ) ) )
        = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_6780_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A2 @ ( suc @ B2 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_6781_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A2 @ ( suc @ B2 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_6782_prod_OatLeastLessThan__Suc,axiom,
    ! [A2: nat,B2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A2 @ ( suc @ B2 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_6783_prod_Olast__plus,axiom,
    ! [M2: nat,N: nat,G: nat > code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_3573771949741848930nteger @ ( G @ N ) @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ) ).

% prod.last_plus
thf(fact_6784_prod_Olast__plus,axiom,
    ! [M2: nat,N: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_assn @ ( G @ N ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ) ).

% prod.last_plus
thf(fact_6785_prod_Olast__plus,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_rat @ ( G @ N ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ) ).

% prod.last_plus
thf(fact_6786_prod_Olast__plus,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_nat @ ( G @ N ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ) ).

% prod.last_plus
thf(fact_6787_prod_Olast__plus,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_int @ ( G @ N ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ) ).

% prod.last_plus
thf(fact_6788_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C2 @ Y )
       => ( ( image_nat_nat
            @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ 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
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_6789_sup__assn__def,axiom,
    ( sup_sup_assn
    = ( ^ [P3: assn,Q3: assn] :
          ( abs_assn
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P3 @ H )
              | ( rep_assn @ Q3 @ H ) ) ) ) ) ).

% sup_assn_def
thf(fact_6790_inf__assn__def,axiom,
    ( inf_inf_assn
    = ( ^ [P3: assn,Q3: assn] :
          ( abs_assn
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P3 @ H )
              & ( rep_assn @ Q3 @ H ) ) ) ) ) ).

% inf_assn_def
thf(fact_6791_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_6792_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_6793_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_6794_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_code(3)
thf(fact_6795_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera5444537566228673987atural @ ( bit1 @ N ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ ( numera5444537566228673987atural @ N ) ) @ one_one_Code_natural ) ) ).

% numeral_code(3)
thf(fact_6796_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_6797_power__numeral__even,axiom,
    ! [Z: assn,W: num] :
      ( ( power_power_assn @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_assn @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_6798_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_6799_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_6800_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_6801_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ Z @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6802_power__numeral__odd,axiom,
    ! [Z: assn,W: num] :
      ( ( power_power_assn @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_assn @ ( times_times_assn @ Z @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6803_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6804_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6805_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6806_prod_Ohead__if,axiom,
    ! [N: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.head_if
thf(fact_6807_prod_Ohead__if,axiom,
    ! [N: nat,M2: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.head_if
thf(fact_6808_prod_Ohead__if,axiom,
    ! [N: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.head_if
thf(fact_6809_prod_Ohead__if,axiom,
    ! [N: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.head_if
thf(fact_6810_prod_Ohead__if,axiom,
    ! [N: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ).

% prod.head_if
thf(fact_6811_bot_Oordering__top__axioms,axiom,
    ( orderi2172309028950807442at_nat
    @ ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ Y2 @ X2 )
    @ ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( ord_le7866589430770878221at_nat @ Y2 @ X2 )
    @ bot_bo2099793752762293965at_nat ) ).

% bot.ordering_top_axioms
thf(fact_6812_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_o
    @ ^ [X2: set_o,Y2: set_o] : ( ord_less_eq_set_o @ Y2 @ X2 )
    @ ^ [X2: set_o,Y2: set_o] : ( ord_less_set_o @ Y2 @ X2 )
    @ bot_bot_set_o ) ).

% bot.ordering_top_axioms
thf(fact_6813_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_int
    @ ^ [X2: set_int,Y2: set_int] : ( ord_less_eq_set_int @ Y2 @ X2 )
    @ ^ [X2: set_int,Y2: set_int] : ( ord_less_set_int @ Y2 @ X2 )
    @ bot_bot_set_int ) ).

% bot.ordering_top_axioms
thf(fact_6814_bot_Oordering__top__axioms,axiom,
    ( ordering_top_assn
    @ ^ [X2: assn,Y2: assn] : ( ord_less_eq_assn @ Y2 @ X2 )
    @ ^ [X2: assn,Y2: assn] : ( ord_less_assn @ Y2 @ X2 )
    @ bot_bot_assn ) ).

% bot.ordering_top_axioms
thf(fact_6815_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_nat
    @ ^ [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ X2 )
    @ ^ [X2: set_nat,Y2: set_nat] : ( ord_less_set_nat @ Y2 @ X2 )
    @ bot_bot_set_nat ) ).

% bot.ordering_top_axioms
thf(fact_6816_bot_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X2 )
    @ ^ [X2: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X2 )
    @ bot_bot_nat ) ).

