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Theorem List for Metamath Proof Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem1lt4 11801 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4

Theorem4lt5 11802 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5

Theorem3lt5 11803 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5

Theorem2lt5 11804 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5

Theorem1lt5 11805 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5

Theorem5lt6 11806 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6

Theorem4lt6 11807 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6

Theorem3lt6 11808 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6

Theorem2lt6 11809 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6

Theorem1lt6 11810 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6

Theorem6lt7 11811 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7

Theorem5lt7 11812 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7

Theorem4lt7 11813 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7

Theorem3lt7 11814 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7

Theorem2lt7 11815 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7

Theorem1lt7 11816 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7

Theorem7lt8 11817 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8

Theorem6lt8 11818 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8

Theorem5lt8 11819 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8

Theorem4lt8 11820 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8

Theorem3lt8 11821 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8

Theorem2lt8 11822 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8

Theorem1lt8 11823 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8

Theorem8lt9 11824 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9

Theorem7lt9 11825 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9

Theorem6lt9 11826 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9

Theorem5lt9 11827 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9

Theorem4lt9 11828 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9

Theorem3lt9 11829 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9

Theorem2lt9 11830 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9

Theorem1lt9 11831 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9

Theorem0ne2 11832 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 2

Theorem1ne2 11833 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
1 ≠ 2

Theorem1le2 11834 1 is less than or equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 2

Theorem2cnne0 11835 2 is a nonzero complex number. (Contributed by David A. Wheeler, 7-Dec-2018.)
(2 ∈ ℂ ∧ 2 ≠ 0)

Theorem2rene0 11836 2 is a nonzero real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 ∈ ℝ ∧ 2 ≠ 0)

Theorem1le3 11837 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 ≤ 3

Theoremneg1mulneg1e1 11838 -1 · -1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1 · -1) = 1

Theoremhalfre 11839 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℝ

Theoremhalfcn 11840 One-half is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 / 2) ∈ ℂ

Theoremhalfgt0 11841 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
0 < (1 / 2)

Theoremhalfge0 11842 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
0 ≤ (1 / 2)

Theoremhalflt1 11843 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
(1 / 2) < 1

Theorem1mhlfehlf 11844 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
(1 − (1 / 2)) = (1 / 2)

Theorem8th4div3 11845 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
((1 / 8) · (4 / 3)) = (1 / 6)

Theoremhalfpm6th 11846 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
(((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3))

Theoremit0e0 11847 i times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(i · 0) = 0

Theorem2mulicn 11848 (2 · i) ∈ ℂ. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ∈ ℂ

Theorem2muline0 11849 (2 · i) ≠ 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · i) ≠ 0

5.4.5  Simple number properties

Theoremhalfcl 11850 Closure of half of a number. (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ)

Theoremrehalfcl 11851 Real closure of half. (Contributed by NM, 1-Jan-2006.)
(𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ)

Theoremhalf0 11852 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
(𝐴 ∈ ℂ → ((𝐴 / 2) = 0 ↔ 𝐴 = 0))

Theorem2halves 11853 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
(𝐴 ∈ ℂ → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)

Theoremhalfpos2 11854 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ 0 < (𝐴 / 2)))

Theoremhalfpos 11855 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ (𝐴 / 2) < 𝐴))

Theoremhalfnneg2 11856 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
(𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 2)))

Theoremhalfaddsubcl 11857 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) / 2) ∈ ℂ ∧ ((𝐴𝐵) / 2) ∈ ℂ))

Theoremhalfaddsub 11858 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐵) / 2) + ((𝐴𝐵) / 2)) = 𝐴 ∧ (((𝐴 + 𝐵) / 2) − ((𝐴𝐵) / 2)) = 𝐵))

Theoremsubhalfhalf 11859 Subtracting the half of a number from the number yields the half of the number. (Contributed by AV, 28-Jun-2021.)
(𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2))

Theoremlt2halves 11860 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < (𝐶 / 2) ∧ 𝐵 < (𝐶 / 2)) → (𝐴 + 𝐵) < 𝐶))

