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Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxi2m1 9601 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 9625. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)

Theoremax1ne0 9602 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1ne0 9626. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)

Theoremax1rid 9603 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, derived from ZF set theory. Weakened from the original axiom in the form of statement in mulid1 9658, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 9627. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)

Theoremaxrnegex 9604* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 9628. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)

Theoremaxrrecex 9605* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rrecex 9629. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)

Theoremaxcnre 9606* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 9630. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theoremaxpre-lttri 9607 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 9723. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 9631. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Theoremaxpre-lttrn 9608 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 9724. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 9632. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Theoremaxpre-ltadd 9609 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axltadd 9725. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 9633. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)

Theoremaxpre-mulgt0 9610 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 9726. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 9634. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theoremaxpre-sup 9611* A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 9727. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9635. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Theoremwuncn 9612 A weak universe containing contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

5.1.3  Real and complex number postulates restated as axioms

Axiomax-cnex 9613 The complex numbers form a set. This axiom is redundant - see cnexALT 11321- but we provide this axiom because the justification theorem axcnex 9589 does not use ax-rep 4508 even though the redundancy proof does. Proofs should normally use cnex 9638 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)

Axiomax-resscn 9614 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 9590. (Contributed by NM, 1-Mar-1995.)

Axiomax-1cn 9615 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 9591. (Contributed by NM, 1-Mar-1995.)

Axiomax-icn 9616 is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 9592. (Contributed by NM, 1-Mar-1995.)

Axiomax-addcl 9617 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 9593. Proofs should normally use addcl 9639 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-addrcl 9618 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 9594. Proofs should normally use readdcl 9640 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-mulcl 9619 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 9595. Proofs should normally use mulcl 9641 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-mulrcl 9620 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 9596. Proofs should normally use remulcl 9642 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-mulcom 9621 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 9597. Proofs should normally use mulcom 9643 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-addass 9622 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 9598. Proofs should normally use addass 9644 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-mulass 9623 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9599. Proofs should normally use mulass 9645 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-distr 9624 Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9600. Proofs should normally use adddi 9646 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-i2m1 9625 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, justified by theorem axi2m1 9601. (Contributed by NM, 29-Jan-1995.)

Axiomax-1ne0 9626 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 9602. (Contributed by NM, 29-Jan-1995.)

Axiomax-1rid 9627 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 9603. Weakened from the original axiom in the form of statement in mulid1 9658, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)

Axiomax-rnegex 9628* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 9604. (Contributed by Eric Schmidt, 21-May-2007.)

Axiomax-rrecex 9629* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 9605. (Contributed by Eric Schmidt, 11-Apr-2007.)

Axiomax-cnre 9630* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, justified by theorem axcnre 9606. For naming consistency, use cnre 9657 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)

Axiomax-pre-lttri 9631 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 9607. Note: The more general version for extended reals is axlttri 9723. Normally new proofs would use xrlttri 11461. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-lttrn 9632 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 9608. Note: The more general version for extended reals is axlttrn 9724. Normally new proofs would use lttr 9728. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-ltadd 9633 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 9609. Normally new proofs would use axltadd 9725. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-mulgt0 9634 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 9610. Normally new proofs would use axmulgt0 9726. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-sup 9635* A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 9611. Note: Normally new proofs would use axsup 9727. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-addf 9636 Addition is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific addcl 9639 should be used. Note that uses of ax-addf 9636 can be eliminated by using the defined operation in place of , from which this axiom (with the defined operation in place of ) follows as a theorem.

This axiom is justified by theorem axaddf 9587. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

Axiomax-mulf 9637 Multiplication is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific ax-mulcl 9619 should be used. Note that uses of ax-mulf 9637 can be eliminated by using the defined operation in place of , from which this axiom (with the defined operation in place of ) follows as a theorem.

This axiom is justified by theorem axmulf 9588. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

5.2  Derive the basic properties from the field axioms

5.2.1  Some deductions from the field axioms for complex numbers

Theoremcnex 9638 Alias for ax-cnex 9613. See also cnexALT 11321. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremaddcl 9639 Alias for ax-addcl 9617, for naming consistency with addcli 9665. Use this theorem instead of ax-addcl 9617 or axaddcl 9593. (Contributed by NM, 10-Mar-2008.)