% bot.ordering_top_axioms
thf(fact_6817_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: $o,U: $o] :
      ( ( ord_less_o @ L @ U )
     => ( ( sup_sup_set_o @ ( insert_o @ L @ bot_bot_set_o ) @ ( set_or1716231572884733764Than_o @ L @ U ) )
        = ( set_or7139685690850216873Than_o @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_6818_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_int @ L @ U )
     => ( ( sup_sup_set_int @ ( insert_int @ L @ bot_bot_set_int ) @ ( set_or5832277885323065728an_int @ L @ U ) )
        = ( set_or4662586982721622107an_int @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_6819_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le6747313008572928689nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( insert_Code_integer @ L @ bot_bo3990330152332043303nteger ) @ ( set_or4266950643985792945nteger @ L @ U ) )
        = ( set_or8404916559141939852nteger @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_6820_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: code_natural,U: code_natural] :
      ( ( ord_le5570908160329646204atural @ L @ U )
     => ( ( sup_su5306851075354978058atural @ ( insert_Code_natural @ L @ bot_bo8448779972843232370atural ) @ ( set_or3090545795742510460atural @ L @ U ) )
        = ( set_or7228511710898657367atural @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_6821_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
        = ( set_or4665077453230672383an_nat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_6822_prod_Oin__pairs,axiom,
    ! [G: nat > assn,M2: nat,N: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6906906614972039071t_assn
        @ ^ [I2: nat] : ( times_times_assn @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_6823_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M2: nat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I2: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_6824_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I2: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_6825_prod_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I2: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_6826_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_6827_uminus__assn__def,axiom,
    ( uminus_uminus_assn
    = ( ^ [P3: assn] :
          ( abs_assn
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( in_range @ H )
              & ~ ( rep_assn @ P3 @ H ) ) ) ) ) ).

% uminus_assn_def
thf(fact_6828_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_6829_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_6830_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_6831_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_6832_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_6833_rat__abs__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( abs_abs_int @ A3 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_abs_code
thf(fact_6834_rat__uminus__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_uminus_code
thf(fact_6835_rat__minus__code,axiom,
    ! [P5: rat,Q5: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P5 @ Q5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ B3 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
            @ ( quotient_of @ Q5 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_minus_code
thf(fact_6836_rat__divide__code,axiom,
    ! [P5: rat,Q5: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P5 @ Q5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ C4 @ B3 ) ) )
            @ ( quotient_of @ Q5 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_divide_code
thf(fact_6837_rat__times__code,axiom,
    ! [P5: rat,Q5: rat] :
      ( ( quotient_of @ ( times_times_rat @ P5 @ Q5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ C4 @ D4 ) ) )
            @ ( quotient_of @ Q5 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_times_code
thf(fact_6838_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q8: nat] : ( product_Pair_nat_nat @ ( suc @ Q8 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_6839_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_6840_rat__plus__code,axiom,
    ! [P5: rat,Q5: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P5 @ Q5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D4 ) @ ( times_times_int @ B3 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
            @ ( quotient_of @ Q5 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_plus_code
thf(fact_6841_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_6842_rat__inverse__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B3 ) @ ( abs_abs_int @ A3 ) ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_inverse_code
thf(fact_6843_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_6844_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_6845_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_6846_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.semilattice_neutr_axioms
thf(fact_6847_Gcd__in,axiom,
    ! [A: set_nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( member_nat @ A5 @ A )
         => ( ( member_nat @ B5 @ A )
           => ( member_nat @ ( gcd_gcd_nat @ A5 @ B5 ) @ A ) ) )
     => ( ( A != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A ) @ A ) ) ) ).

% Gcd_in
thf(fact_6848_uminus__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
    @ uminus_uminus_rat ) ).

% uminus_rat.transfer
thf(fact_6849_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_6850_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X2: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_6851_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X2: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_6852_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V2 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) )
          @ Xa
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_6853_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y3: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ).

% eq_Abs_Integ
thf(fact_6854_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y4: int,Z4: int] : Y4 = Z4
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y4: int,Z4: int] : Y4 = Z4
      @ ratrel )
    @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) )
    @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ).

% Fract.rsp
thf(fact_6855_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X2: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) )
    @ ^ [X2: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ).

% times_rat.rsp
thf(fact_6856_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V2 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_6857_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_6858_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X2: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) )
    @ ^ [X2: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ).

% plus_rat.rsp
thf(fact_6859_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V2 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_6860_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_6861_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_6862_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y4: nat,Z4: nat] : Y4 = Z4
    @ pcr_int
    @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_6863_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_6864_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_6865_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_6866_uminus__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
    @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ).

% uminus_rat.rsp
thf(fact_6867_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V2 ) ) )
          @ Xa
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_6868_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) )
          @ Xa
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_6869_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ).

% inverse_rat.rsp
thf(fact_6870_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K4 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_6871_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K4 @ L2 ) @ ( modulo364778990260209775nteger @ K4 @ L2 ) ) ) ) ).

% divmod_integer_def
thf(fact_6872_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V2 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V2 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_6873_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_6874_zero__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).

% zero_int.rsp
thf(fact_6875_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) ) ) ).

% uminus_int.rsp
thf(fact_6876_one__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).

% one_int.rsp
thf(fact_6877_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_6878_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V2 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_6879_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_6880_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X7: set_nat] : ( if_nat @ ( X7 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).