Theoremaddltmul 11861 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (2 < 𝐴 ∧ 2 < 𝐵)) → (𝐴 + 𝐵) < (𝐴 · 𝐵))

Theoremnominpos 11862* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
¬ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ¬ ∃𝑦 ∈ ℝ (0 < 𝑦𝑦 < 𝑥))

Theoremavglt1 11863 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < ((𝐴 + 𝐵) / 2)))

Theoremavglt2 11864 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ((𝐴 + 𝐵) / 2) < 𝐵))

Theoremavgle1 11865 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵𝐴 ≤ ((𝐴 + 𝐵) / 2)))

Theoremavgle2 11866 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ((𝐴 + 𝐵) / 2) ≤ 𝐵))

Theoremavgle 11867 The average of two numbers is less than or equal to at least one of them. (Contributed by NM, 9-Dec-2005.) (Revised by Mario Carneiro, 28-May-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵))

Theorem2timesd 11868 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (2 · 𝐴) = (𝐴 + 𝐴))

Theoremtimes2d 11869 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴))

Theoremhalfcld 11870 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 2) ∈ ℂ)

Theorem2halvesd 11871 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((𝐴 / 2) + (𝐴 / 2)) = 𝐴)

Theoremrehalfcld 11872 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 / 2) ∈ ℝ)

Theoremlt2halvesd 11873 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐶 / 2))    &   (𝜑𝐵 < (𝐶 / 2))       (𝜑 → (𝐴 + 𝐵) < 𝐶)

Theoremrehalfcli 11874 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       (𝐴 / 2) ∈ ℝ

Theoremlt2addmuld 11875 If two real numbers are less than a third real number, the sum of the two real numbers is less than twice the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵 < 𝐶)       (𝜑 → (𝐴 + 𝐵) < (2 · 𝐶))

Theoremadd1p1 11876 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
(𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))

Theoremsub1m1 11877 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
(𝑁 ∈ ℂ → ((𝑁 − 1) − 1) = (𝑁 − 2))

Theoremcnm2m1cnm3 11878 Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(𝐴 ∈ ℂ → ((𝐴 − 2) − 1) = (𝐴 − 3))

Theoremxp1d2m1eqxm1d2 11879 A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
(𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2))

Theoremdiv4p1lem1div2 11880 An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))

5.4.6  The Archimedean property

Theoremnnunb 11881* The set of positive integers is unbounded above. Theorem I.28 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (𝑦 < 𝑥𝑦 = 𝑥)

Theoremarch 11882* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
(𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)

Theoremnnrecl 11883* There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴)

Theorembndndx 11884* A bounded real sequence 𝐴(𝑘) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
(∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴𝑥) → ∃𝑘 ∈ ℕ 𝐴𝑘)

5.4.7  Nonnegative integers (as a subset of complex numbers)

Syntaxcn0 11885 Extend class notation to include the class of nonnegative integers.
class 0

Definitiondf-n0 11886 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
0 = (ℕ ∪ {0})

Theoremelnn0 11887 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
(𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))

Theoremnnssnn0 11888 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
ℕ ⊆ ℕ0

Theoremnn0ssre 11889 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
0 ⊆ ℝ

Theoremnn0sscn 11890 Nonnegative integers are a subset of the complex numbers. (Contributed by NM, 9-May-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.)
0 ⊆ ℂ

Theoremnn0ex 11891 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
0 ∈ V

Theoremnnnn0 11892 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)

Theoremnnnn0i 11893 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
𝑁 ∈ ℕ       𝑁 ∈ ℕ0

Theoremnn0re 11894 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℝ)

Theoremnn0cn 11895 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
(𝐴 ∈ ℕ0𝐴 ∈ ℂ)

Theoremnn0rei 11896 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
𝐴 ∈ ℕ0       𝐴 ∈ ℝ

Theoremnn0cni 11897 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.)
𝐴 ∈ ℕ0       𝐴 ∈ ℂ

Theoremdfn2 11898 The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
ℕ = (ℕ0 ∖ {0})

Theoremelnnne0 11899 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
(𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0𝑁 ≠ 0))

Theorem0nn0 11900 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
0 ∈ ℕ0

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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 40101-40200 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 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45272
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