P. 255 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 25401-25500 *Has distinct variable group(s)
Type	Label	Description
Statement

19.4.3 Properties of pre-Hilbert spaces

Theorem	isph 25401*	The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
_CV $/\$

Theorem	phpar2 25402	The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
_CV $/\$ $/\$

Theorem	phpar 25403	The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
_CV $/\$ $/\$

Theorem	ip0i 25404	A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
_CV

Theorem	ip1ilem 25405	Lemma for ip1i 25406. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
_CV

Theorem	ip1i 25406	Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)


Theorem	ip2i 25407	Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)


Theorem	ipdirilem 25408	Lemma for ipdiri 25409. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)


Theorem	ipdiri 25409	Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
$/\$ $/\$

Theorem	ipasslem1 25410	Lemma for ipassi 25420. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
$/\$

Theorem	ipasslem2 25411	Lemma for ipassi 25420. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
$/\$

Theorem	ipasslem3 25412	Lemma for ipassi 25420. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
$/\$

Theorem	ipasslem4 25413	Lemma for ipassi 25420. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
$/\$

Theorem	ipasslem5 25414	Lemma for ipassi 25420. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
$/\$

Theorem	ipasslem7 25415*	Lemma for ipassi 25420. Show that is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
ℂ_fld

Theorem	ipasslem8 25416*	Lemma for ipassi 25420. By ipasslem5 25414, is 0 for all ; since it is continuous and is dense in by qdensere2 21032, we conclude is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)


Theorem	ipasslem9 25417	Lemma for ipassi 25420. Conclude from ipasslem8 25416 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)


Theorem	ipasslem10 25418	Lemma for ipassi 25420. Show the inner product associative law for the imaginary number . (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
_CV

Theorem	ipasslem11 25419	Lemma for ipassi 25420. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)


Theorem	ipassi 25420	Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
$/\$ $/\$

Theorem	dipdir 25421	Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	dipdi 25422	Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	ip2dii 25423	Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)


Theorem	dipass 25424	Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	dipassr 25425	"Associative" law for second argument of inner product (compare dipass 25424). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	dipassr2 25426	"Associative" law for inner product. Conjugate version of dipassr 25425. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	dipsubdir 25427	Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	dipsubdi 25428	Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pythi 25429	The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space . The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
_CV

Theorem	siilem1 25430	Lemma for sii 25433. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
_CV

Theorem	siilem2 25431	Lemma for sii 25433. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
_CV $/\$ $/\$

Theorem	siii 25432	Inference from sii 25433. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
_CV

Theorem	sii 25433	Schwarz inequality. Part of Lemma 3-2.1(a) of [Kreyszig] p. 137. This is also called the Cauchy-Schwarz inequality by some authors and Bunjakovaskij-Cauchy-Schwarz inequality by others. See also theorems bcseqi 25701, bcsiALT 25760, bcsiHIL 25761, csbren 21556. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
_CV $/\$

Theorem	sspph 25434	A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
$/\$

Theorem	ipblnfi 25435*	A function generated by varying the first argument of an inner product (with its second argument a fixed vector ) is a bounded linear functional, i.e. a bounded linear operator from the vector space to . (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)


Theorem	ip2eqi 25436*	Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
$/\$

Theorem	phoeqi 25437*	A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
$/\$

Theorem	ajmoi 25438*	Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
$/\$

Theorem	ajfuni 25439	The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)


Theorem	ajfun 25440	The adjoint function is a function. This is not immediately apparent from df-aj 25329 but results from the uniqueness shown by ajmoi 25438. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
$/\$

Theorem	ajval 25441*	Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
$/\$ $/\$ $/\$

19.5 Complex Banach spaces

19.5.1 Definition and basic properties

Syntax	ccbn 25442	Extend class notation with the class of all complex Banach spaces.


Definition	df-cbn 25443	Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)


Theorem	iscbn 25444	A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)
$/\$

Theorem	cbncms 25445	The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)


Theorem	bnnv 25446	Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)


Theorem	bnrel 25447	The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)


Theorem	bnsscmcl 25448	A subspace of a Banach space is a Banach space iff it is closed in the norm-induced metric of the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
$/\$

19.5.2 Examples of complex Banach spaces

Theorem	cnbn 25449	The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.)


19.5.3 Uniform Boundedness Theorem

Theorem	ubthlem1 25450*	Lemma for ubth 25453. The function exhibits a countable collection of sets that are closed, being the inverse image under of the closed ball of radius , and by assumption they cover . Thus, by the Baire Category theorem bcth2 21499, for some the set has an interior, meaning that there is a closed ball in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
_CV

Theorem	ubthlem2 25451*	Lemma for ubth 25453. Given that there is a closed ball in , for any , we have and , so both of these have and so , which is our desired uniform bound. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
_CV

Theorem	ubthlem3 25452*	Lemma for ubth 25453. Prove the reverse implication, using nmblolbi 25379. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
_CV

