id sid tid token lemma pos iajs-1977 1 1 139 139 NUM iajs-1977 1 2 ibn ibn PROPN iajs-1977 1 3 al al PROPN iajs-1977 1 4 - - PUNCT iajs-1977 1 5 haitham haitham PROPN iajs-1977 1 6 jour jour X iajs-1977 1 7 . . PROPN iajs-1977 1 8 for for ADP iajs-1977 1 9 pure pure ADJ iajs-1977 1 10 & & CCONJ iajs-1977 1 11 appl appl PROPN iajs-1977 1 12 . . PUNCT iajs-1977 2 1 sc sc PROPN iajs-1977 2 2 i. i. PROPN iajs-1977 2 3 ihjpas ihjpas PROPN iajs-1977 2 4 https://doi.org/10.30526/32.1.1997 https://doi.org/10.30526/32.1.1997 PROPN iajs-1977 2 5 vol vol NOUN iajs-1977 2 6 . . PUNCT iajs-1977 3 1 32 32 NUM iajs-1977 3 2 ( ( PUNCT iajs-1977 3 3 1 1 NUM iajs-1977 3 4 ) ) PUNCT iajs-1977 3 5 2019 2019 NUM iajs-1977 3 6 ๐…๐ข๐ฑ๐ž๐ ๐…๐ข๐ฑ๐ž๐ PROPN iajs-1977 3 7 ๐๐จ๐ข๐ง๐ญ๐ฌ ๐๐จ๐ข๐ง๐ญ๐ฌ PROPN iajs-1977 3 8 ๐‘๐ž๐ฌ๐ฎ๐ฅ๐ญ๐ฌ ๐‘๐ž๐ฌ๐ฎ๐ฅ๐ญ๐ฌ PROPN iajs-1977 3 9 ๐ข๐ง ๐ข๐ง ADP iajs-1977 3 10 ๐‘ฎ ๐‘ฎ NOUN iajs-1977 3 11 โˆ’ โˆ’ PROPN iajs-1977 3 12 ๐Œ๐ž๐ญ๐ซ๐ข๐œ ๐Œ๐ž๐ญ๐ซ๐ข๐œ PROPN iajs-1977 3 13 ๐’๐ฉ๐š๐œ๐ž๐ฌ ๐’๐ฉ๐š๐œ๐ž๐ฌ PROPN iajs-1977 3 14 anaam anaam NOUN iajs-1977 3 15 neamah neamah PROPN iajs-1977 3 16 faraj faraj PROPN iajs-1977 3 17 anaamnema1@gmail.com anaamnema1@gmail.com X iajs-1977 3 18 salwa salwa PROPN iajs-1977 3 19 salman salman PROPN iajs-1977 3 20 abed abe VERB iajs-1977 3 21 salwaalbundi@yahoo.com salwaalbundi@yahoo.com X iajs-1977 3 22 departmen departman NOUN iajs-1977 3 23 of of ADP iajs-1977 3 24 mathmatics mathmatics PROPN iajs-1977 3 25 , , PUNCT iajs-1977 3 26 college college NOUN iajs-1977 3 27 of of ADP iajs-1977 3 28 education education NOUN iajs-1977 3 29 for for ADP iajs-1977 3 30 pure pure ADJ iajs-1977 3 31 sciences science NOUN iajs-1977 3 32 , , PUNCT iajs-1977 3 33 ibn ibn PROPN iajs-1977 3 34 al al PROPN iajs-1977 3 35 haitham haitham PROPN iajs-1977 3 36 , , PUNCT iajs-1977 3 37 university university PROPN iajs-1977 3 38 of of ADP iajs-1977 3 39 baghdad baghdad PROPN iajs-1977 3 40 , , PUNCT iajs-1977 3 41 baghdad baghdad PROPN iajs-1977 3 42 , , PUNCT iajs-1977 3 43 iraq iraq PROPN iajs-1977 3 44 . . PUNCT iajs-1977 4 1 article article NOUN iajs-1977 4 2 history history NOUN iajs-1977 4 3 : : PUNCT iajs-1977 4 4 received receive VERB iajs-1977 4 5 11 11 NUM iajs-1977 4 6 october october PROPN iajs-1977 4 7 2018,accepted 2018,accepted NUM iajs-1977 4 8 26 26 NUM iajs-1977 4 9 november november PROPN iajs-1977 4 10 2018,publish 2018,publish NUM iajs-1977 4 11 february february PROPN iajs-1977 4 12 2019 2019 NUM iajs-1977 4 13 abstract abstract NOUN iajs-1977 4 14 in in ADP iajs-1977 4 15 this this DET iajs-1977 4 16 paper paper NOUN iajs-1977 4 17 , , PUNCT iajs-1977 4 18 the the DET iajs-1977 4 19 concept concept NOUN iajs-1977 4 20 of of ADP iajs-1977 4 21 ๐น ๐น PROPN iajs-1977 4 22 โˆ’contraction โˆ’contraction NUM iajs-1977 4 23 mapping mapping NOUN iajs-1977 4 24 on on ADP iajs-1977 4 25 a a DET iajs-1977 4 26 ๐บ-metric ๐บ-metric ADJ iajs-1977 4 27 space space NOUN iajs-1977 4 28 is be AUX iajs-1977 4 29 extended extend VERB iajs-1977 4 30 with with ADP iajs-1977 4 31 a a DET iajs-1977 4 32 consideration consideration NOUN iajs-1977 4 33 on on ADP iajs-1977 4 34 local local ADJ iajs-1977 4 35 ๐น ๐น PROPN iajs-1977 4 36 โˆ’ โˆ’ PROPN iajs-1977 4 37 contraction contraction NOUN iajs-1977 4 38 . . PUNCT iajs-1977 5 1 as as ADP iajs-1977 5 2 a a DET iajs-1977 5 3 result result NOUN iajs-1977 5 4 , , PUNCT iajs-1977 5 5 two two NUM iajs-1977 5 6 fixed fix VERB iajs-1977 5 7 point point NOUN iajs-1977 5 8 theorems theorem NOUN iajs-1977 5 9 were be AUX iajs-1977 5 10 proved prove VERB iajs-1977 5 11 for for ADP iajs-1977 5 12 ๐น ๐น PROPN iajs-1977 5 13 โˆ’ โˆ’ PROPN iajs-1977 5 14 contraction contraction NOUN iajs-1977 5 15 on on ADP iajs-1977 5 16 a a DET iajs-1977 