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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtgcgrcomr 24601 Congruence commutes on the RHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
Itv       TarskiG

Theoremtgcgrcoml 24602 Congruence commutes on the LHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
Itv       TarskiG

Theoremtgcgrcomlr 24603 Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

Theoremtgcgreqb 24604 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Itv       TarskiG

Theoremtgcgreq 24605 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Itv       TarskiG

Theoremtgcgrneq 24606 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Itv       TarskiG

Theoremtgcgrtriv 24607 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

Theoremtgcgrextend 24608 Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.)
Itv       TarskiG

Theoremtgsegconeq 24609 Two points that satisfy the conclusion of axtgsegcon 24591 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

15.2.2  Betweenness

Theoremtgbtwntriv2 24610 Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Itv       TarskiG

Theoremtgbtwncom 24611 Betweenness commutes. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Itv       TarskiG

Theoremtgbtwncomb 24612 Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Itv       TarskiG

Theoremtgbtwnne 24613 Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.)
Itv       TarskiG

Theoremtgbtwntriv1 24614 Betweenness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Itv       TarskiG

Theoremtgbtwnswapid 24615 If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 16-Mar-2019.)
Itv       TarskiG

Theoremtgbtwnintr 24616 Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
Itv       TarskiG

Theoremtgbtwnexch3 24617 Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
Itv       TarskiG

Theoremtgbtwnouttr2 24618 Outer transitivity law for betweenness. Left-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
Itv       TarskiG

Theoremtgbtwnexch2 24619 Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

Theoremtgbtwnouttr 24620 Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

Theoremtgbtwnexch 24621 Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

Theoremtgtrisegint 24622* A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

15.2.3  Dimension

Theoremtglowdim1 24623* Lower dimension axiom for one dimension. In dimension at least 1, there are at least two distinct points. The condition "the space is of dimension 1 or more" is written here as to avoid a new definition, but a different convention could be chosen. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

Theoremtglowdim1i 24624* Lower dimension axiom for one dimension. (Contributed by Thierry Arnoux, 28-May-2019.)
Itv       TarskiG

Theoremtgldimor 24625 Excluded-middle like statement allowing to treat dimension zero as a special case. (Contributed by Thierry Arnoux, 11-Apr-2019.)

Theoremtgldim0eq 24626 In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 11-Apr-2019.)

Theoremtgldim0itv 24627 In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 12-Apr-2019.)
Itv       TarskiG

Theoremtgldim0cgr 24628 In dimension zero, any two pairs of points are congruent. (Contributed by Thierry Arnoux, 12-Apr-2019.)
Itv       TarskiG

Theoremtgbtwndiff 24629* There is always a distinct from such that lies between and . Theorem 3.14 of [Schwabhauser] p. 32. The condition "the space is of dimension 1 or more" is written here as for simplicity. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Itv       TarskiG

Theoremtgdim01 24630 In geometries of dimension lower than 2, all points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.)
Itv              DimTarskiG

Theoremnehash2 24631 The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019.)

15.2.4  Betweenness and Congruence

Theoremtgifscgr 24632 Inner five segment congruence. Take two triangles, and , with between and and between and . If the other components of the triangles are congruent, then so are and . Theorem 4.2 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 24-Mar-2019.)
Itv       TarskiG

Theoremtgcgrsub 24633 Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Itv       TarskiG

15.2.5  Congruence of a series of points

Syntaxccgrg 24634 Declare the constant for the congruence between shapes relation.
cgrG

Definitiondf-cgrg 24635* Define the relation congruence bewteen shapes. Definition 4.4 of [Schwabhauser] p. 35. Ideally, we would define this for functions of any set, but we will used words (functions over ) in most cases. (Contributed by Thierry Arnoux, 3-Apr-2019.)
cgrG

Theoremiscgrg 24636* The congruence property for sequences of points. (Contributed by Thierry Arnoux, 3-Apr-2019.)
cgrG