Theorem	ubth 25453*	Uniform Boundedness Theorem, also called the Banach-Steinhaus Theorem. Let be a collection of bounded linear operators on a Banach space. If, for every vector , the norms of the operators' values are bounded, then the operators' norms are also bounded. Theorem 4.7-3 of [Kreyszig] p. 249. See also http://en.wikipedia.org/wiki/Uniform_boundedness_principle. (Contributed by NM, 7-Nov-2007.) (Proof shortened by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
_CV $/\$ $/\$

19.5.4 Minimizing Vector Theorem

Theorem	minvecolem1 25454*	Lemma for minveco 25464. The set of all distances from points of to are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.)
_CV $/\$ $/\$

Theorem	minvecolem2 25455*	Lemma for minveco 25464. Any two points and in are close to each other if they are close to the infimum of distance to . (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
_CV

Theorem	minvecolem3 25456*	Lemma for minveco 25464. The sequence formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.)
_CV $/\$

Theorem	minvecolem4a 25457*	Lemma for minveco 25464. is convergent in the subspace topology on . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
_CV $/\$

Theorem	minvecolem4b 25458*	Lemma for minveco 25464. The convergent point of the cauchy sequence is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
_CV $/\$

Theorem	minvecolem4c 25459*	Lemma for minveco 25464. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
_CV $/\$

Theorem	minvecolem4 25460*	Lemma for minveco 25464. The convergent point of the cauchy sequence attains the minimum distance, and so is closer to than any other point in . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
_CV $/\$

Theorem	minvecolem5 25461*	Lemma for minveco 25464. Discharge the assumption about the sequence by applying countable choice ax-cc 8806. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
_CV

Theorem	minvecolem6 25462*	Lemma for minveco 25464. Any minimal point is less than away from . (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
_CV $/\$

Theorem	minvecolem7 25463*	Lemma for minveco 25464. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
_CV

Theorem	minveco 25464*	Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace that minimizes the distance to an arbitrary vector in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
_CV

19.6 Complex Hilbert spaces

19.6.1 Definition and basic properties

Syntax	chlo 25465	Extend class notation with the class of all complex Hilbert spaces.


Definition	df-hlo 25466	Define the class of all complex Hilbert spaces. A Hilbert space is a Banach space which is also an inner product space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)


Theorem	ishlo 25467	The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
$/\$

Theorem	hlobn 25468	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)


Theorem	hlph 25469	Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.)


Theorem	hlrel 25470	The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)


Theorem	hlnv 25471	Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)


Theorem	hlnvi 25472	Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)


Theorem	hlvc 25473	Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)


Theorem	hlcmet 25474	The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)


Theorem	hlmet 25475	The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)


Theorem	hlpar2 25476	The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
_CV $/\$ $/\$

Theorem	hlpar 25477	The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
_CV $/\$ $/\$

19.6.2 Standard axioms for a complex Hilbert space

Theorem	hlex 25478	The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)


Theorem	hladdf 25479	Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)


Theorem	hlcom 25480	Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$

Theorem	hlass 25481	Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hl0cl 25482	The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)


Theorem	hladdid 25483	Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
$/\$

Theorem	hlmulf 25484	Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)


Theorem	hlmulid 25485	Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
$/\$

Theorem	hlmulass 25486	Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hldi 25487	Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hldir 25488	Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hlmul0 25489	Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
$/\$

Theorem	hlipf 25490	Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)


Theorem	hlipcj 25491	Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$

Theorem	hlipdir 25492	Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hlipass 25493	Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	hlipgt0 25494	The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
$/\$ $/\$

Theorem	hlcompl 25495	Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
$/\$

19.6.3 Examples of complex Hilbert spaces

Theorem	cnchl 25496	The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)


19.6.4 Subspaces

Theorem	ssphl 25497	A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)
$/\$ $/\$

19.6.5 Hellinger-Toeplitz Theorem

Theorem	htthlem 25498*	Lemma for htth 25499. The collection , which consists of functions for each in the unit ball, is a collection of bounded linear functions by ipblnfi 25435, so by the Uniform Boundedness theorem ubth 25453, there is a uniform bound on for all in the unit ball. Then , so and is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
_CV

Theorem	htth 25499*	Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
$/\$ $/\$

PART 20 COMPLEX HILBERT SPACE EXPLORER (DEPRECATED) This part contains the definitions and theorems used by the Hilbert Space Explorer (HSE), http://us.metamath.org/mpeuni/mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should instead work with general Hilbert spaces as defined by df-hil 18497; note that df-hil 18497 uses extensible structures. The intent is for this deprecated section to be deleted once all its theorems have been translated into extensible structure versions (or are not useful). Many of the theorems in this section have already been translated to extensible structure versions, but there is still a lot in this section that might be useful for future reference. It is much easier to translate these by hand from this section than to start from scratch from textbook proofs, since the HSE omits no details.

20.1 Axiomatization of complex pre-Hilbert spaces

20.1.1 Basic Hilbert space definitions

Syntax	chil 25500	Extend class notation with Hilbert vector space.

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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 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800

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