Theoremreaddcl 9640 Alias for ax-addrcl 9618, for naming consistency with readdcli 9674. (Contributed by NM, 10-Mar-2008.)

Theoremmulcl 9641 Alias for ax-mulcl 9619, for naming consistency with mulcli 9666. (Contributed by NM, 10-Mar-2008.)

Theoremremulcl 9642 Alias for ax-mulrcl 9620, for naming consistency with remulcli 9675. (Contributed by NM, 10-Mar-2008.)

Theoremmulcom 9643 Alias for ax-mulcom 9621, for naming consistency with mulcomi 9667. (Contributed by NM, 10-Mar-2008.)

Theoremaddass 9644 Alias for ax-addass 9622, for naming consistency with addassi 9669. (Contributed by NM, 10-Mar-2008.)

Theoremmulass 9645 Alias for ax-mulass 9623, for naming consistency with mulassi 9670. (Contributed by NM, 10-Mar-2008.)

Theoremadddi 9646 Alias for ax-distr 9624, for naming consistency with adddii 9671. (Contributed by NM, 10-Mar-2008.)

Theoremrecn 9647 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)

Theoremreex 9648 The real numbers form a set. See also reexALT 11319. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremreelprrecn 9649 Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremcnelprrecn 9650 Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremelimne0 9651 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 15-May-1999.)

Theoremadddir 9652 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)

Theorem0cn 9653 0 is a complex number. See also 0cnALT 9884. (Contributed by NM, 19-Feb-2005.)

Theorem0cnd 9654 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremc0ex 9655 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1ex 9656 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theoremcnre 9657* Alias for ax-cnre 9630, for naming consistency. (Contributed by NM, 3-Jan-2013.)

Theoremmulid1 9658 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmulid2 9659 Identity law for multiplication. Note: see mulid1 9658 for commuted version. (Contributed by NM, 8-Oct-1999.)

Theorem1re 9660 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 9615, by exploiting properties of the imaginary unit . (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)

Theorem0re 9661 is a real number. See also 0reALT 9991. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)

Theorem0red 9662 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)

Theoremmulid1i 9663 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)

Theoremmulid2i 9664 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)

Theoremaddcli 9665 Closure law for addition. (Contributed by NM, 23-Nov-1994.)

Theoremmulcli 9666 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulcomi 9667 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulcomli 9668 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremaddassi 9669 Associative law for addition. (Contributed by NM, 23-Nov-1994.)

Theoremmulassi 9670 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremadddii 9671 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)

Theoremadddiri 9672 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)

Theoremrecni 9673 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)

Theoremreaddcli 9674 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)

Theoremremulcli 9675 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)

Theorem1red 9676 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)

Theorem1cnd 9677 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)

Theoremmulid1d 9678 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulid2d 9679 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcld 9680 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcld 9681 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcomd 9682 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddassd 9683 Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulassd 9684 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadddid 9685 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadddird 9686 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecnd 9687 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)

Theoremreaddcld 9688 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremremulcld 9689 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)

5.2.2  Infinity and the extended real number system

Syntaxcpnf 9690 Plus infinity.

Syntaxcmnf 9691 Minus infinity.

Syntaxcxr 9692 The set of extended reals (includes plus and minus infinity).

Syntaxclt 9693 'Less than' predicate (extended to include the extended reals).

Syntaxcle 9694 Extend wff notation to include the 'less than or equal to' relation.

Definitiondf-pnf 9695 Define plus infinity. Note that the definition is arbitrary, requiring only that be a set not in and different from (df-mnf 9696). We use to make it independent of the construction of , and Cantor's Theorem will show that it is different from any member of and therefore . See pnfnre 9700, mnfnre 9701, and pnfnemnf 11440.

A simpler possibility is to define as and as , but that approach requires the Axiom of Regularity to show that and are different from each other and from all members of . (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Definitiondf-mnf 9696 Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that be a set not in and different from (see mnfnre 9701 and pnfnemnf 11440). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Definitiondf-xr 9697 Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.)

Definitiondf-ltxr 9698* Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, is primitive and not necessarily a relation on . (Contributed by NM, 13-Oct-2005.)

Definitiondf-le 9699 Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 9738 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)

Theorempnfnre 9700 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

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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 32101-32200 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 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
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