Theoremiscgrgd 24637* The property for two sequences and of points to be congruent. (Contributed by Thierry Arnoux, 3-Apr-2019.)
cgrG

Theoremiscgrglt 24638* The property for two sequences and of points to be congruent, where the congruence is only required for indices verifying a less-than relation. (Contributed by Thierry Arnoux, 7-Oct-2020.)
cgrG       TarskiG

Theoremtrgcgrg 24639 The property for two triangles to be congruent to each other. (Contributed by Thierry Arnoux, 3-Apr-2019.)
cgrG       TarskiG

Theoremtrgcgr 24640 Triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
cgrG       TarskiG

Theoremercgrg 24641 The shape congruence relation is an equivalence relation. Statement 4.4 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.)
TarskiG cgrG

Theoremtgcgrxfr 24642* A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.)
Itv       cgrG       TarskiG

Theoremcgr3id 24643 Reflexivity law for three-place congruence. (Contributed by Thierry Arnoux, 28-Apr-2019.)
Itv       cgrG       TarskiG

Theoremcgr3simp1 24644 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Itv       cgrG       TarskiG

Theoremcgr3simp2 24645 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Itv       cgrG       TarskiG

Theoremcgr3simp3 24646 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Itv       cgrG       TarskiG

Theoremcgr3swap12 24647 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Itv       cgrG       TarskiG

Theoremcgr3swap23 24648 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Itv       cgrG       TarskiG

Theoremcgr3swap13 24649 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 3-Oct-2020.)
Itv       cgrG       TarskiG

Theoremcgr3rotr 24650 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020.)
Itv       cgrG       TarskiG

Theoremcgr3rotl 24651 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020.)
Itv       cgrG       TarskiG

Theoremtrgcgrcom 24652 Commutative law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Itv       cgrG       TarskiG

Theoremcgr3tr 24653 Transitivity law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Itv       cgrG       TarskiG

Theoremtgbtwnxfr 24654 A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Itv       cgrG       TarskiG

Theoremtgcgr4 24655 Two quadrilaterals to be congruent to each other if one triangle formed by their vertices is, and the additional points are equidistant too. (Contributed by Thierry Arnoux, 8-Oct-2020.)
Itv       cgrG       TarskiG

15.2.6  Motions

Syntaxcismt 24656 Declare the constant for the isometry builder.
Ismt

Definitiondf-ismt 24657* Define the set of isometries between two structures. Definition 4.8 of [Schwabhauser] p. 36. See isismt 24658. (Contributed by Thierry Arnoux, 13-Dec-2019.)
Ismt

Theoremisismt 24658* Property of being an isometry. Compare with isismty 32197. (Contributed by Thierry Arnoux, 13-Dec-2019.)
Ismt

Theoremismot 24659* Property of being an isometry mapping to the same space. In geometry, this is also called a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt

Theoremmotcgr 24660 Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt

Theoremidmot 24661 The identity is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt

Theoremmotf1o 24662 Motions are bijections. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt

Theoremmotcl 24663 Closure of motions. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt

Theoremmotco 24664 The composition of two motions is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt       Ismt       Ismt

Theoremcnvmot 24665 The converse of a motion is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt       Ismt

Theoremmotplusg 24666* The operation for motions is their composition. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt Ismt Ismt        Ismt       Ismt

Theoremmotgrp 24667* The motions of a geometry form a group with respect to function composition, called the Isometry group. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt Ismt Ismt

Theoremmotcgrg 24668* Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Ismt Ismt Ismt        cgrG       Word        Ismt

Theoremmotcgr3 24669 Property of a motion: distances are preserved, special case of triangles. (Contributed by Thierry Arnoux, 15-Dec-2019.)
cgrG       TarskiG                                                 Ismt