5 17 closed closed ADJ iajs-1977 5 18 ball ball NOUN iajs-1977 5 19 in in ADP iajs-1977 5 20 a a DET iajs-1977 5 21 complete complete ADJ iajs-1977 5 22 ๐บ-metric ๐บ-metric ADJ iajs-1977 5 23 space space NOUN iajs-1977 5 24 . . PUNCT iajs-1977 6 1 keywords keyword NOUN iajs-1977 6 2 : : PUNCT iajs-1977 6 3 ๐บ ๐บ PROPN iajs-1977 6 4 โˆ’ โˆ’ NOUN iajs-1977 6 5 metric metric ADJ iajs-1977 6 6 spaces space NOUN iajs-1977 6 7 , , PUNCT iajs-1977 6 8 local local ADJ iajs-1977 6 9 fixed fix VERB iajs-1977 6 10 points point NOUN iajs-1977 6 11 , , PUNCT iajs-1977 6 12 ๐น ๐น PROPN iajs-1977 6 13 โˆ’ โˆ’ NOUN iajs-1977 6 14 contractions contraction NOUN iajs-1977 6 15 . . PUNCT iajs-1977 7 1 1.introduction 1.introduction NUM iajs-1977 7 2 and and CCONJ iajs-1977 7 3 preliminaries preliminary NOUN iajs-1977 7 4 bapure bapure NOUN iajs-1977 7 5 dhage dhage NOUN iajs-1977 7 6 in in ADP iajs-1977 7 7 his his PRON iajs-1977 7 8 phd phd NOUN iajs-1977 7 9 thesis thesis NOUN iajs-1977 7 10 [ [ X iajs-1977 7 11 1992 1992 NUM iajs-1977 7 12 ] ] PUNCT iajs-1977 7 13 introduced introduce VERB iajs-1977 7 14 a a DET iajs-1977 7 15 new new ADJ iajs-1977 7 16 class class NOUN iajs-1977 7 17 of of ADP iajs-1977 7 18 generalized generalized ADJ iajs-1977 7 19 metric metric ADJ iajs-1977 7 20 spaces space NOUN iajs-1977 7 21 , , PUNCT iajs-1977 7 22 named name VERB iajs-1977 7 23 d d PROPN iajs-1977 7 24 metric metric ADJ iajs-1977 7 25 spaces space NOUN iajs-1977 7 26 . . PUNCT iajs-1977 8 1 mustafa mustafa PROPN iajs-1977 8 2 and and CCONJ iajs-1977 8 3 sims sim NOUN iajs-1977 8 4 proved prove VERB iajs-1977 8 5 that that SCONJ iajs-1977 8 6 most most ADJ iajs-1977 8 7 of of ADP iajs-1977 8 8 the the DET iajs-1977 8 9 claims claim NOUN iajs-1977 8 10 concerning concern VERB iajs-1977 8 11 the the DET iajs-1977 8 12 fundamental fundamental ADJ iajs-1977 8 13 structures structure NOUN iajs-1977 8 14 on on ADP iajs-1977 8 15 d d X iajs-1977 8 16 metric metric ADJ iajs-1977 8 17 spaces space NOUN iajs-1977 8 18 are be AUX iajs-1977 8 19 incorrect incorrect ADJ iajs-1977 8 20 and and CCONJ iajs-1977 8 21 introduced introduce VERB iajs-1977 8 22 an an DET iajs-1977 8 23 appropriate appropriate ADJ iajs-1977 8 24 notion notion NOUN iajs-1977 8 25 of of ADP iajs-1977 8 26 d d PROPN iajs-1977 8 27 metric metric ADJ iajs-1977 8 28 space space NOUN iajs-1977 8 29 , , PUNCT iajs-1977 8 30 named name VERB iajs-1977 8 31 g g NOUN iajs-1977 8 32 - - PUNCT iajs-1977 8 33 metric metric ADJ iajs-1977 8 34 spaces space NOUN iajs-1977 8 35 . . PUNCT iajs-1977 9 1 in in ADP iajs-1977 9 2 fact fact NOUN iajs-1977 9 3 , , PUNCT iajs-1977 9 4 mustafa mustafa NOUN iajs-1977 9 5 , , PUNCT iajs-1977 9 6 sims sim NOUN iajs-1977 9 7 and and CCONJ iajs-1977 9 8 other other ADJ iajs-1977 9 9 authors author NOUN iajs-1977 9 10 introduced introduce VERB iajs-1977 9 11 many many ADJ iajs-1977 9 12 fixed fix VERB iajs-1977 9 13 point point NOUN iajs-1977 9 14 results result NOUN iajs-1977 9 15 for for ADP iajs-1977 9 16 self self NOUN iajs-1977 9 17 mappings mapping NOUN iajs-1977 9 18 in in ADP iajs-1977 9 19 g g PROPN iajs-1977 9 20 metric metric ADJ iajs-1977 9 21 spaces space NOUN iajs-1977 9 22 under under ADP iajs-1977 9 23 certain certain ADJ iajs-1977 9 24 conditions condition NOUN iajs-1977 9 25 . . PUNCT iajs-1977 10 1 actually actually ADV iajs-1977 10 2 , , PUNCT iajs-1977 10 3 the the DET iajs-1977 10 4 method method NOUN iajs-1977 10 5 is be AUX iajs-1977 10 6 used use VERB iajs-1977 10 7 in in ADP iajs-1977 10 8 the the DET iajs-1977 10 9 study study NOUN iajs-1977 10 10 of of ADP iajs-1977 10 11 fixed fix VERB iajs-1977 10 12 points point NOUN iajs-1977 10 13 in in ADP iajs-1977 10 14 metric metric ADJ iajs-1977 