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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdecaddci2 9201 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  e.  NN0   &    |-  M  = ; A B   &    |-  ( A  +  1 )  =  D   &    |-  ( B  +  N )  = ; 1 0   =>    |-  ( M  +  N )  = ; D 0
 
Theoremdecsubi 9202 Difference between a numeral  M and a nonnegative integer  N (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  e.  NN0   &    |-  M  = ; A B   &    |-  ( A  +  1 )  =  D   &    |-  ( B  -  N )  =  C   =>    |-  ( M  -  N )  = ; A C
 
Theoremdecmul1 9203 The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
 |-  P  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  = ; A B   &    |-  D  e.  NN0   &    |-  ( A  x.  P )  =  C   &    |-  ( B  x.  P )  =  D   =>    |-  ( N  x.  P )  = ; C D
 
Theoremdecmul1c 9204 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  P  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  = ; A B   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  ( ( A  x.  P )  +  E )  =  C   &    |-  ( B  x.  P )  = ; E D   =>    |-  ( N  x.  P )  = ; C D
 
Theoremdecmul2c 9205 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
 |-  P  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  N  = ; A B   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  ( ( P  x.  A )  +  E )  =  C   &    |-  ( P  x.  B )  = ; E D   =>    |-  ( P  x.  N )  = ; C D
 
Theoremdecmulnc 9206 The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
 |-  N  e.  NN0   &    |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |-  ( N  x. ; A B )  = ; ( N  x.  A ) ( N  x.  B )
 
Theorem11multnc 9207 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
 |-  N  e.  NN0   =>    |-  ( N  x. ; 1 1 )  = ; N N
 
Theoremdecmul10add 9208 A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  M  e.  NN0   &    |-  E  =  ( M  x.  A )   &    |-  F  =  ( M  x.  B )   =>    |-  ( M  x. ; A B )  =  (; E 0  +  F )
 
Theorem6p5lem 9209 Lemma for 6p5e11 9212 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  D  e.  NN0   &    |-  E  e.  NN0   &    |-  B  =  ( D  +  1 )   &    |-  C  =  ( E  +  1 )   &    |-  ( A  +  D )  = ; 1 E   =>    |-  ( A  +  B )  = ; 1 C
 
Theorem5p5e10 9210 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
 |-  ( 5  +  5 )  = ; 1 0
 
Theorem6p4e10 9211 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  +  4 )  = ; 1 0
 
Theorem6p5e11 9212 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  +  5 )  = ; 1 1
 
Theorem6p6e12 9213 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  +  6 )  = ; 1 2
 
Theorem7p3e10 9214 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
 |-  ( 7  +  3 )  = ; 1 0
 
Theorem7p4e11 9215 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 7  +  4 )  = ; 1 1
 
Theorem7p5e12 9216 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  5 )  = ; 1 2
 
Theorem7p6e13 9217 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  6 )  = ; 1 3
 
Theorem7p7e14 9218 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  +  7 )  = ; 1 4
 
Theorem8p2e10 9219 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
 |-  ( 8  +  2 )  = ; 1 0
 
Theorem8p3e11 9220 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 8  +  3 )  = ; 1 1
 
Theorem8p4e12 9221 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  4 )  = ; 1 2
 
Theorem8p5e13 9222 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  5 )  = ; 1 3
 
Theorem8p6e14 9223 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  6 )  = ; 1 4
 
Theorem8p7e15 9224 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  7 )  = ; 1 5
 
Theorem8p8e16 9225 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  +  8 )  = ; 1 6
 
Theorem9p2e11 9226 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 9  +  2 )  = ; 1 1
 
Theorem9p3e12 9227 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  3 )  = ; 1 2
 
Theorem9p4e13 9228 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  4 )  = ; 1 3
 
Theorem9p5e14 9229 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  5 )  = ; 1 4
 
Theorem9p6e15 9230 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  6 )  = ; 1 5
 