% Sup_nat_def
thf(fact_6881_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_6882_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_6883_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_6884_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_6885_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K4: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_6886_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ ( code_divmod_abs @ K4 @ 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 @ K4 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K4 @ 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 @ K4 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_6887_Gcd__eq__Max,axiom,
    ! [M4: set_nat] :
      ( ( finite_finite_nat @ M4 )
     => ( ( M4 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M4 )
         => ( ( gcd_Gcd_nat @ M4 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M: nat] :
                      ( collect_nat
                      @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) )
                  @ M4 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_6888_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_6889_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_6890_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K4 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K4 @ 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 @ K4 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_6891_Rep__unit__induct,axiom,
    ! [Y: $o,P: $o > $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
       => ( P @ Y ) ) ) ).

% Rep_unit_induct
thf(fact_6892_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_6893_Abs__unit__induct,axiom,
    ! [P: product_unit > $o,X: product_unit] :
      ( ! [Y3: $o] :
          ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P @ ( product_Abs_unit @ Y3 ) ) )
     => ( P @ X ) ) ).

% Abs_unit_induct
thf(fact_6894_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_6895_Rep__unit,axiom,
    ! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_6896_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_6897_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_6898_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_6899_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S2: nat,T: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ S2 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_6900_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) ).

% pair_less_iff1
thf(fact_6901_pair__lessI1,axiom,
    ! [A2: nat,B2: nat,S2: nat,T: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_6902_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_6903_pair__leqI2,axiom,
    ! [A2: nat,B2: nat,S2: nat,T: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ S2 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_6904_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_6905_pair__leqI1,axiom,
    ! [A2: nat,B2: nat,S2: nat,T: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_6906_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_6907_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_6908_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y6: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y6 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y6 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X6 ) @ Y6 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_6909_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_6910_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_6911_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_6912_smax__emptyI,axiom,
    ! [Y6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y6 )
     => ( ( Y6 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y6 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_6913_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_6914_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X6 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X6 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_6915_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_6916_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% pred_nat_trancl_eq_le
thf(fact_6917_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X2: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X2 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_6918_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% less_eq
thf(fact_6919_uminus__rat__def,axiom,
    ( uminus_uminus_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ) ).

% uminus_rat_def
thf(fact_6920_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X2: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y2 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ) ).

% plus_rat_def
thf(fact_6921_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X2: product_prod_int_int,Y2: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y2 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y2 ) ) ) ) ) ).

% times_rat_def
thf(fact_6922_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_6923_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_6924_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M3: nat] :
          ( X
         != ( product_Pair_nat_nat @ K2 @ M3 ) ) ).

% prod_decode_aux.cases
thf(fact_6925_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K4: nat,M: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M @ K4 ) @ ( product_Pair_nat_nat @ M @ ( minus_minus_nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus_nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_6926_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_6927_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_6928_normalize__stable,axiom,
    ! [Q5: int,P5: int] :
      ( ( ord_less_int @ zero_zero_int @ Q5 )
     => ( ( algebr932160517623751201me_int @ P5 @ Q5 )
       => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q5 ) )
          = ( product_Pair_int_int @ P5 @ Q5 ) ) ) ) ).

% normalize_stable
thf(fact_6929_quotient__of__coprime,axiom,
    ! [R2: rat,P5: int,Q5: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P5 @ Q5 ) )
     => ( algebr932160517623751201me_int @ P5 @ Q5 ) ) ).

% quotient_of_coprime
thf(fact_6930_normalize__coprime,axiom,
    ! [R2: product_prod_int_int,P5: int,Q5: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P5 @ Q5 ) )
     => ( algebr932160517623751201me_int @ P5 @ Q5 ) ) ).

% normalize_coprime
thf(fact_6931_max__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.semilattice_neutr_axioms
thf(fact_6932_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_6933_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y4: int,Z4: int] : Y4 = Z4
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y4: int,Z4: int] : Y4 = Z4
      @ pcr_rat )
    @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_6934_quotient__of__eq,axiom,
    ! [A2: int,B2: int,P5: int,Q5: int] :
      ( ( ( quotient_of @ ( fract @ A2 @ B2 ) )
        = ( product_Pair_int_int @ P5 @ Q5 ) )
     => ( ( fract @ P5 @ Q5 )
        = ( fract @ A2 @ B2 ) ) ) ).

% quotient_of_eq
thf(fact_6935_normalize__eq,axiom,
    ! [A2: int,B2: int,P5: int,Q5: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A2 @ B2 ) )
        = ( product_Pair_int_int @ P5 @ Q5 ) )
     => ( ( fract @ P5 @ Q5 )
        = ( fract @ A2 @ B2 ) ) ) ).

% normalize_eq
thf(fact_6936_quotient__of__Fract,axiom,
    ! [A2: int,B2: int] :
      ( ( quotient_of @ ( fract @ A2 @ B2 ) )
      = ( normalize @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ).