10 15 spaces space NOUN iajs-1977 10 16 , , PUNCT iajs-1977 10 17 and and CCONJ iajs-1977 10 18 symmetric symmetric ADJ iajs-1977 10 19 spaces space NOUN iajs-1977 10 20 . . PUNCT iajs-1977 11 1 in in ADP iajs-1977 11 2 this this DET iajs-1977 11 3 paper paper NOUN iajs-1977 11 4 , , PUNCT iajs-1977 11 5 a a DET iajs-1977 11 6 general general ADJ iajs-1977 11 7 fixed fix VERB iajs-1977 11 8 point point NOUN iajs-1977 11 9 theorem theorem NOUN iajs-1977 11 10 for for ADP iajs-1977 11 11 pairs pair NOUN iajs-1977 11 12 of of ADP iajs-1977 11 13 non non PRON iajs-1977 11 14 weakly weakly ADJ iajs-1977 11 15 compatible compatible ADJ iajs-1977 11 16 mappings mapping NOUN iajs-1977 11 17 in in ADP iajs-1977 11 18 g g PROPN iajs-1977 11 19 metric metric ADJ iajs-1977 11 20 space space NOUN iajs-1977 11 21 is be AUX iajs-1977 11 22 proved prove VERB iajs-1977 11 23 . . PUNCT iajs-1977 12 1 in in ADP iajs-1977 12 2 the the DET iajs-1977 12 3 case case NOUN iajs-1977 12 4 of of ADP iajs-1977 12 5 a a DET iajs-1977 12 6 single single ADJ iajs-1977 12 7 mapping mapping NOUN iajs-1977 12 8 some some DET iajs-1977 12 9 results result NOUN iajs-1977 12 10 . . PUNCT iajs-1977 13 1 in in ADP iajs-1977 13 2 2012 2012 NUM iajs-1977 13 3 , , PUNCT iajs-1977 13 4 wardowski wardowski PROPN iajs-1977 13 5 introduced introduce VERB iajs-1977 13 6 a a DET iajs-1977 13 7 new new ADJ iajs-1977 13 8 concept concept NOUN iajs-1977 13 9 for for ADP iajs-1977 13 10 contraction contraction NOUN iajs-1977 13 11 mappings mapping NOUN iajs-1977 13 12 as as SCONJ iajs-1977 13 13 called call VERB iajs-1977 13 14 f f NOUN iajs-1977 13 15 - - PUNCT iajs-1977 13 16 contraction contraction NOUN iajs-1977 13 17 by by ADP iajs-1977 13 18 considering consider VERB iajs-1977 13 19 a a DET iajs-1977 13 20 class class NOUN iajs-1977 13 21 of of ADP iajs-1977 13 22 real real ADJ iajs-1977 13 23 valued value VERB iajs-1977 13 24 functions function NOUN iajs-1977 13 25 . . PUNCT iajs-1977 14 1 let let VERB iajs-1977 14 2 โ„ณ โ„ณ PRON iajs-1977 14 3 be be AUX iajs-1977 14 4 a a DET iajs-1977 14 5 nonempty nonempty ADV iajs-1977 14 6 set set VERB iajs-1977 14 7 and and CCONJ iajs-1977 14 8 ฯ… ฯ… NOUN iajs-1977 14 9 : : PUNCT iajs-1977 14 10 โ„ณร— โ„ณร— PROPN iajs-1977 14 11 โ„ณ โ„ณ PROPN iajs-1977 14 12 ร— ร— NOUN iajs-1977 14 13 โ„ณ โ„ณ NOUN iajs-1977 14 14 โ†’ โ†’ PUNCT iajs-1977 14 15 โ„+ โ„+ PUNCT iajs-1977 14 16 be be AUX iajs-1977 14 17 a a DET iajs-1977 14 18 function function NOUN iajs-1977 14 19 satisfying satisfy VERB iajs-1977 14 20 the the DET iajs-1977 14 21 following follow VERB iajs-1977 14 22 condition condition NOUN iajs-1977 14 23 : : PUNCT iajs-1977 15 1 1๐›ถ 1๐›ถ PROPN iajs-1977 15 2 ( ( PUNCT iajs-1977 15 3 ๐‘ž ๐‘ž PROPN iajs-1977 15 4 , , PUNCT iajs-1977 15 5 ๐‘ข ๐‘ข PROPN iajs-1977 15 6 , , PUNCT iajs-1977 15 7 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 15 8 ) ) PUNCT iajs-1977 15 9 = = SYM iajs-1977 15 10 0 0 PUNCT iajs-1977 16 1 if if SCONJ iajs-1977 16 2 and and CCONJ iajs-1977 16 3 only only ADV iajs-1977 16 4 if if SCONJ iajs-1977 16 5 ๐‘ž ๐‘ž X iajs-1977 16 6 = = X iajs-1977 16 7 ๐‘ข ๐‘ข PROPN iajs-1977 16 8 = = SYM iajs-1977 16 9 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 16 10 , , PUNCT iajs-1977 16 11 2 2 NUM iajs-1977 16 12 0 0 NUM iajs-1977 16 13 < < X iajs-1977 16 14 ๐›ถ ๐›ถ PROPN iajs-1977 16 15 ( ( PUNCT iajs-1977 16 16 ๐‘ž ๐‘ž PROPN iajs-1977 16 17 , , PUNCT iajs-1977 16 18 ๐‘ž ๐‘ž PROPN iajs-1977 16 19 , , PUNCT iajs-1977 16 20 ๐‘ข ๐‘ข NOUN iajs-1977 16 21 ) ) PUNCT iajs-1977 16 22 , , PUNCT iajs-1977 16 23 โˆ€ โˆ€ X iajs-1977 16 24 ๐‘ž ๐‘ž NOUN iajs-1977 16 25 , , PUNCT iajs-1977 16 26 ๐‘ข ๐‘ข PROPN iajs-1977 16 27 โˆˆ โˆˆ PROPN iajs-1977 16 28 โ„ณ โ„ณ NOUN