Home Metamath Proof ExplorerTheorem List (p. 238 of 322) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21498) Hilbert Space Explorer (21499-23021) Users' Mathboxes (23022-32154)

Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremderangsn 23701* The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangenlem 23702* One half of derangen 23703. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremderangen 23703* The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsubfacval 23704* The subfactorial is defined as the number of derangements (see derangval 23698) of the set . (Contributed by Mario Carneiro, 21-Jan-2015.)

Theoremderangen2 23705* Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsubfacf 23706* The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfaclefac 23707* The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfac0 23708* The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfac1 23709* The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfacp1lem1 23710* Lemma for subfacp1 23717. The set together with partitions the set . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem2a 23711* Lemma for subfacp1 23717. Properties of a bijection on augmented with the two-element flip to get a bijection on . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem2b 23712* Lemma for subfacp1 23717. Properties of a bijection on augmented with the two-element flip to get a bijection on . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem3 23713* Lemma for subfacp1 23717. In subfacp1lem6 23716 we cut up the set of all derangements on first according to the value at , and then by whether or not . In this lemma, we show that the subset of all derangements that satisfy this for fixed is in bijection with derangements, by simply dropping the and points from the function to get a derangement on . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem4 23714* Lemma for subfacp1 23717. The function , which swaps with and leaves all other elements alone, is a bijection of order , i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfacp1lem5 23715* Lemma for subfacp1 23717. In subfacp1lem6 23716 we cut up the set of all derangements on first according to the value at , and then by whether or not . In this lemma, we show that the subset of all derangements with for fixed is in bijection with derangements of , because pre-composing with the function swaps and and turns the function into a bijection with and for all other , so dropping the point at yields a derangement on the remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem6 23716* Lemma for subfacp1 23717. By induction, we cut up the set of all derangements on according to the possible values of (since ), and for each set for fixed , the subset of derangements with has size (by subfacp1lem3 23713), while the subset with has size (by subfacp1lem5 23715). Adding it all up yields the desired equation for the number of derangements on . (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsubfacp1 23717* A two-term recurrence for the subfactorial. This theorem allows us to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 23708, subfac1 23709. (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacval2 23718* A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfaclim 23719* The subfactorial converges rapidly to . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacval3 23720* Another closed form expression for the subfactorial. The expression is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremderangfmla 23721* The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)

18.4.5  The Erdős-Szekeres theorem

Theoremerdszelem1 23722* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem2 23723* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem3 23724* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem4 23725* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem5 23726* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem6 23727* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem7 23728* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem8 23729* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem9 23730* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem10 23731* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem11 23732* Lemma for erdsze 23733. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze 23733* The Erdős-Szekeres theorem. For any injective sequence on the reals of length at least , there is either a subsequence of length at least on which is increasing (i.e. a order isomorphism) or a subsequence of length at least on which is decreasing (i.e. a order isomorphism, recalling that is the greater-than relation). (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2lem1 23734* Lemma for erdsze2 23736. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2lem2 23735* Lemma for erdsze2 23736. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2 23736* Generalize the statement of the Erdős-Szekeres theorem erdsze 23733 to "sequences" indexed by an arbitrary subset of , which can be infinite. (Contributed by Mario Carneiro, 22-Jan-2015.)

18.4.6  The Kuratowski closure-complement theorem

Theoremkur14lem1 23737 Lemma for kur14 23747. (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremkur14lem2 23738 Lemma for kur14 23747. Write interior in terms of closure and complement: where is complement and is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem3 23739 Lemma for kur14 23747. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem4 23740 Lemma for kur14 23747. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem5 23741 Lemma for kur14 23747. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem6 23742 Lemma for kur14 23747. If is the complementation operator and is the closure operator, this expresses the identity for any subset of the topological space. This is the key result that lets us cut down long enough sequences of that arise when applying closure and complement repeatedly to , and explains why we end up with a number as large as , yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem7 23743 Lemma for kur14 23747: main proof. The set here contains all the distinct combinations of and that can arise, and we prove here that applying or to any element of yields another elemnt of . In operator shorthand, we have . From the identities and , we can reduce any operator combination containing two adjacent identical operators, which is why the list only contains alternating sequences. The reason the sequences don't keep going after a certain point is due to the identity , proved in kur14lem6 23742. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem8 23744 Lemma for kur14 23747. Show that the set contains at most elements. (It could be less if some of the operators take the same value for a given set, but Kuratowski showed that this upper bound of is tight in the sense that there exist topological spaces and subsets of these spaces for which all generated sets are distinct, and indeed the real numbers form such a topological space.) (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14lem9 23745* Lemma for kur14 23747. Since the set is closed under closure and complement, it contains the minimal set as a subset, so also has at most elements. (Indeed , and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14lem10 23746* Lemma for kur14 23747. Discharge the set . (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14 23747* Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
;

