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Theorem List for Metamath Proof Explorer - 38301-38400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvolicon0 38301 The measure of a nonempty left-closed, right-open interval. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhsphoif 38302* is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29 (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoidmvval 38303* The dimensional volume of a multidimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoissrrn2 38304* A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhsphoival 38305* is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29 (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoiprodcl3 38306* The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremvolicore 38307 The Lebesgue measure of a left-closed right-open interval is a real number. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoidmvcl 38308* The dimensional volume of a multidimensional half-open interval is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoidmv0val 38309* The dimensional volume of a 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoidmvn0val 38310* The dimensional volume of a non 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhsphoidmvle2 38311* The dimensional volume of a half-open interval intersected with a two half-spaces. Used in the last inequality of step (c) of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhsphoidmvle 38312* The dimensional volume of a half-open interval intersected with a half-space, is less than or equal to the dimensional volume of the original half-open interval. Used in the last inequality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoidmvval0 38313* The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoiprodp1 38314* The dimensional volume of a half-open interval with dimension . Used in the first equality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremsge0hsphoire 38315* If the generalized sum of dimensional volumes of n-dimensional half-open intervals is finite, then the sum stays finite if every half-open interval is intersected with a half-space. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^                      Σ^

Theoremhoidmvval0b 38316* The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Theoremhoidmv1lelem1 38317* The supremum of belongs to . This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23 (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^        Σ^

Theoremhoidmv1lelem2 38318* This is the contradiction proven in step (c) in the proof of Lemma 114B of [Fremlin1] p. 23 (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^        Σ^

Theoremhoidmv1lelem3 38319* The dimensional volume of a 1-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is the non-empty, finite generalized sum, sub case in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^        Σ^               Σ^

Theoremhoidmv1le 38320* The dimensional volume of a 1-dimensional half-open interval is less than or equal to the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is one of the two base cases of the induction of Lemma 115B of [Fremlin1] p. 29 (the other base case is the 0-dimensional case). This proof of the 1-dimensional case is given in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^

Theoremhoidmvlelem1 38321* The supremum of belongs to . Step (c) in the proof of Lemma 115B of [Fremlin1] p. 29 (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^                             Σ^

Theoremhoidmvlelem2 38322* This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^                             Σ^                                                         inf

Theoremhoidmvlelem3 38323* This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^                             Σ^                             Σ^

Theoremhoidmvlelem4 38324* The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29, case nonempty interval and dimension of the space greater than . (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^                             Σ^               Σ^               Σ^

Theoremhoidmvlelem5 38325* The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^                      Σ^

Theoremhoidmvle 38326* The dimensional volume of a n-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^

Theoremovnhoilem1 38327* The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. First part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^               voln*

Theoremovnhoilem2 38328* The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. Second part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^                      voln*

Theoremovnhoi 38329* The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
voln*

21.31  Mathbox for Saveliy Skresanov

21.31.1  Ceva's theorem

Theoremsigarval 38330* Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarim 38331* Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarac 38332* Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigaraf 38333* Signed area is additive by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarmf 38334* Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigaras 38335* Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarms 38336* Signed area is additive (with respect to subtraction) by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarls 38337* Signed area is linear by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarid 38338* Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigarexp 38339* Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigarperm 38340* Signed area acts as a double area of a triangle . Here we prove that cyclically permuting the vertices doesn't change the area. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigardiv 38341* If signed area between vectors and is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Theoremsigarimcd 38342* Signed area takes value in complex numbers. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremsigariz 38343* If signed area is zero, the signed area with swapped arguments is also zero. Deduction version. ( Contributed by Saveliy Skresanov, 23-Sep-2017.) (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremsigarcol 38344* Given three points , and such that , the point lies on the line going through and iff the corresponding signed area is zero. That justifies the usage of signed area as a collinearity indicator. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Theoremsharhght 38345* Let be a triangle, and let lie on the line . Then (doubled) areas of triangles and relate as lengths of corresponding bases and . (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremsigaradd 38346* Subtracting (double) area of from yields the (double) area of . (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremcevathlem1 38347 Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremcevathlem2 38348* Ceva's theorem second lemma. Relate (doubled) areas of triangles and with of segments and . (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremcevath 38349* Ceva's theorem. Let be a triangle and let points , and lie on sides , , correspondingly. Suppose that cevians , and intersect at one point . Then triangle's sides are partitioned into segments and their lengths satisfy a certain identity. Here we obtain a bit stronger version by using complex numbers themselves instead of their absolute values.

The proof goes by applying cevathlem2 38348 three times and then using cevathlem1 38347 to multiply obtained identities and prove the theorem.