iajs-1977 16 29 ๐‘ค๐‘–๐‘กโ„Ž ๐‘ค๐‘–๐‘กโ„Ž NOUN iajs-1977 16 30 ๐‘ž ๐‘ž X iajs-1977 16 31 โ‰  โ‰  PROPN iajs-1977 16 32 ๐‘ข ๐‘ข PROPN iajs-1977 16 33 , , PUNCT iajs-1977 16 34 3๐›ถ 3๐›ถ PROPN iajs-1977 16 35 ( ( PUNCT iajs-1977 16 36 ๐‘ž ๐‘ž PROPN iajs-1977 16 37 , , PUNCT iajs-1977 16 38 ๐‘ž ๐‘ž PROPN iajs-1977 16 39 , , PUNCT iajs-1977 16 40 ๐‘ข ๐‘ข NOUN iajs-1977 16 41 ) ) PUNCT iajs-1977 16 42 โ‰ค โ‰ค PUNCT iajs-1977 16 43 ๐›ถ ๐›ถ PROPN iajs-1977 16 44 ( ( PUNCT iajs-1977 16 45 ๐‘ž ๐‘ž PROPN iajs-1977 16 46 , , PUNCT iajs-1977 16 47 ๐‘ข ๐‘ข PROPN iajs-1977 16 48 , , PUNCT iajs-1977 16 49 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 16 50 ) ) PUNCT iajs-1977 16 51 , , PUNCT iajs-1977 16 52 โˆ€ โˆ€ X iajs-1977 16 53 ๐‘ž ๐‘ž PROPN iajs-1977 16 54 , , PUNCT iajs-1977 16 55 ๐‘ข ๐‘ข PROPN iajs-1977 16 56 , , PUNCT iajs-1977 16 57 ๐‘ฃ ๐‘ฃ PRON iajs-1977 16 58 โˆˆ โˆˆ PROPN iajs-1977 16 59 โ„ณ โ„ณ NOUN iajs-1977 16 60 ๐‘ค๐‘–๐‘กโ„Ž ๐‘ค๐‘–๐‘กโ„Ž NOUN iajs-1977 16 61 ๐‘ข ๐‘ข PRON iajs-1977 16 62 โ‰  โ‰  PROPN iajs-1977 16 63 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 16 64 , , PUNCT iajs-1977 16 65 4๐›ถ 4๐›ถ NOUN iajs-1977 16 66 ( ( PUNCT iajs-1977 16 67 ๐‘ž ๐‘ž PROPN iajs-1977 16 68 , , PUNCT iajs-1977 16 69 ๐‘ข ๐‘ข PROPN iajs-1977 16 70 , , PUNCT iajs-1977 16 71 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 16 72 ) ) PUNCT iajs-1977 16 73 = = SYM iajs-1977 16 74 ๐›ถ(๐‘ž ๐›ถ(๐‘ž PROPN iajs-1977 16 75 , , PUNCT iajs-1977 16 76 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 16 77 , , PUNCT iajs-1977 16 78 ๐‘ข ๐‘ข X iajs-1977 16 79 ) ) PUNCT iajs-1977 16 80 = = SYM iajs-1977 16 81 โ€ฆ โ€ฆ PUNCT iajs-1977 16 82 , , PUNCT iajs-1977 16 83 ( ( PUNCT iajs-1977 16 84 symmetry symmetry NOUN iajs-1977 16 85 in in ADP iajs-1977 16 86 all all DET iajs-1977 16 87 three three NUM iajs-1977 16 88 vairables vairable NOUN iajs-1977 16 89 ) ) PUNCT iajs-1977 16 90 , , PUNCT iajs-1977 16 91 5๐›ถ 5๐›ถ PROPN iajs-1977 16 92 ( ( PUNCT iajs-1977 16 93 ๐‘ž ๐‘ž PROPN iajs-1977 16 94 , , PUNCT iajs-1977 16 95 ๐‘ข ๐‘ข PROPN iajs-1977 16 96 , , PUNCT iajs-1977 16 97 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 16 98 ) ) PUNCT iajs-1977 16 99 โ‰ค โ‰ค NOUN iajs-1977 16 100 ๐›ถ ๐›ถ PROPN iajs-1977 16 101 ( ( PUNCT iajs-1977 16 102 ๐‘ž ๐‘ž PROPN iajs-1977 16 103 , , PUNCT iajs-1977 16 104 ๐‘Ž ๐‘Ž PROPN iajs-1977 16 105 , , PUNCT iajs-1977 16 106 ๐‘Ž ๐‘Ž NOUN iajs-1977 16 107 ) ) PUNCT iajs-1977 16 108 + + CCONJ iajs-1977 16 109 ๐›ถ ๐›ถ PROPN iajs-1977 16 110 ( ( PUNCT iajs-1977 16 111 ๐‘Ž ๐‘Ž X iajs-1977 16 112 , , PUNCT iajs-1977 16 113 ๐‘ข ๐‘ข X iajs-1977 16 114 , , PUNCT iajs-1977 16 115 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 16 116 ) ) PUNCT iajs-1977 16 117 , , PUNCT iajs-1977 16 118 โˆ€ โˆ€ X iajs-1977 16 119 ๐‘ž ๐‘ž NOUN iajs-1977 16 120 , , PUNCT iajs-1977 16 121 ๐‘ข ๐‘ข PROPN iajs-1977 16 122 , , PUNCT iajs-1977 16 123 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 16 124 , , PUNCT iajs-1977 16 125 ๐‘Ž ๐‘Ž PROPN iajs-1977 16 126 โˆˆ โˆˆ NOUN iajs-1977 16 127 โ„ณ. โ„ณ. NOUN iajs-1977 16 128 then then ADV iajs-1977 16 129 the the DET iajs-1977 16 130 function function NOUN iajs-1977 16 131 ๐›ถ ๐›ถ PROPN iajs-1977 16 132 is be AUX iajs-1977 16 133 called call VERB iajs-1977 16 134 generalized generalized ADJ iajs-1977 16 135 metric metric NOUN iajs-1977 16 136 on on ADP iajs-1977 16 137 โ„ณ โ„ณ PROPN iajs-1977 16 138 [ [ NOUN iajs-1977 16 139 1 1 NUM iajs-1977 16 140 ] ] PUNCT iajs-1977 16 141 and and CCONJ iajs-1977 16 142 the the DET iajs-1977 16 143 pair pair NOUN iajs-1977 16 144 ( ( PUNCT iajs-1977 16 145 โ„ณ,๐›ถ โ„ณ,๐›ถ ADJ iajs-1977 16 146 ) ) PUNCT iajs-1977 16 147 is be AUX iajs-1977 16 148 called call VERB iajs-1977 16 149 a a