% quotient_of_Fract
thf(fact_6937_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa3: int,X2: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa3 @ X2 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_6938_smsI,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A ) @ ( set_ms8126754132646256062at_nat @ B ) ) @ fun_max_strict )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A ) @ ( plus_p7104986032573967614at_nat @ Z5 @ B ) ) @ ms_strict ) ) ).

% smsI
thf(fact_6939_wmsI,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
      ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A ) @ ( set_ms8126754132646256062at_nat @ B ) ) @ fun_max_strict )
        | ( ( A = zero_z1048942125864253310at_nat )
          & ( B = zero_z1048942125864253310at_nat ) ) )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A ) @ ( plus_p7104986032573967614at_nat @ Z5 @ B ) ) @ ms_weak ) ) ).

% wmsI
thf(fact_6940_ms__reduction__pair,axiom,
    fun_re7357418987779152907at_nat @ ( produc5245064249948416855at_nat @ ms_strict @ ms_weak ) ).

% ms_reduction_pair
thf(fact_6941_ms__weakI1,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z5 @ Z6 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A ) @ ( set_ms8126754132646256062at_nat @ B ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A ) @ ( plus_p7104986032573967614at_nat @ Z6 @ B ) ) @ ms_weak ) ) ) ).

% ms_weakI1
thf(fact_6942_ms__strictI,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z5 @ Z6 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A ) @ ( set_ms8126754132646256062at_nat @ B ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ A ) @ ( plus_p7104986032573967614at_nat @ Z6 @ B ) ) @ ms_strict ) ) ) ).

% ms_strictI
thf(fact_6943_pw__leq__split,axiom,
    ! [X6: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( ( pw_leq @ X6 @ Y6 )
     => ? [A7: multis2468970476368604999at_nat,B8: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat] :
          ( ( X6
            = ( plus_p7104986032573967614at_nat @ A7 @ Z7 ) )
          & ( Y6
            = ( plus_p7104986032573967614at_nat @ B8 @ Z7 ) )
          & ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A7 ) @ ( set_ms8126754132646256062at_nat @ B8 ) ) @ fun_max_strict )
            | ( ( B8 = zero_z1048942125864253310at_nat )
              & ( A7 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
thf(fact_6944_ms__weakI2,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z5 @ Z6 )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z5 @ zero_z1048942125864253310at_nat ) @ ( plus_p7104986032573967614at_nat @ Z6 @ zero_z1048942125864253310at_nat ) ) @ ms_weak ) ) ).

% ms_weakI2
thf(fact_6945_pw__leq_Ocases,axiom,
    ! [A12: multis2468970476368604999at_nat,A23: multis2468970476368604999at_nat] :
      ( ( pw_leq @ A12 @ A23 )
     => ( ( ( A12 = zero_z1048942125864253310at_nat )
         => ( A23 != zero_z1048942125864253310at_nat ) )
       => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,X8: multis2468970476368604999at_nat] :
              ( ( A12
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X3 @ zero_z1048942125864253310at_nat ) @ X8 ) )
             => ! [Y7: multis2468970476368604999at_nat] :
                  ( ( A23
                    = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y3 @ zero_z1048942125864253310at_nat ) @ Y7 ) )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
                   => ~ ( pw_leq @ X8 @ Y7 ) ) ) ) ) ) ).

% pw_leq.cases
thf(fact_6946_pw__leq__lstep,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
     => ( pw_leq @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) ) ) ).

% pw_leq_lstep
thf(fact_6947_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X6: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
     => ( ( pw_leq @ X6 @ Y6 )
       => ( pw_leq @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ X6 ) @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) @ Y6 ) ) ) ) ).

% pw_leq_step
thf(fact_6948_pw__leq_Osimps,axiom,
    ( pw_leq
    = ( ^ [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
          ( ( ( A1 = zero_z1048942125864253310at_nat )
            & ( A22 = zero_z1048942125864253310at_nat ) )
          | ? [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,X7: multis2468970476368604999at_nat,Y8: multis2468970476368604999at_nat] :
              ( ( A1
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X2 @ zero_z1048942125864253310at_nat ) @ X7 ) )
              & ( A22
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y2 @ zero_z1048942125864253310at_nat ) @ Y8 ) )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ fun_pair_leq )
              & ( pw_leq @ X7 @ Y8 ) ) ) ) ) ).

% pw_leq.simps
thf(fact_6949_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc8763457246119570046nteger] :
      ~ ! [F2: code_integer > option6357759511663192854e_term,D: code_integer,I3: code_integer] :
          ( X
         != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D @ I3 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_6950_exhaustive__int_H_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,D: int,I3: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D @ I3 ) ) ) ).

% exhaustive_int'.cases
thf(fact_6951_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc2285326912895808259nt_int] :
      ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D: int,I3: int] :
          ( X
         != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D @ I3 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_6952_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc1908205239877642774nteger] :
      ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D: code_integer,I3: code_integer] :
          ( X
         != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D @ I3 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_6953_Lcm__nat__empty,axiom,
    ( ( gcd_Lcm_nat @ bot_bot_set_nat )
    = one_one_nat ) ).