15.2.7  Colinearity

Theoremtglng 24670* Lines of a Tarski Geometry. This relates to both Definition 4.10 of [Schwabhauser] p. 36. and Definition 6.14 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 28-Mar-2019.)
LineG       Itv       TarskiG

Theoremtglnfn 24671 Lines as functions. (Contributed by Thierry Arnoux, 25-May-2019.)
LineG       Itv       TarskiG

Theoremtglnunirn 24672 Lines are sets of points. (Contributed by Thierry Arnoux, 25-May-2019.)
LineG       Itv       TarskiG

Theoremtglnpt 24673 Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.)
LineG       Itv       TarskiG

Theoremtglngne 24674 It takes two different points to form a line. (Contributed by Thierry Arnoux, 6-Aug-2019.)
LineG       Itv       TarskiG

Theoremtglngval 24675* The line going through points and . (Contributed by Thierry Arnoux, 28-Mar-2019.)
LineG       Itv       TarskiG

Theoremtglnssp 24676 Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.)
LineG       Itv       TarskiG

Theoremtgellng 24677 Property of lying on the line going through points and . Definition 4.10 of [Schwabhauser] p. 36. We choose the notation LineG instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-2019.)
LineG       Itv       TarskiG

Theoremtgcolg 24678 We choose the notation instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
LineG       Itv       TarskiG

Theorembtwncolg1 24679 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
LineG       Itv       TarskiG

Theorembtwncolg2 24680 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
LineG       Itv       TarskiG

Theorembtwncolg3 24681 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
LineG       Itv       TarskiG

Theoremcolcom 24682 Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
LineG       Itv       TarskiG

Theoremcolrot1 24683 Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
LineG       Itv       TarskiG

Theoremcolrot2 24684 Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
LineG       Itv       TarskiG

Theoremncolcom 24685 Swapping non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
LineG       Itv       TarskiG

Theoremncolrot1 24686 Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
LineG       Itv       TarskiG

Theoremncolrot2 24687 Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
LineG       Itv       TarskiG

Theoremtgdim01ln 24688 In geometries of dimension lower than 2, any 3 points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.)
LineG       Itv       TarskiG                            DimTarskiG

Theoremncoltgdim2 24689 If there are 3 non-colinear points, dimension must be 2 or more. tglowdim2l 24774 converse. (Contributed by Thierry Arnoux, 23-Feb-2020.)
LineG       Itv       TarskiG                                   DimTarskiG

Theoremlnxfr 24690 Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
LineG       Itv       TarskiG                            cgrG

Theoremlnext 24691* Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
LineG       Itv       TarskiG                            cgrG

Theoremtgfscgr 24692 Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
LineG       Itv       TarskiG                            cgrG

Theoremlncgr 24693 Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 28-Apr-2019.)
LineG       Itv       TarskiG                            cgrG

Theoremlnid 24694 Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.)
LineG       Itv       TarskiG                            cgrG

Theoremtgidinside 24695 Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.)
LineG       Itv       TarskiG                            cgrG

15.2.8  Connectivity of betweenness

Theoremtgbtwnconn1lem1 24696 Lemma for tgbtwnconn1 24699. (Contributed by Thierry Arnoux, 30-Apr-2019.)
Itv       TarskiG

Theoremtgbtwnconn1lem2 24697 Lemma for tgbtwnconn1 24699. (Contributed by Thierry Arnoux, 30-Apr-2019.)
Itv       TarskiG

Theoremtgbtwnconn1lem3 24698 Lemma for tgbtwnconn1 24699. (Contributed by Thierry Arnoux, 30-Apr-2019.)
Itv       TarskiG

Theoremtgbtwnconn1 24699 Connectivity law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. In earlier presentations of Tarski's axioms, this theorem appeared as an additional axiom. It was derived from the other axioms by Gupta, 1965. (Contributed by Thierry Arnoux, 30-Apr-2019.)
Itv       TarskiG

Theoremtgbtwnconn2 24700 Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.)
Itv       TarskiG

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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 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