DET iajs-1977 16 150 ๐บmetric ๐บmetric ADJ iajs-1977 16 151 space space NOUN iajs-1977 16 152 . . PUNCT iajs-1977 17 1 a a DET iajs-1977 17 2 ๐บ ๐บ PROPN iajs-1977 17 3 -metric -metric ADJ iajs-1977 17 4 space space NOUN iajs-1977 17 5 โ„ณ โ„ณ NOUN iajs-1977 17 6 is be AUX iajs-1977 17 7 called call VERB iajs-1977 17 8 a a DET iajs-1977 17 9 symmetric symmetric ADJ iajs-1977 17 10 [ [ X iajs-1977 17 11 2 2 NUM iajs-1977 17 12 ] ] X iajs-1977 17 13 if if SCONJ iajs-1977 17 14 โˆ€ โˆ€ X iajs-1977 17 15 ๐‘ž ๐‘ž X iajs-1977 17 16 , , PUNCT iajs-1977 17 17 ๐‘ข ๐‘ข PROPN iajs-1977 17 18 , , PUNCT iajs-1977 17 19 ๐‘ฃ ๐‘ฃ PRON iajs-1977 17 20 โˆˆ โˆˆ PROPN iajs-1977 17 21 โ„ณ โ„ณ PROPN iajs-1977 17 22 ๐›ถ ๐›ถ PROPN iajs-1977 17 23 ( ( PUNCT iajs-1977 17 24 ๐‘ž ๐‘ž PROPN iajs-1977 17 25 , , PUNCT iajs-1977 17 26 ๐‘ข ๐‘ข PROPN iajs-1977 17 27 , , PUNCT iajs-1977 17 28 ๐‘ข ๐‘ข NOUN iajs-1977 17 29 ) ) PUNCT iajs-1977 17 30 = = SYM iajs-1977 17 31 ๐›ถ ๐›ถ PROPN iajs-1977 17 32 ( ( PUNCT iajs-1977 17 33 ๐‘ž ๐‘ž PROPN iajs-1977 17 34 , , PUNCT iajs-1977 17 35 ๐‘ž ๐‘ž PROPN iajs-1977 17 36 , , PUNCT iajs-1977 17 37 ๐‘ข ๐‘ข NOUN iajs-1977 17 38 ) ) PUNCT iajs-1977 17 39 many many ADJ iajs-1977 17 40 results result NOUN iajs-1977 17 41 and and CCONJ iajs-1977 17 42 examples example NOUN iajs-1977 17 43 about about ADP iajs-1977 17 44 ๐›ถ-metric ๐›ถ-metric ADJ iajs-1977 17 45 space space NOUN iajs-1977 17 46 and and CCONJ iajs-1977 17 47 its its PRON iajs-1977 17 48 generalization generalization NOUN iajs-1977 17 49 one one PRON iajs-1977 17 50 can can AUX iajs-1977 17 51 found find VERB iajs-1977 17 52 in in ADP iajs-1977 17 53 [ [ PUNCT iajs-1977 17 54 2 2 NUM iajs-1977 17 55 - - SYM iajs-1977 17 56 10 10 NUM iajs-1977 17 57 ] ] PUNCT iajs-1977 17 58 . . PUNCT iajs-1977 18 1 proposition proposition NOUN iajs-1977 18 2 1 1 NUM iajs-1977 19 1 [ [ X iajs-1977 19 2 5 5 NUM iajs-1977 19 3 ] ] PUNCT iajs-1977 19 4 : : PUNCT iajs-1977 19 5 let let VERB iajs-1977 19 6 ( ( PUNCT iajs-1977 19 7 โ„ณ โ„ณ PROPN iajs-1977 19 8 , , PUNCT iajs-1977 19 9 ๐›ถ ๐›ถ PROPN iajs-1977 19 10 ) ) PUNCT iajs-1977 19 11 be be AUX iajs-1977 19 12 a a DET iajs-1977 19 13 ๐บ-metric ๐บ-metric ADJ iajs-1977 19 14 space space NOUN iajs-1977 19 15 , , PUNCT iajs-1977 19 16 then then ADV iajs-1977 19 17 the the DET iajs-1977 19 18 following following ADJ iajs-1977 19 19 statements statement NOUN iajs-1977 19 20 are be AUX iajs-1977 19 21 equivalent equivalent ADJ iajs-1977 19 22 : : PUNCT iajs-1977 19 23 1-(โ„ณ 1-(โ„ณ NUM iajs-1977 19 24 , , PUNCT iajs-1977 19 25 ๐›ถ ๐›ถ PROPN iajs-1977 19 26 ) ) PUNCT iajs-1977 19 27 is be AUX iajs-1977 19 28 symmetric symmetric ADJ iajs-1977 19 29 . . PUNCT iajs-1977 20 1 2-๐›ถ 2-๐›ถ NUM iajs-1977 20 2 ( ( PUNCT iajs-1977 20 3 ๐‘ž ๐‘ž NOUN iajs-1977 20 4 , , PUNCT iajs-1977 20 5 ๐‘ข ๐‘ข PROPN iajs-1977 20 6 , , PUNCT iajs-1977 20 7 ๐‘ข ๐‘ข NOUN iajs-1977 20 8 ) ) PUNCT iajs-1977 20 9 โ‰ค โ‰ค PUNCT iajs-1977 21 1 ๐›ถ ๐›ถ PROPN iajs-1977 21 2 ( ( PUNCT iajs-1977 21 3 ๐‘ž ๐‘ž PROPN iajs-1977 21 4 , , PUNCT iajs-1977 21 5 ๐‘ข ๐‘ข PROPN iajs-1977 21 6 , , PUNCT iajs-1977 21 7 ๐‘Ž ๐‘Ž NOUN iajs-1977 21 8 ) ) PUNCT iajs-1977 21 9 for for ADP iajs-1977 21 10 all all DET iajs-1977 21 11 ๐‘ž ๐‘ž PROPN iajs-1977 21 12 , , PUNCT iajs-1977 21 13 ๐‘ข ๐‘ข PROPN iajs-1977 21 14 , , PUNCT iajs-1977 21 15 ๐‘Ž ๐‘Ž PROPN iajs-1977 21 16 โˆˆ โˆˆ PROPN iajs-1977 21 17 โ„ณ โ„ณ PROPN iajs-1977 21 18 , , PUNCT iajs-1977 21 19 3-๐›ถ 3-๐›ถ NUM iajs-1977 21 20 ( ( PUNCT iajs-1977 21 21 ๐‘ž ๐‘ž PROPN iajs-1977 21 22 , , PUNCT iajs-1977 21 23 ๐‘ข ๐‘ข PROPN iajs-1977 21 24 , , PUNCT iajs-1977 21 25 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 21 26 ) ) PUNCT iajs-1977 21 27 โ‰ค โ‰ค NOUN iajs-1977 21 28 ๐›ถ ๐›ถ PROPN iajs-1977 21 29 ( ( PUNCT iajs-1977 21 30 ๐‘ž ๐‘ž PROPN iajs-1977 21 31 , , PUNCT iajs-1977 21 32 ๐‘ข ๐‘ข PROPN iajs-1977 21 33 , , PUNCT iajs-1977 21 34 ๐‘Ž ๐‘Ž NOUN iajs-1977 21 35 ) ) PUNCT iajs-1977 21 36 + + CCONJ iajs-1977 21 37 ๐›ถ(๐‘ฃ ๐›ถ(๐‘ฃ NOUN iajs-1977 21 38 , , PUNCT iajs-1977 21 39 ๐‘ข ๐‘ข X iajs-1977 21 40 , , PUNCT iajs-1977 21 41 ๐‘ ๐‘ NOUN iajs-1977 21 42 ) ) PUNCT iajs-1977 21 43 for for ADP iajs-1977 21 44 all all DET iajs-1977 21 45 ๐‘ž ๐‘ž PROPN iajs-1977 21 46 , , PUNCT iajs-1977 21 47 ๐‘ข ๐‘ข X iajs-1977 21 48 , , PUNCT iajs-1977 21 49 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 21 50 , , PUNCT iajs-1977 21 51 ๐‘Ž ๐‘Ž NOUN iajs-1977 21 52 , , PUNCT iajs-1977 21 53 ๐‘ ๐‘ PROPN iajs-1977 21 54 โˆˆ โˆˆ PROPN iajs-1977 21 55 โ„ณ. โ„ณ. PROPN iajs-1977 21 56 https://doi.org/10.30526/32.1.1997 https://doi.org/10.30526/32.1.1997 VERB iajs-1977 21 57 mailto:anaamnema1@gmail.com mailto:anaamnema1@gmail.com PROPN iajs-1977 21 58 mailto:salwaalbundi@yahoo.com mailto:salwaalbundi@yahoo.com X iajs-1977 21 59 ย  ย  ย  ย  SPACE iajs-1977 21 60 140 140 NUM iajs-1977 21 61 ibn ibn PROPN iajs-1977 21 62 al al PROPN iajs-1977 21 63 - - PUNCT iajs-1977 21 64 haitham haitham PROPN iajs-1977 21 65 jour jour X iajs-1977 21 66 . . PROPN iajs-1977 22 1 for for ADP iajs-1977 22 2 pure pure ADJ iajs-1977 22 3 & & CCONJ iajs-1977 22 4 appl appl PROPN iajs-1977 22 5 . . PUNCT iajs-1977 23 1 sc sc PROPN iajs-1977 23 2 i. i. PROPN iajs-1977 23 3 ihjpas ihjpas PROPN iajs-1977 23 4 ย  ย  SPACE iajs-1977 23 5 https://doi.org/10.30526/32.1.1977 https://doi.org/10.30526/32.1.1977 NOUN iajs-1977 23 6 ย  ย  SPACE iajs-1977 23 7 vol vol NOUN iajs-1977 23 8 . . PUNCT iajs-1977 24 1 32 32 NUM iajs-1977 24 2 ( ( PUNCT iajs-1977 24 3 1 1 NUM iajs-1977 24 4 ) ) PUNCT iajs-1977 24 5 2019 2019 NUM iajs-1977 24 6 the the DET iajs-1977 24 7 ฯ… ฯ… NOUN iajs-1977 24 8 - - NOUN iajs-1977 24 9 ball ball NOUN iajs-1977 24 10 with with ADP iajs-1977 24 11 center center NOUN iajs-1977 24 12 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 24 13 and and CCONJ iajs-1977 24 14 radius radius NOUN iajs-1977 24 15 ๐œ– ๐œ– PROPN iajs-1977 24 16 0 0 PUNCT iajs-1977 24 17 is be AUX iajs-1977 24 18 ๐ต ๐ต NOUN iajs-1977 24 19 ( ( PUNCT iajs-1977 24 20 ๐‘Ÿ ๐‘Ÿ X iajs-1977 24 21 , , PUNCT iajs-1977 24 22 ๐œ– ๐œ– PROPN iajs-1977 24 23 ) ) PUNCT iajs-1977 25 1 [ [ X iajs-1977 25 2 10 10 NUM iajs-1977 25 3 ] ] PUNCT iajs-1977 25 4 is be AUX iajs-1977 25 5 : : PUNCT iajs-1977 25 6 ๐ต ๐ต NOUN iajs-1977 25 7 ( ( PUNCT iajs-1977 25 8 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 25 9 , , PUNCT iajs-1977 25 10 ๐œ– ๐œ– PROPN iajs-1977 25 11 ) ) PUNCT iajs-1977 25 12 ย  ย  SPACE iajs-1977 26 1 = = NOUN iajs-1977 26 2 ย  ย  SPACE iajs-1977 26 3 { { PUNCT iajs-1977 26 4 s s NOUN iajs-1977 26 5 โˆˆ โˆˆ PROPN iajs-1977 26 6 โ„ณ โ„ณ PROPN iajs-1977 26 7 โˆถ โˆถ NOUN iajs-1977 26 8 ย  ย  SPACE iajs-1977 26 9 ฯ… ฯ… NOUN iajs-1977 26 10 ( ( PUNCT iajs-1977 26 11 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 26 12 , , PUNCT iajs-1977 26 13 ย  ย  SPACE iajs-1977 26 14 s s PROPN iajs-1977 26 15 , , PUNCT iajs-1977 26 16 ย  ย  SPACE iajs-1977 26 17 and and CCONJ iajs-1977 26 18 ย  ย  SPACE iajs-1977 26 19 s s PART iajs-1977 26 20 ) ) PUNCT iajs-1977 26 21 ๐œ– ๐œ– PROPN iajs-1977 26 22 . . PUNCT iajs-1977 26 23 ย  ย  SPACE iajs-1977 26 24 the the DET iajs-1977 26 25 sequence sequence NOUN iajs-1977 26 26 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 26 27 in in ADP iajs-1977 26 28 a a DET iajs-1977 26 29 ๐บ ๐บ PROPN iajs-1977 26 30 metric metric ADJ iajs-1977 26 31 space space NOUN iajs-1977 26 32 โ„ณ โ„ณ PROPN iajs-1977 26 33 , , PUNCT iajs-1977 26 34 ๐›ถ ๐›ถ PROPN iajs-1977 26 35 is be AUX