% Lcm_nat_empty
thf(fact_6954_Lcm__eq__Max__nat,axiom,
    ! [M4: set_nat] :
      ( ( finite_finite_nat @ M4 )
     => ( ( M4 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M4 )
         => ( ! [M3: nat,N4: nat] :
                ( ( member_nat @ M3 @ M4 )
               => ( ( member_nat @ N4 @ M4 )
                 => ( member_nat @ ( gcd_lcm_nat @ M3 @ N4 ) @ M4 ) ) )
           => ( ( gcd_Lcm_nat @ M4 )
              = ( lattic8265883725875713057ax_nat @ M4 ) ) ) ) ) ) ).

% Lcm_eq_Max_nat
thf(fact_6955_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M4: set_nat] :
      ( ( finite_finite_nat @ M4 )
     => ( ( M4 != bot_bot_set_nat )
       => ( ! [M3: nat,N4: nat] :
              ( ( member_nat @ M3 @ M4 )
             => ( ( member_nat @ N4 @ M4 )
               => ( member_nat @ ( gcd_lcm_nat @ M3 @ N4 ) @ M4 ) ) )
         => ( member_nat @ ( gcd_Lcm_nat @ M4 ) @ M4 ) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
thf(fact_6956_in__range_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ As )
           => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).

% in_range.simps
thf(fact_6957_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ( Y
              = ( ~ ! [X2: nat] :
                      ( ( member_nat @ X2 @ As3 )
                     => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H3 ) ) ) ) ) ) ) ).

% in_range.elims(1)
thf(fact_6958_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ~ ! [X4: nat] :
                  ( ( member_nat @ X4 @ As3 )
                 => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H3 ) ) ) ) ) ).

% in_range.elims(2)
thf(fact_6959_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
           => ! [X3: nat] :
                ( ( member_nat @ X3 @ As3 )
               => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H3 ) ) ) ) ) ).

% in_range.elims(3)
thf(fact_6960_ctwo__Cnotzero,axiom,
    ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ bNF_Cardinal_ctwo @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
    & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).

% ctwo_Cnotzero
thf(fact_6961_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_6962_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_6963_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_6964_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D: int,I3: int] :
          ( X
         != ( product_Pair_int_int @ D @ I3 ) ) ).

% small_lazy'.cases
thf(fact_6965_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ).

% le_prod_encode_2
thf(fact_6966_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ).

% le_prod_encode_1
thf(fact_6967_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
     => ( ( ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_6968_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_6969_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_6970_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs2 ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_6971_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_6972_sum__list__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M2 @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ).

% sum_list_upt
thf(fact_6973_Fract__def,axiom,
    ( fract
    = ( map_fu7831380289885515383nt_rat @ id_int @ ( map_fu3424225382358772769nt_rat @ id_int @ abs_Rat )
      @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ) ).

% Fract_def
thf(fact_6974_relH__def,axiom,
    ( relH
    = ( ^ [As4: set_nat,H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit] :
          ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) )
          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
          & ! [T3: typerep,X2: nat] :
              ( ( member_nat @ X2 @ As4 )
             => ( ( ( refs_Product_unit @ H @ T3 @ X2 )
                  = ( refs_Product_unit @ H6 @ T3 @ X2 ) )
                & ( ( arrays_Product_unit @ H @ T3 @ X2 )
                  = ( arrays_Product_unit @ H6 @ T3 @ X2 ) ) ) ) ) ) ) ).

% relH_def
thf(fact_6975_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( Y
                  = ( ! [X2: nat] :
                        ( ( member_nat @ X2 @ As3 )
                       => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H3 ) ) ) ) )
               => ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ) ) ).

% in_range.pelims(1)
thf(fact_6976_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
               => ~ ! [X4: nat] :
                      ( ( member_nat @ X4 @ As3 )
                     => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H3 ) ) ) ) ) ) ) ).

% in_range.pelims(2)
thf(fact_6977_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
               => ! [X3: nat] :
                    ( ( member_nat @ X3 @ As3 )
                   => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H3 ) ) ) ) ) ) ) ).

% in_range.pelims(3)
thf(fact_6978_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 )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( Y
                  = ( As3 = bot_bot_set_nat ) )
               => ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ) ) ).

% one_assn_raw.pelims(1)
thf(fact_6979_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
               => ( As3 != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
thf(fact_6980_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As3 ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
               => ( As3 = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
thf(fact_6981_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V2 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_6982_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V2 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_6983_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V2 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_6984_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) ) ) ) ).

% uminus_int_def
thf(fact_6985_next_Osimps,axiom,
    ! [V: code_natural,W: code_natural] :
      ( ( next @ ( produc3574140220909816553atural @ V @ W ) )
      = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ 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 @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ 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 @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% next.simps
thf(fact_6986_log_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ~ ! [B5: code_natural,I3: code_natural] :
          ( X
         != ( produc3574140220909816553atural @ B5 @ I3 ) ) ).