In the theorem statement we are using function as a collinearity indicator. For justification of that use, see sigarcol 38344. This is Metamath 100 proof #61. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

21.32  Mathbox for Jarvin Udandy

TheoremhirstL-ax3 38350 The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)

Theoremax3h 38351 Recovery of ax-3 8 from hirstL-ax3 38350. (Contributed by Jarvin Udandy, 3-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremaibandbiaiffaiffb 38352 A closed form showing (a implies b and b implies a) same-as (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremaibandbiaiaiffb 38353 A closed form showing (a implies b and b implies a) implies (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremnotatnand 38354 Do not use. Use intnanr instead. Given not a, there exists a proof for not (a and b). (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaistia 38355 Given a is equivalent to , there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaisfina 38356 Given a is equivalent to , there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theorembothtbothsame 38357 Given both a, b are equivalent to , there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorembothfbothsame 38358 Given both a, b are equivalent to , there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaiffbbtat 38359 Given a is equivalent to b, b is equivalent to there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaisbbisfaisf 38360 Given a is equivalent to b, b is equivalent to there exists a proof for a is equivalent to F. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaxorbtnotaiffb 38361 Given a is exclusive to b, there exists a proof for (not (a if-and-only-if b)); df-xor 1401 is a closed form of this. (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaiffnbandciffatnotciffb 38362 Given a is equivalent to (not b), c is equivalent to a, there exists a proof for ( not ( c iff b ) ). (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaxorbciffatcxorb 38363 Given a is equivalent to (not b), c is equivalent to a. there exists a proof for ( c xor b ) . (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaibnbna 38364 Given a implies b, (not b), there exists a proof for (not a). (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaibnbaif 38365 Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaiffbtbat 38366 Given a is equivalent to b, T. is equivalent to b. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremastbstanbst 38367 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaistbistaandb 38368 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for (a and b). (Contributed by Jarvin Udandy, 9-Sep-2016.)

Theoremaisbnaxb 38369 Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremiatbtatnnb 38370 Given a implies b, there exists a proof for a implies not not b. (Contributed by Jarvin Udandy, 2-Sep-2016.)

Theorematbiffatnnb 38371 If a implies b, then a implies not not b (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theorembisaiaisb 38372 Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorematbiffatnnbalt 38373 If a implies b, then a implies not not b. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremabnotbtaxb 38374 Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremabnotataxb 38375 Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremconimpf 38376 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremconimpfalt 38377 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaistbisfiaxb 38378 Given a is equivalent to T., Given b is equivalent to F. there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaisfbistiaxb 38379 Given a is equivalent to F., Given b is equivalent to T., there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaifftbifffaibif 38380 Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a implies b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Theoremaifftbifffaibifff 38381 Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a iff b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Theorematnaiana 38382 Given a, it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Theoremainaiaandna 38383 Given a, a implies it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Theoremabcdta 38384 Given (((a and b) and c) and d), there exists a proof for a (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtb 38385 Given (((a and b) and c) and d), there exists a proof for b (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtc 38386 Given (((a and b) and c) and d), there exists a proof for c (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtd 38387 Given (((a and b) and c) and d), there exists a proof for d (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabciffcbatnabciffncba 38388 Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. Closed form. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Theoremabciffcbatnabciffncbai 38389 Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Theoremnabctnabc 38390 not ( a -> ( b /\ c ) ) we can show: not a implies ( b /\ c ) (Contributed by Jarvin Udandy, 7-Sep-2020.)

Theoremjabtaib 38391 For when pm3.4 lacks a pm3.4i. (Contributed by Jarvin Udandy, 9-Sep-2020.)

Theoremonenotinotbothi 38392 From one negated implication it is not the case its non negated form and a random others are both true. (Contributed by Jarvin Udandy, 11-Sep-2020.)

Theoremtwonotinotbothi 38393 From these two negated implications it is not the case their non negated forms are both true. (Contributed by Jarvin Udandy, 11-Sep-2020.)

Theoremclifte 38394 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)

Theoremcliftet 38395 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)

Theoremclifteta 38396 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)

Theoremcliftetb 38397 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)

Theoremconfun 38398 Given the hypotheses there exists a proof for (c implies ( d iff a ) ) (Contributed by Jarvin Udandy, 6-Sep-2020.)

Theoremconfun2 38399 Confun simplified to two propositions. (Contributed by Jarvin Udandy, 6-Sep-2020.)

Theoremconfun3 38400 Confun's more complex form where both a,d have been "defined". (Contributed by Jarvin Udandy, 6-Sep-2020.)

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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 32101-32200 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 40101-40162
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