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Theorem List for Metamath Proof Explorer - 40001-40100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcrctcshtrl 40001* Cyclically shifting the indices of a circuit results in a trail . (Contributed by AV, 14-Mar-2021.)
Vtx       iEdg       CircuitS              ..^       cyclShift               TrailS

Theoremcrctcsh 40002* Cyclically shifting the indices of a circuit results in a circuit . (Contributed by AV, 10-Mar-2021.)
Vtx       iEdg       CircuitS              ..^       cyclShift               CircuitS

21.33.8.20  Examples for walks, trails and paths

Theorem0ewlk 40003 The empty set (empty sequence of edges) is an s-walk of edges for all s. (Contributed by AV, 4-Jan-2021.)
NN0* EdgWalks

Theorem1ewlk 40004 A sequence of 1 edge is an s-walk of edges for all s. (Contributed by AV, 5-Jan-2021.)
NN0* iEdg EdgWalks

Theorem01wlk 40005 A pair of an empty set (of edges) and a second set (of vertices) is a walk iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 3-Jan-2021.)
Vtx       1Walks

Theoremis01wlklem 40006 Lemma for is01wlk 40007 and is0Trl 40012. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx

Theoremis01wlk 40007 A pair of an empty set (of edges) and a sequence of one vertex is a walk (of length 0). (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx       1Walks

Theorem0wlkOnlem1 40008 Lemma 1 for 0wlkOn 40010 and 0TrlOn 40013. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx

Theorem0wlkOnlem2 40009 Lemma 2 for 0wlkOn 40010 and 0TrlOn 40013. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx

Theorem0wlkOn 40010 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx       WalksOn

Theorem0Trl 40011 A pair of an empty set (of edges) and a second set (of vertices) is a trail iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 7-Jan-2021.)
Vtx       TrailS

Theoremis0Trl 40012 A pair of an empty set (of edges) and a sequence of one vertex is a trail (of length 0). (Contributed by AV, 7-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx       TrailS

Theorem0TrlOn 40013 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 8-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx       TrailsOn

Theorem0pth-av 40014 A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 19-Jan-2021.)
Vtx       PathS

Theorem0spth-av 40015 A pair of an empty set (of edges) and a second set (of vertices) is a simple path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 18-Jan-2021.)
Vtx       SPathS

Theorem0pthon-av 40016 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.)
Vtx       PathsOn

Theorem0pthon1-av 40017 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.)
Vtx       PathsOn

Theorem0pthonv-av 40018* For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 21-Jan-2021.)
Vtx       PathsOn

Theorem0clWlk 40019 A pair of an empty set (of edges) and a second set (of vertices) is a closed walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 17-Feb-2021.)
Vtx       ClWalkS

Theorem0clwlk0 40020 There is no closed walk in the empty set (i.e. the null graph). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
ClWalkS

Theorem0Crct 40021 A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CircuitS Vtx

Theorem0Cycl 40022 A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CycleS Vtx

Theorem1pthdlem1 40023 Lemma 1 for 1pthd 40031. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
..^

Theorem1pthdlem2 40024 Lemma 2 for 1pthd 40031. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
..^

Theorem11wlkdlem1 40025 Lemma 1 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)

Theorem11wlkdlem2 40026 Lemma 2 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)

Theorem11wlkdlem3 40027 Lemma 3 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)
Word

Theorem11wlkdlem4 40028* Lemma 4 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)
..^if-

Theorem11wlkd 40029 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx       iEdg       1Walks

Theorem1trld 40030 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx       iEdg       TrailS

Theorem1pthd 40031 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx       iEdg       PathS

Theorem1pthond 40032 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
Vtx       iEdg       PathsOn

Theoremupgr11wlkdlem1 40033 Lemma 1 for upgr11wlkd 40035. (Contributed by AV, 22-Jan-2021.)
Vtx       Vtx       iEdg        iEdg

Theoremupgr11wlkdlem2 40034 Lemma 2 for upgr11wlkd 40035. (Contributed by AV, 22-Jan-2021.)
Vtx       Vtx       iEdg        iEdg

Theoremupgr11wlkd 40035 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
Vtx       Vtx       iEdg        UPGraph        1Walks

Theoremupgr1trld 40036 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
Vtx       Vtx       iEdg        UPGraph        TrailS

Theoremupgr1pthd 40037 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
Vtx       Vtx       iEdg        UPGraph        PathS

Theoremupgr1pthond 40038 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
Vtx       Vtx       iEdg        UPGraph        PathsOn

Theoremlppthon 40039 A loop (which is an edge at index ) induces a path of length 1 from a vertex to itself in a hypergraph. (Contributed by AV, 1-Feb-2021.)
iEdg       UHGraph PathsOn

Theoremlp1cycl 40040 A loop (which is an edge at index ) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.)
iEdg       UHGraph CycleS

Theorem1pthon2v-av 40041* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
Vtx       Edg       UHGraph PathsOn

Theorem1pthon2ve 40042* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Proof shortened by AV, 15-Feb-2021.)
Vtx       Edg       UHGraph PathsOn

Theorem1wlk2v2elem1 40043 Lemma 1 for 1wlk2v2e 40045: is a length 2 word of over , the domain of the singleton word . (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
Word

Theorem1wlk2v2elem2 40044* Lemma 2 for 1wlk2v2e 40045: The values of after are edges between two vertices enumerated by . (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
..^

Theorem1wlk2v2e 40045 In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that is a simple graph (without loops) only if . (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
1Walks

Theoremntrl2v2e 40046 A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk, see 1wlk2v2e 40045, but not a trail. Notice that is a simple graph (without loops) only if . (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
TrailS

Theorem21wlkdlem1 40047 Lemma 1 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)

Theorem21wlkdlem2 40048 Lemma 2 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
..^

Theorem21wlkdlem3 40049 Lemma 3 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)

Theorem21wlkdlem4 40050* Lemma 4 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)

Theorem21wlkdlem5 40051* Lemma 5 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
..^

Theorem2pthdlem1 40052* Lemma 1 for 2pthd 40062. (Contributed by AV, 14-Feb-2021.)
..^ ..^

Theorem21wlkdlem6 40053 Lemma 6 for 21wlkd 40058. (Contributed by AV, 23-Jan-2021.)