% log.cases
thf(fact_6987_exhaustive__natural_H_Ocases,axiom,
    ! [X: produc8731074985263844745atural] :
      ~ ! [F2: code_natural > option6357759511663192854e_term,D: code_natural,I3: code_natural] :
          ( X
         != ( produc2252593628808123835atural @ F2 @ ( produc3574140220909816553atural @ D @ I3 ) ) ) ).

% exhaustive_natural'.cases
thf(fact_6988_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: produc989692990947075319atural] :
      ~ ! [F2: produc4972180933644002618e_term > option6357759511663192854e_term,D: code_natural,I3: code_natural] :
          ( X
         != ( produc3831813291587773865atural @ F2 @ ( produc3574140220909816553atural @ D @ I3 ) ) ) ).

% full_exhaustive_natural'.cases
thf(fact_6989_split__seed__def,axiom,
    ( split_seed
    = ( ^ [S5: produc7822875418678951345atural] :
          ( produc8282080750456430313atural
          @ ^ [V2: code_natural,W2: code_natural] :
              ( produc8282080750456430313atural
              @ ^ [V3: code_natural,W3: 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 ) @ W3 ) @ ( produc3574140220909816553atural @ V3 @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W2 ) ) )
              @ ( produc6591912806276919810atural @ ( next @ S5 ) ) )
          @ S5 ) ) ) ).

% split_seed_def
thf(fact_6990_list__decode_Opinduct,axiom,
    ! [A0: nat,P: nat > $o] :
      ( ( accp_nat @ nat_list_decode_rel @ A0 )
     => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
         => ( P @ zero_zero_nat ) )
       => ( ! [N4: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N4 ) )
             => ( ! [X4: nat,Y5: nat] :
                    ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
                      = ( nat_prod_decode @ N4 ) )
                   => ( P @ Y5 ) )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_6991_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_6992_Random_Orange__def,axiom,
    ( range
    = ( ^ [K4: code_natural] :
          ( produc5538323210962509403atural
          @ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
            @ ^ [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 @ K4 ) ) ) ) ) ).

% Random.range_def
thf(fact_6993_cone__def,axiom,
    ( bNF_Cardinal_cone
    = ( bNF_Ca7083678892733797993t_unit @ ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ) ).

% cone_def
thf(fact_6994_UNIV__unit,axiom,
    ( top_to1996260823553986621t_unit
    = ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).

% UNIV_unit
thf(fact_6995_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
thf(fact_6996_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [M3: num] :
                  ( ( Xa
                    = ( bit0 @ M3 ) )
                 => ( ( Y
                      = ( bit1 @ M3 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M3 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [M3: num] :
                    ( ( Xa
                      = ( bit1 @ M3 ) )
                   => ( ( Y
                        = ( bit1 @ M3 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M3 ) ) ) ) ) )
             => ( ! [N4: num] :
                    ( ( X
                      = ( bit0 @ N4 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ one ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X
                        = ( bit0 @ N4 ) )
                     => ! [M3: num] :
                          ( ( Xa
                            = ( bit0 @ M3 ) )
                         => ( ( Y
                              = ( bitM @ ( bit_or_not_num_neg @ N4 @ M3 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit0 @ M3 ) ) ) ) ) )
                 => ( ! [N4: num] :
                        ( ( X
                          = ( bit0 @ N4 ) )
                       => ! [M3: num] :
                            ( ( Xa
                              = ( bit1 @ M3 ) )
                           => ( ( Y
                                = ( bit0 @ ( bit_or_not_num_neg @ N4 @ M3 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit1 @ M3 ) ) ) ) ) )
                   => ( ! [N4: num] :
                          ( ( X
                            = ( bit1 @ N4 ) )
                         => ( ( Xa = one )
                           => ( ( Y = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ one ) ) ) ) )
                     => ( ! [N4: num] :
                            ( ( X
                              = ( bit1 @ N4 ) )
                           => ! [M3: num] :
                                ( ( Xa
                                  = ( bit0 @ M3 ) )
                               => ( ( Y
                                    = ( bitM @ ( bit_or_not_num_neg @ N4 @ M3 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit0 @ M3 ) ) ) ) ) )
                       => ~ ! [N4: num] :
                              ( ( X
                                = ( bit1 @ N4 ) )
                             => ! [M3: num] :
                                  ( ( Xa
                                    = ( bit1 @ M3 ) )
                                 => ( ( Y
                                      = ( bitM @ ( bit_or_not_num_neg @ N4 @ M3 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit1 @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_6997_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 )
             => ! [N4: num] :
                  ( ( Xa
                    = ( bit0 @ N4 ) )
                 => ( ( Y = none_num )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N4: num] :
                    ( ( Xa
                      = ( bit1 @ N4 ) )
                   => ( ( Y
                        = ( some_num @ one ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ( ( Xa = one )
                     => ( ( Y = none_num )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit0 @ N4 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X
                          = ( bit0 @ M3 ) )
                       => ! [N4: num] :
                            ( ( Xa
                              = ( bit1 @ N4 ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ one ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) ) ) ) )
                     => ( ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit0 @ N4 ) )
                               => ( ( Y
                                    = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X
                                = ( bit1 @ M3 ) )
                             => ! [N4: num] :
                                  ( ( Xa
                                    = ( bit1 @ N4 ) )
                                 => ( ( Y
                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                        @ ^ [N5: num] : ( some_num @ ( bit1 @ N5 ) )
                                        @ ( bit_un7362597486090784418nd_num @ M3 @ N4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
thf(fact_6998_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 )
             => ! [N4: num] :
                  ( ( Xa
                    = ( bit0 @ N4 ) )
                 => ( ( Y
                      = ( some_num @ one ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N4: num] :
                    ( ( Xa
                      = ( bit1 @ N4 ) )
                   => ( ( Y = none_num )
                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( some_num @ ( bit0 @ M3 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit0 @ N4 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X
                          = ( bit0 @ M3 ) )
                       => ! [N4: num] :
                            ( ( Xa
                              = ( bit1 @ N4 ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M3 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) ) ) ) )
                     => ( ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit0 @ N4 ) )
                               => ( ( Y
                                    = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                      @ ^ [N5: num] : ( some_num @ ( bit1 @ N5 ) )
                                      @ ( bit_and_not_num @ M3 @ N4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X
                                = ( bit1 @ M3 ) )
                             => ! [N4: num] :
                                  ( ( Xa
                                    = ( bit1 @ N4 ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
thf(fact_6999_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N4: num] :
                  ( ( Xa
                    = ( bit0 @ N4 ) )
                 => ( ( Y
                      = ( some_num @ ( bit1 @ N4 ) ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N4: num] :
                    ( ( Xa
                      = ( bit1 @ N4 ) )
                   => ( ( Y
                        = ( some_num @ ( bit0 @ N4 ) ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( some_num @ ( bit1 @ M3 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit0 @ N4 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X
                          = ( bit0 @ M3 ) )
                       => ! [N4: num] :
                            ( ( Xa
                              = ( bit1 @ N4 ) )
                           => ( ( Y
                                = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N4 ) ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M3 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) ) ) ) )
                     => ( ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit0 @ N4 ) )
                               => ( ( Y
                                    = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N4 ) ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X
                                = ( bit1 @ M3 ) )
                             => ! [N4: num] :
                                  ( ( Xa
                                    = ( bit1 @ N4 ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
thf(fact_7000_take__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N2: nat,M: num] :
          ( produc478579273971653890on_num
          @ ^ [A3: nat,X2: 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 ) ) )
                  @ X2 )
              @ A3 )
          @ ( product_Pair_nat_num @ N2 @ M ) ) ) ) ).