iajs-1977 26 36 said say VERB iajs-1977 26 37 to to PART iajs-1977 26 38 be be AUX iajs-1977 26 39 ย ย  ย ย  SPACE iajs-1977 26 40 1๐›ถ 1๐›ถ ADJ iajs-1977 26 41 convergent convergent NOUN iajs-1977 26 42 to to ADP iajs-1977 26 43 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 26 44 if if SCONJ iajs-1977 26 45 โˆƒ๐‘˜ โˆƒ๐‘˜ PROPN iajs-1977 26 46 โˆˆ โˆˆ PROPN iajs-1977 26 47 ๐‘ ๐‘ PROPN iajs-1977 26 48 , , PUNCT iajs-1977 26 49 ๐œ– ๐œ– PROPN iajs-1977 26 50 0 0 NUM iajs-1977 26 51 for for ADP iajs-1977 26 52 ๐‘Ž๐‘™๐‘™ ๐‘Ž๐‘™๐‘™ NOUN iajs-1977 26 53 ๐‘š ๐‘š PROPN iajs-1977 26 54 , , PUNCT iajs-1977 26 55 ๐‘› ๐‘› PRON iajs-1977 26 56 ๐‘˜ ๐‘˜ NOUN iajs-1977 26 57 such such ADJ iajs-1977 26 58 that that SCONJ iajs-1977 26 59 ๐›ถ ๐›ถ PROPN iajs-1977 26 60 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 26 61 , , PUNCT iajs-1977 26 62 ๐‘Ÿ ๐‘Ÿ X iajs-1977 26 63 , , PUNCT iajs-1977 26 64 ๐‘Ÿ ๐‘Ÿ X iajs-1977 26 65 ๐œ–. ๐œ–. ADJ iajs-1977 26 66 2๐›ถ 2๐›ถ PROPN iajs-1977 26 67 โ€“ โ€“ PUNCT iajs-1977 26 68 cauchy cauchy PROPN iajs-1977 26 69 if if SCONJ iajs-1977 26 70 โˆƒ โˆƒ PROPN iajs-1977 26 71 ๐‘˜ ๐‘˜ X iajs-1977 26 72 โˆˆ โˆˆ PROPN iajs-1977 26 73 ๐‘ ๐‘ PROPN iajs-1977 26 74 , , PUNCT iajs-1977 26 75 ๐œ– ๐œ– PROPN iajs-1977 26 76 0 0 NUM iajs-1977 26 77 for for ADP iajs-1977 26 78 ๐‘Ž๐‘™๐‘™ ๐‘Ž๐‘™๐‘™ NOUN iajs-1977 26 79 ๐‘š ๐‘š PROPN iajs-1977 26 80 , , PUNCT iajs-1977 26 81 ๐‘› ๐‘› PROPN iajs-1977 26 82 , , PUNCT iajs-1977 26 83 ๐‘™ ๐‘™ PROPN iajs-1977 26 84 ๐‘˜ ๐‘˜ INTJ iajs-1977 26 85 such such ADJ iajs-1977 26 86 that that SCONJ iajs-1977 26 87 ฯ… ฯ… NOUN iajs-1977 26 88 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 26 89 , , PUNCT iajs-1977 26 90 ๐‘Ÿ ๐‘Ÿ X iajs-1977 26 91 , , PUNCT iajs-1977 26 92 ๐‘Ÿ ๐‘Ÿ X iajs-1977 26 93 ) ) PUNCT iajs-1977 26 94 ๐œ– ๐œ– PROPN iajs-1977 26 95 . . PUNCT iajs-1977 27 1 a a DET iajs-1977 27 2 ๐บโ€“metric ๐บโ€“metric ADJ iajs-1977 27 3 space space NOUN iajs-1977 27 4 โ„ณ โ„ณ PROPN iajs-1977 27 5 , , PUNCT iajs-1977 27 6 ๐›ถ ๐›ถ PROPN iajs-1977 27 7 is be AUX iajs-1977 27 8 complete complete ADJ iajs-1977 27 9 if if SCONJ iajs-1977 27 10 every every DET iajs-1977 27 11 ๐›ถ ๐›ถ PROPN iajs-1977 27 12 -cauchy -cauchy ADJ iajs-1977 27 13 sequence sequence NOUN iajs-1977 27 14 ( ( PUNCT iajs-1977 27 15 โ„ณ โ„ณ PROPN iajs-1977 27 16 , , PUNCT iajs-1977 27 17 ฯ… ฯ… NOUN iajs-1977 27 18 ) ) PUNCT iajs-1977 27 19 is be AUX iajs-1977 27 20 ฯ…convergent ฯ…convergent VERB iajs-1977 27 21 in in ADP iajs-1977 27 22 ( ( PUNCT iajs-1977 27 23 โ„ณ โ„ณ PROPN iajs-1977 27 24 , , PUNCT iajs-1977 27 25 ฯ… ฯ… NOUN iajs-1977 27 26 1 1 NUM iajs-1977 27 27 . . PUNCT iajs-1977 28 1 proposition proposition NOUN iajs-1977 28 2 2 2 NUM iajs-1977 29 1 [ [ X iajs-1977 29 2 11 11 NUM iajs-1977 29 3 ] ] PUNCT iajs-1977 29 4 : : PUNCT iajs-1977 29 5 let let VERB iajs-1977 29 6 โ„ณ โ„ณ PROPN iajs-1977 29 7 , , PUNCT iajs-1977 29 8 ๐›ถ ๐›ถ PROPN iajs-1977 29 9 be be VERB iajs-1977 29 10 a a DET iajs-1977 29 11 ๐บ-metric ๐บ-metric ADJ iajs-1977 29 12 space space NOUN iajs-1977 29 13 the the DET iajs-1977 29 14 following follow VERB iajs-1977 29 15 statements statement NOUN iajs-1977 29 16 are be AUX iajs-1977 29 17 equivalent equivalent ADJ iajs-1977 29 18 1๐‘Ÿ 1๐‘Ÿ NUM iajs-1977 29 19 is be AUX iajs-1977 29 20 ๐›ถ-convergent ๐›ถ-convergent ADJ iajs-1977 29 21 to to ADP iajs-1977 29 22 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 29 23 , , PUNCT iajs-1977 30 1 if if SCONJ iajs-1977 30 2 and and CCONJ iajs-1977 30 3 only only ADV iajs-1977 30 4 if if SCONJ iajs-1977 30 5 ๐›ถ ๐›ถ PROPN iajs-1977 30 6 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 30 7 , , PUNCT iajs-1977 30 8 ๐‘Ÿ ๐‘Ÿ X iajs-1977 30 9 , , PUNCT iajs-1977 30 10 ๐‘Ÿ ๐‘Ÿ X