18.4.7  Retracts and sections

Syntaxcretr 23748 Extend class notation with the retract relation.
Retr

Definitiondf-retr 23749* Define the set of retractions on two topological spaces. We say that is a retraction from to . or Retr iff there is an such that are continuous functions called the retraction and section respectively, and their composite is homotopic to the identity map. If a retraction exists, we say is a retract of . (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.)
Retr Htpy

18.4.8  Path-connected and simply connected spaces

Syntaxcpcon 23750 Extend class notation with the class of path-connected topologies.
PCon

Syntaxcscon 23751 Extend class notation with the class of simply connected topologies.
SCon

Definitiondf-pcon 23752* Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the unit interval) that goes from to for any points in the space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Definitiondf-scon 23753* Define the class of simply connected topologies. A topology is simply connected if it is path-connected and every loop (continuous path with identical start and endpoint) is contractible to a point (path-homotopic to a constant function). (Contributed by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.)
SCon PCon

Theoremispcon 23754* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempconcn 23755* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempcontop 23756 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremisscon 23757* The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremsconpcon 23758 A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremscontop 23759 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremsconpht 23760 A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremcnpcon 23761 An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon PCon

Theorempconcon 23762 A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremtxpcon 23763 The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon PCon PCon

Theoremptpcon 23764 The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.)
PCon PCon

Theoremindispcon 23765 The indiscrete topology (or trivial topology) on any set is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)
PCon

Theoremconpcon 23766 A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)
𝑛Locally PCon PCon

Theoremqtoppcon 23767 A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon qTop PCon

Theorempconpi1 23768 All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon 𝑔

Theoremsconpht2 23769 Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015.)
SCon

Theoremsconpi1 23770 A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon SCon

Theoremtxsconlem 23771 Lemma for txscon 23772. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremtxscon 23772 The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
SCon SCon SCon

Theoremcvxpcon 23773* A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld       t        PCon

Theoremcvxscon 23774* A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld       t        SCon

Theoremblscon 23775 An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld              t        SCon

Theoremcnllyscon 23776 The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.)
fld       Locally SCon

Theoremrescon 23777 A subset of is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t        SCon

Theoremiooscon 23778 An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t SCon

Theoremiccscon 23779 An closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t SCon

Theoremretopscon 23780 The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
SCon

Theoremiccllyscon 23781 An closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
t Locally SCon

Theoremrellyscon 23782 The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally SCon

Theoremiiscon 23783 The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
SCon

Theoremiillyscon 23784 The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally SCon

Theoremiinllycon 23785 The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝑛Locally

18.4.9  Covering maps

Syntaxccvm 23786 Extend class notation with the class of covering maps.
CovMap

Definitiondf-cvm 23787* Define the class of covering maps on two topological spaces. A function is a covering map if it is continuous and for every point in the target space there is a neighborhood of and a decomposition of the preimage of as a disjoint union such that is a homeomorphism of each set onto . (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap t t

Theoremfncvm 23788 Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmscbv 23789* Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.)
t t        t t

Theoremiscvm 23790* The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t               CovMap

Theoremcvmtop1 23791 Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
CovMap

Theoremcvmtop2 23792 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcn 23793 A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcov 23794* Property of a covering map. In order to make the covering property more manageable, we define here the set of all even coverings of an open set in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t               CovMap

Theoremcvmsrcl 23795* Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsi 23796* One direction of cvmsval 23797. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsval 23797* Elementhood in the set of all even coverings of an open set in . is an even covering of if it is a nonempty collection of disjoint open sets in whose union is the preimage of , such that each set is homeomorphic under to . (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsss 23798* An even covering is a subset of the topology of the domain (i.e. a collection of open sets). (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsn0 23799* An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsuni 23800* An even covering of has union equal to the preimage of by . (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32154
 Copyright terms: Public domain < Previous  Next >