% take_bit_num_code
thf(fact_7001_or__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( ( bit_un6697907153464112080or_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N4: num] :
                  ( ( Xa
                    = ( bit0 @ N4 ) )
                 => ( ( Y
                      = ( bit1 @ N4 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N4: num] :
                    ( ( Xa
                      = ( bit1 @ N4 ) )
                   => ( ( Y
                        = ( bit1 @ N4 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( bit1 @ M3 ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N4: num] :
                          ( ( Xa
                            = ( bit0 @ N4 ) )
                         => ( ( Y
                              = ( bit0 @ ( bit_un6697907153464112080or_num @ M3 @ N4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X
                          = ( bit0 @ M3 ) )
                       => ! [N4: num] :
                            ( ( Xa
                              = ( bit1 @ N4 ) )
                           => ( ( Y
                                = ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( bit1 @ M3 ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) ) ) ) )
                     => ( ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N4: num] :
                                ( ( Xa
                                  = ( bit0 @ N4 ) )
                               => ( ( Y
                                    = ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N4 ) ) ) ) ) )
                       => ~ ! [M3: num] :
                              ( ( X
                                = ( bit1 @ M3 ) )
                             => ! [N4: num] :
                                  ( ( Xa
                                    = ( bit1 @ N4 ) )
                                 => ( ( Y
                                      = ( bit1 @ ( bit_un6697907153464112080or_num @ M3 @ N4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_num.pelims
thf(fact_7002_unit__pred__cases,axiom,
    ! [P: pred_Product_unit > $o,Q: pred_Product_unit] :
      ( ( P @ bot_bo2717538794563056311t_unit )
     => ( ( P @ ( single_Product_unit @ product_Unity ) )
       => ( P @ Q ) ) ) ).

% unit_pred_cases
thf(fact_7003_if__pred__eq,axiom,
    ( if_pred
    = ( ^ [B3: $o] : ( if_pred_Product_unit @ B3 @ ( single_Product_unit @ product_Unity ) @ bot_bo2717538794563056311t_unit ) ) ) ).

% if_pred_eq
thf(fact_7004_Random__Pred_Onot__randompred__def,axiom,
    ( random6974930770145893639ompred
    = ( ^ [P3: produc7822875418678951345atural > produc4675096598859438275atural,S5: produc7822875418678951345atural] :
          ( produc6665183775751917029atural
          @ ^ [P8: pred_Product_unit,S6: produc7822875418678951345atural] : ( if_Pro3444522238938527101atural @ ( eval_Product_unit @ P8 @ product_Unity ) @ ( produc5069803637994805237atural @ bot_bo2717538794563056311t_unit @ S6 ) @ ( produc5069803637994805237atural @ ( single_Product_unit @ product_Unity ) @ S6 ) )
          @ ( P3 @ S5 ) ) ) ) ).