Theorem9p7e16 9231 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  7 )  = ; 1 6
 
Theorem9p8e17 9232 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  8 )  = ; 1 7
 
Theorem9p9e18 9233 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  +  9 )  = ; 1 8
 
Theorem10p10e20 9234 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  (; 1 0  + ; 1 0 )  = ; 2
 0
 
Theorem10m1e9 9235 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
 |-  (; 1 0  -  1
 )  =  9
 
Theorem4t3lem 9236 Lemma for 4t3e12 9237 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  =  ( B  +  1 )   &    |-  ( A  x.  B )  =  D   &    |-  ( D  +  A )  =  E   =>    |-  ( A  x.  C )  =  E
 
Theorem4t3e12 9237 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  3
 )  = ; 1 2
 
Theorem4t4e16 9238 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 4  x.  4
 )  = ; 1 6
 
Theorem5t2e10 9239 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
 |-  ( 5  x.  2
 )  = ; 1 0
 
Theorem5t3e15 9240 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 5  x.  3
 )  = ; 1 5
 
Theorem5t4e20 9241 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 5  x.  4
 )  = ; 2 0
 
Theorem5t5e25 9242 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 5  x.  5
 )  = ; 2 5
 
Theorem6t2e12 9243 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  2
 )  = ; 1 2
 
Theorem6t3e18 9244 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  3
 )  = ; 1 8
 
Theorem6t4e24 9245 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 6  x.  4
 )  = ; 2 4
 
Theorem6t5e30 9246 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  x.  5
 )  = ; 3 0
 
Theorem6t6e36 9247 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 6  x.  6
 )  = ; 3 6
 
Theorem7t2e14 9248 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  2
 )  = ; 1 4
 
Theorem7t3e21 9249 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  3
 )  = ; 2 1
 
Theorem7t4e28 9250 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  4
 )  = ; 2 8
 
Theorem7t5e35 9251 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  5
 )  = ; 3 5
 
Theorem7t6e42 9252 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  6
 )  = ; 4 2
 
Theorem7t7e49 9253 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 7  x.  7
 )  = ; 4 9
 
Theorem8t2e16 9254 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  2
 )  = ; 1 6
 
Theorem8t3e24 9255 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  3
 )  = ; 2 4
 
Theorem8t4e32 9256 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  4
 )  = ; 3 2
 
Theorem8t5e40 9257 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 8  x.  5
 )  = ; 4 0
 
Theorem8t6e48 9258 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
 |-  ( 8  x.  6
 )  = ; 4 8
 
Theorem8t7e56 9259 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  7
 )  = ; 5 6
 
Theorem8t8e64 9260 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 8  x.  8
 )  = ; 6 4
 
Theorem9t2e18 9261 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  2
 )  = ; 1 8
 
Theorem9t3e27 9262 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  3
 )  = ; 2 7
 
Theorem9t4e36 9263 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  4
 )  = ; 3 6
 
Theorem9t5e45 9264 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  5
 )  = ; 4 5
 
Theorem9t6e54 9265 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  6
 )  = ; 5 4
 
Theorem9t7e63 9266 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  7
 )  = ; 6 3
 
Theorem9t8e72 9267 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  8
 )  = ; 7 2
 
Theorem9t9e81 9268 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  ( 9  x.  9
 )  = ; 8 1
 
Theorem9t11e99 9269 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
 |-  ( 9  x. ; 1 1 )  = ; 9
 9
 
Theorem9lt10 9270 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
 |-  9  < ; 1 0
 
Theorem8lt10 9271 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
 |-  8  < ; 1 0
 
Theorem7lt10 9272 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
 |-  7  < ; 1 0
 
Theorem6lt10 9273 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
 |-  6  < ; 1 0
 
Theorem5lt10 9274 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
 |-  5  < ; 1 0
 
Theorem4lt10 9275 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
 |-  4  < ; 1 0
 
