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Type | Label | Description |
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Statement | ||
Theorem | xp1d2m1eqxm1d2 9201 | 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.) |
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Theorem | div4p1lem1div2 9202 | 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.) |
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Theorem | arch 9203* | 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.) |
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Theorem | nnrecl 9204* | 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.) |
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Theorem | bndndx 9205* |
A bounded real sequence ![]() ![]() ![]() ![]() |
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Syntax | cn0 9206 | Extend class notation to include the class of nonnegative integers. |
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Definition | df-n0 9207 | Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | elnn0 9208 | Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | nnssnn0 9209 | Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | nn0ssre 9210 | Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | nn0sscn 9211 | Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.) |
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Theorem | nn0ex 9212 | The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
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Theorem | nnnn0 9213 | A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.) |
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Theorem | nnnn0i 9214 | A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.) |
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Theorem | nn0re 9215 | A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.) |
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Theorem | nn0cn 9216 | A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.) |
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Theorem | nn0rei 9217 | A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
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Theorem | nn0cni 9218 | A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) |
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Theorem | dfn2 9219 | 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.) |
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Theorem | elnnne0 9220 | The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
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Theorem | 0nn0 9221 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | 1nn0 9222 | 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | 2nn0 9223 | 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | 3nn0 9224 | 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
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Theorem | 4nn0 9225 | 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
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Theorem | 5nn0 9226 | 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
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Theorem | 6nn0 9227 | 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
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Theorem | 7nn0 9228 | 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
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Theorem | 8nn0 9229 | 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
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Theorem | 9nn0 9230 | 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
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Theorem | nn0ge0 9231 | A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
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Theorem | nn0nlt0 9232 | A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | nn0ge0i 9233 | Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | nn0le0eq0 9234 | A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.) |
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Theorem | nn0p1gt0 9235 | A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
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Theorem | nnnn0addcl 9236 | A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
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Theorem | nn0nnaddcl 9237 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
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Theorem | 0mnnnnn0 9238 | The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.) |
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Theorem | un0addcl 9239 |
If ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | un0mulcl 9240 |
If ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nn0addcl 9241 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
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Theorem | nn0mulcl 9242 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
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Theorem | nn0addcli 9243 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | nn0mulcli 9244 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | nn0p1nn 9245 | A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.) |
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Theorem | peano2nn0 9246 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
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Theorem | nnm1nn0 9247 | A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.) |
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Theorem | elnn0nn 9248 | The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
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Theorem | elnnnn0 9249 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
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Theorem | elnnnn0b 9250 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
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Theorem | elnnnn0c 9251 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
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Theorem | nn0addge1 9252 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
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Theorem | nn0addge2 9253 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
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Theorem | nn0addge1i 9254 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
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Theorem | nn0addge2i 9255 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
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Theorem | nn0le2xi 9256 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | nn0lele2xi 9257 | 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
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Theorem | nn0supp 9258 |
Two ways to write the support of a function on ![]() |
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Theorem | nnnn0d 9259 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nn0red 9260 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nn0cnd 9261 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nn0ge0d 9262 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nn0addcld 9263 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nn0mulcld 9264 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nn0readdcl 9265 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
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Theorem | nn0ge2m1nn 9266 | If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.) |
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Theorem | nn0ge2m1nn0 9267 | If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) |
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Theorem | nn0nndivcl 9268 | Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
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The function values of the hash (set size) function are either nonnegative
integers or positive infinity. To avoid the need to distinguish between
finite and infinite sets (and therefore if the set size is a nonnegative
integer or positive infinity), it is useful to provide a definition of the
set of nonnegative integers extended by positive infinity, analogously to
the extension of the real numbers | ||
Syntax | cxnn0 9269 | The set of extended nonnegative integers. |
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Definition | df-xnn0 9270 |
Define the set of extended nonnegative integers that includes positive
infinity. Analogue of the extension of the real numbers ![]() |
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Theorem | elxnn0 9271 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0ssxnn0 9272 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0xnn0 9273 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0xr 9274 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
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Theorem | 0xnn0 9275 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
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Theorem | pnf0xnn0 9276 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0nepnf 9277 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0xnn0d 9278 | A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
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Theorem | nn0nepnfd 9279 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0nemnf 9280 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0xrnemnf 9281 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
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Theorem | xnn0nnn0pnf 9282 | An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
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Syntax | cz 9283 | Extend class notation to include the class of integers. |
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Definition | df-z 9284 | Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.) |
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Theorem | elz 9285 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
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Theorem | nnnegz 9286 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
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Theorem | zre 9287 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
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Theorem | zcn 9288 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
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Theorem | zrei 9289 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
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Theorem | zssre 9290 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
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Theorem | zsscn 9291 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
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Theorem | zex 9292 | The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
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Theorem | elnnz 9293 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
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Theorem | 0z 9294 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
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Theorem | 0zd 9295 | Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | elnn0z 9296 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
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Theorem | elznn0nn 9297 | Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
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Theorem | elznn0 9298 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
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Theorem | elznn 9299 | Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.) |
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Theorem | nnssz 9300 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) |
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