% Random_Pred.not_randompred_def
thf(fact_7005_not__pred__eq,axiom,
    ( not_pred
    = ( ^ [P3: pred_Product_unit] : ( if_pred_Product_unit @ ( eval_Product_unit @ P3 @ product_Unity ) @ bot_bo2717538794563056311t_unit @ ( single_Product_unit @ product_Unity ) ) ) ) ).

% not_pred_eq
thf(fact_7006_wait__def,axiom,
    ( heap_Time_wait
    = ( ^ [N2: nat] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H: heap_e7401611519738050253t_unit] : ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ product_Unity @ ( produc584006145561248582it_nat @ H @ N2 ) ) ) ) ) ) ).

% wait_def
thf(fact_7007_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N6: nat] :
          ! [M: nat] :
            ( ( ord_less_eq_nat @ N6 @ M )
           => ! [N2: nat] :
                ( ( ord_less_eq_nat @ N6 @ N2 )
               => ( P @ ( product_Pair_nat_nat @ N2 @ M ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_7008_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_7009_random__aux__set_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ( ! [J2: code_natural] :
          ( X
         != ( produc3574140220909816553atural @ zero_z2226904508553997617atural @ J2 ) )
     => ~ ! [I3: code_natural,J2: code_natural] :
            ( X
           != ( produc3574140220909816553atural @ ( code_Suc @ I3 ) @ J2 ) ) ) ).

% random_aux_set.cases
thf(fact_7010_nth__item_Opinduct,axiom,
    ! [A0: nat,P: nat > $o] :
      ( ( accp_nat @ nth_item_rel @ A0 )
     => ( ( ( accp_nat @ nth_item_rel @ zero_zero_nat )
         => ( P @ zero_zero_nat ) )
       => ( ! [N4: nat] :
              ( ( accp_nat @ nth_item_rel @ ( suc @ N4 ) )
             => ( ! [A8: nat,Aa: nat] :
                    ( ( ( nat_sum_decode @ N4 )
                      = ( sum_Inl_nat_nat @ A8 ) )
                   => ( ( ( nat_sum_decode @ A8 )
                        = ( sum_Inl_nat_nat @ Aa ) )
                     => ( P @ Aa ) ) )
               => ( ! [A8: nat,B9: nat] :
                      ( ( ( nat_sum_decode @ N4 )
                        = ( sum_Inl_nat_nat @ A8 ) )
                     => ( ( ( nat_sum_decode @ A8 )
                          = ( sum_Inr_nat_nat @ B9 ) )
                       => ( P @ B9 ) ) )
                 => ( ! [B9: nat,Ba: nat,X4: nat,Y5: nat] :
                        ( ( ( nat_sum_decode @ N4 )
                          = ( sum_Inr_nat_nat @ B9 ) )
                       => ( ( ( nat_sum_decode @ B9 )
                            = ( sum_Inr_nat_nat @ Ba ) )
                         => ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
                              = ( nat_prod_decode @ Ba ) )
                           => ( P @ X4 ) ) ) )
                   => ( ! [B9: nat,Ba: nat,X4: nat,Y5: nat] :
                          ( ( ( nat_sum_decode @ N4 )
                            = ( sum_Inr_nat_nat @ B9 ) )
                         => ( ( ( nat_sum_decode @ B9 )
                              = ( sum_Inr_nat_nat @ Ba ) )
                           => ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
                                = ( nat_prod_decode @ Ba ) )
                             => ( P @ Y5 ) ) ) )
                     => ( P @ ( suc @ N4 ) ) ) ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% nth_item.pinduct

% Helper facts (29)
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__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( if_Code_natural @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( if_Code_natural @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Predicate__Opred_It__Product____Type__Ounit_J_T,axiom,
    ! [X: pred_Product_unit,Y: pred_Product_unit] :
      ( ( if_pred_Product_unit @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Predicate__Opred_It__Product____Type__Ounit_J_T,axiom,
    ! [X: pred_Product_unit,Y: pred_Product_unit] :
      ( ( if_pred_Product_unit @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__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_3_1_If_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_T,axiom,
    ! [X: produc4675096598859438275atural,Y: produc4675096598859438275atural] :
      ( ( if_Pro3444522238938527101atural @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_T,axiom,
    ! [X: produc4675096598859438275atural,Y: produc4675096598859438275atural] :
      ( ( if_Pro3444522238938527101atural @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    entails @ ( times_times_assn @ ( sup_sup_assn @ a @ b ) @ c ) @ ( sup_sup_assn @ ( times_times_assn @ a @ c ) @ ( times_times_assn @ b @ c ) ) ).

%------------------------------------------------------------------------------