Theorem3lt10 9276 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
 |-  3  < ; 1 0
 
Theorem2lt10 9277 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
 |-  2  < ; 1 0
 
Theorem1lt10 9278 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
 |-  1  < ; 1 0
 
Theoremdecbin0 9279 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   =>    |-  ( 4  x.  A )  =  ( 2  x.  ( 2  x.  A ) )
 
Theoremdecbin2 9280 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   =>    |-  ( ( 4  x.  A )  +  2 )  =  ( 2  x.  ( ( 2  x.  A )  +  1 ) )
 
Theoremdecbin3 9281 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN0   =>    |-  ( ( 4  x.  A )  +  3 )  =  ( ( 2  x.  ( ( 2  x.  A )  +  1 ) )  +  1 )
 
4.4.11  Upper sets of integers
 
Syntaxcuz 9282 Extend class notation with the upper integer function. Read " ZZ>= `  M " as "the set of integers greater than or equal to  M."
 class  ZZ>=
 
Definitiondf-uz 9283* Define a function whose value at  j is the semi-infinite set of contiguous integers starting at  j, which we will also call the upper integers starting at  j. Read " ZZ>= `  M " as "the set of integers greater than or equal to  M." See uzval 9284 for its value, uzssz 9301 for its relationship to  ZZ, nnuz 9317 and nn0uz 9316 for its relationships to  NN and  NN0, and eluz1 9286 and eluz2 9288 for its membership relations. (Contributed by NM, 5-Sep-2005.)
 |- 
 ZZ>=  =  ( j  e. 
 ZZ  |->  { k  e.  ZZ  |  j  <_  k }
 )
 
Theoremuzval 9284* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( N  e.  ZZ  ->  ( ZZ>= `  N )  =  { k  e.  ZZ  |  N  <_  k }
 )
 
Theoremuzf 9285 The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |- 
 ZZ>= : ZZ --> ~P ZZ
 
Theoremeluz1 9286 Membership in the upper set of integers starting at  M. (Contributed by NM, 5-Sep-2005.)
 |-  ( M  e.  ZZ  ->  ( N  e.  ( ZZ>=
 `  M )  <->  ( N  e.  ZZ  /\  M  <_  N ) ) )
 
Theoremeluzel2 9287 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  e.  ZZ )
 
Theoremeluz2 9288 Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show  M  e.  ZZ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( N  e.  ( ZZ>=
 `  M )  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N )
 )
 
Theoremeluz1i 9289 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
 |-  M  e.  ZZ   =>    |-  ( N  e.  ( ZZ>= `  M )  <->  ( N  e.  ZZ  /\  M  <_  N ) )
 
Theoremeluzuzle 9290 An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  ( ( B  e.  ZZ  /\  B  <_  A )  ->  ( C  e.  ( ZZ>= `  A )  ->  C  e.  ( ZZ>= `  B ) ) )
 
Theoremeluzelz 9291 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  N  e.  ZZ )
 
Theoremeluzelre 9292 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  N  e.  RR )
 
Theoremeluzelcn 9293 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  N  e.  CC )
 
Theoremeluzle 9294 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  <_  N )
 
Theoremeluz 9295 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  ( ZZ>= `  M )  <->  M 
 <_  N ) )
 
Theoremuzid 9296 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M ) )
 
Theoremuzn0 9297 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
 |-  ( M  e.  ran  ZZ>=  ->  M  =/=  (/) )
 
Theoremuztrn 9298 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
 |-  ( ( M  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  N ) )  ->  M  e.  ( ZZ>= `  N ) )
 
Theoremuztrn2 9299 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  K )   =>    |-  ( ( N  e.  Z  /\  M  e.  ( ZZ>=
 `  N ) ) 
 ->  M  e.  Z )
 
Theoremuzneg 9300 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  -u M  e.  ( ZZ>= `  -u N ) )
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