Theorem21wlkdlem7 40054 Lemma 7 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)

Theorem21wlkdlem8 40055 Lemma 8 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)

Theorem21wlkdlem9 40056 Lemma 9 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)

Theorem21wlkdlem10 40057* Lemma 10 for 31wlkd 40084. (Contributed by AV, 14-Feb-2021.)
..^

Theorem21wlkd 40058 Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018.) (Revised by AV, 23-Jan-2021.) (Proof shortened by AV, 14-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg       1Walks

Theorem21wlkond 40059 A 1-walk of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg       WalksOn

Theorem2trld 40060 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg              TrailS

Theorem2trlond 40061 A trail of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg              TrailsOn

Theorem2pthd 40062 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg              PathS

Theorem2spthd 40063 A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg                     SPathS

Theorem2pthond 40064 A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Proof shortened by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg                     SPathsOn

Theorem2pthon3v-av 40065* For a vertex adjacent to two other vertices there is a simple path of length 2 between these other vertices in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.)
Vtx       Edg       UHGraph SPathsOn

Theoremumgr2adedgwlklem 40066 Lemma for umgr2adedgwlk 40067, umgr2adedgspth 40070, etc. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 29-Jan-2021.)
Edg       UMGraph Vtx Vtx Vtx

Theoremumgr2adedgwlk 40067 In a multigraph, two adjacent edges form a walk of length 2. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 29-Jan-2021.)
Edg       iEdg                     UMGraph                             1Walks

Theoremumgr2adedgwlkon 40068 In a multigraph, two adjacent edges form a walk between two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
Edg       iEdg                     UMGraph                             WalksOn

Theoremumgr2adedgwlkonALT 40069 Alternate proof for umgr2adedgwlkon 40068, using umgr2adedgwlk 40067, but with a much longer proof! In a multigraph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Edg       iEdg                     UMGraph                             WalksOn

Theoremumgr2adedgspth 40070 In a multigraph, two adjacent edges with different endvertices form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 29-Jan-2021.)
Edg       iEdg                     UMGraph                                    SPathS

Theoremumgr2wlk 40071* In a multigraph, there is a 1-walk of length 2 for each pair of adjacent edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
Edg       UMGraph 1Walks

Theoremumgr2wlkon 40072* For each pair of adjacent edges in a multigraph, there is a 1-walk of length 2 between the not common vertices of the edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
Edg       UMGraph WalksOn

Theorem31wlkdlem1 40073 Lemma 1 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)

Theorem31wlkdlem2 40074 Lemma 2 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)
..^

Theorem31wlkdlem3 40075 Lemma 3 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.)

Theorem31wlkdlem4 40076* Lemma 4 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)

Theorem31wlkdlem5 40077* Lemma 5 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
..^

Theorem3pthdlem1 40078* Lemma 1 for 3pthd 40088. (Contributed by AV, 9-Feb-2021.)
..^ ..^

Theorem31wlkdlem6 40079 Lemma 6 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)

Theorem31wlkdlem7 40080 Lemma 7 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)

Theorem31wlkdlem8 40081 Lemma 8 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)

Theorem31wlkdlem9 40082 Lemma 9 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)

Theorem31wlkdlem10 40083* Lemma 10 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
..^

Theorem31wlkd 40084 Construction of a walk from two given edges in a graph. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg       1Walks

Theorem31wlkond 40085 A 1-walk of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg       WalksOn

Theorem3trld 40086 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg              TrailS

Theorem3trlond 40087 A trail of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg              TrailsOn

Theorem3pthd 40088 A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg              PathS

Theorem3pthond 40089 A path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg              PathsOn

Theorem3spthd 40090 A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg                     SPathS

Theorem3spthond 40091 A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg                     SPathsOn

Theorem3cycld 40092 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg                     CycleS

Theorem3cyclpd 40093 Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
Vtx       iEdg                     CycleS

Theoremupgr3v3e3cycl 40094* If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Edg       Vtx       UPGraph CycleS

Theoremuhgr3cyclexlem 40095 Lemma for uhgr3cyclex 40096. (Contributed by AV, 12-Feb-2021.)
Vtx       Edg       iEdg

Theoremuhgr3cyclex 40096* If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
Vtx       Edg       UHGraph CycleS

Theoremumgr3cyclex 40097* If there are three (different) vertices in a multigraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
Vtx       Edg       UMGraph CycleS

Theoremumgr3v3e3cycl 40098* If and only if there is a 3-cycle in a multigraph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.) (Revised by AV, 12-Feb-2021.)
Vtx       Edg       UMGraph CycleS

Theoremupgr4cycl4dv4e 40099* If there is a cycle of length 4 in a pseudograph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 13-Feb-2021.)
Vtx       Edg       UPGraph CycleS

21.33.8.21  Connected graphs

Syntaxcconngr 40100 Extend class notation with connected graphs.
ConnGraph

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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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