iajs-1977 30 11 โ†’ โ†’ SYM iajs-1977 30 12 0 0 NUM iajs-1977 30 13 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN iajs-1977 30 14 ๐‘› ๐‘› PROPN iajs-1977 30 15 โ†’ โ†’ SYM iajs-1977 30 16 โˆž โˆž PROPN iajs-1977 30 17 , , PUNCT iajs-1977 30 18 2is 2is ADJ iajs-1977 30 19 ฯ… ฯ… NOUN iajs-1977 31 1 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 31 2 , , PUNCT iajs-1977 31 3 ๐‘Ÿ ๐‘Ÿ X iajs-1977 31 4 , , PUNCT iajs-1977 31 5 ๐‘Ÿ)โ†’ ๐‘Ÿ)โ†’ NUM iajs-1977 31 6 0 0 NUM iajs-1977 31 7 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN iajs-1977 31 8 ๐‘› ๐‘› PROPN iajs-1977 31 9 โ†’ โ†’ SYM iajs-1977 31 10 โˆž โˆž PROPN iajs-1977 31 11 if if SCONJ iajs-1977 31 12 and and CCONJ iajs-1977 31 13 only only ADV iajs-1977 31 14 if if SCONJ iajs-1977 31 15 ฯ… ฯ… NOUN iajs-1977 31 16 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 31 17 , , PUNCT iajs-1977 31 18 ๐‘Ÿ ๐‘Ÿ X iajs-1977 31 19 , , PUNCT iajs-1977 31 20 ๐‘Ÿ)โ†’ ๐‘Ÿ)โ†’ X iajs-1977 31 21 0 0 NUM iajs-1977 31 22 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN iajs-1977 31 23 ๐‘š ๐‘š PROPN iajs-1977 31 24 , , PUNCT iajs-1977 31 25 ๐‘› ๐‘› PROPN iajs-1977 31 26 โ†’ โ†’ SYM iajs-1977 31 27 โˆž. โˆž. PROPN iajs-1977 31 28 proposition proposition NOUN iajs-1977 31 29 3 3 NUM iajs-1977 32 1 [ [ X iajs-1977 32 2 6 6 NUM iajs-1977 32 3 ] ] PUNCT iajs-1977 32 4 : : PUNCT iajs-1977 32 5 let let VERB iajs-1977 32 6 ๐‘ž ๐‘ž PRON iajs-1977 32 7 and and CCONJ iajs-1977 32 8 ๐‘ข ๐‘ข PRON iajs-1977 32 9 be be VERB iajs-1977 32 10 a a DET iajs-1977 32 11 sequence sequence NOUN iajs-1977 32 12 in in ADP iajs-1977 32 13 a a DET iajs-1977 32 14 ๐บ ๐บ PROPN iajs-1977 32 15 metric metric ADJ iajs-1977 32 16 space space NOUN iajs-1977 32 17 ( ( PUNCT iajs-1977 32 18 โ„ณ โ„ณ PROPN iajs-1977 32 19 , , PUNCT iajs-1977 32 20 ฯ… ฯ… PROPN iajs-1977 32 21 ) ) PUNCT iajs-1977 32 22 if if SCONJ iajs-1977 32 23 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 32 24 converges converge VERB iajs-1977 32 25 to to ADP iajs-1977 32 26 ๐‘ž ๐‘ž PRON iajs-1977 32 27 and and CCONJ iajs-1977 32 28 ๐‘ข ๐‘ข PRON iajs-1977 32 29 converge converge VERB iajs-1977 32 30 to to ADP iajs-1977 32 31 ๐‘ข. ๐‘ข. NOUN iajs-1977 32 32 then then ADV iajs-1977 32 33 ฯ… ฯ… PROPN iajs-1977 32 34 ๐‘ž ๐‘ž PROPN iajs-1977 32 35 , , PUNCT iajs-1977 32 36 ๐‘ž ๐‘ž X iajs-1977 32 37 , , PUNCT iajs-1977 32 38 ๐‘ข ๐‘ข PROPN iajs-1977 32 39 converges converge NOUN iajs-1977 32 40 to to ADP iajs-1977 32 41 ฯ… ฯ… PROPN iajs-1977 32 42 ๐‘ž ๐‘ž PROPN iajs-1977 32 43 , , PUNCT iajs-1977 32 44 ๐‘ž ๐‘ž PROPN iajs-1977 32 45 , , PUNCT iajs-1977 32 46 ๐‘ข ๐‘ข NOUN iajs-1977 32 47 . . PUNCT iajs-1977 33 1 the the DET iajs-1977 33 2 selfmapping selfmappe VERB iajs-1977 33 3 ๐‘“ ๐‘“ PRON iajs-1977 33 4 on on ADP iajs-1977 33 5 a a DET iajs-1977 33 6 g g NOUN iajs-1977 33 7 - - PUNCT iajs-1977 33 8 metric metric ADJ iajs-1977 33 9 space space NOUN iajs-1977 33 10 โ„ณ โ„ณ PROPN iajs-1977 33 11 , , PUNCT iajs-1977 33 12 ๐›ถ ๐›ถ PROPN iajs-1977 33 13 is be AUX iajs-1977 33 14 ฯ… ฯ… PRON iajs-1977 33 15 continuous continuous ADJ iajs-1977 33 16 at at ADP iajs-1977 33 17 ๐‘Ÿ ๐‘Ÿ X iajs-1977 33 18 โˆˆ โˆˆ PROPN iajs-1977 33 19 โ„ณ โ„ณ PROPN iajs-1977 33 20 [ [ NOUN iajs-1977 33 21 9 9 NUM iajs-1977 33 22 ] ] PUNCT iajs-1977 33 23 iff iff NOUN iajs-1977 33 24 every every DET iajs-1977 33 25 sequence sequence NOUN iajs-1977 33 26 ๐‘Ÿ ๐‘Ÿ X iajs-1977 33 27 โŠ‚ โŠ‚ PROPN iajs-1977 33 28 โ„ณ โ„ณ PROPN iajs-1977 33 29 , , PUNCT iajs-1977 33 30 with with ADP iajs-1977 33 31 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 33 32 โ†’ โ†’ SYM iajs-1977 33 33 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 33 34 , , PUNCT iajs-1977 33 35 we we PRON iajs-1977 33 36 have have VERB iajs-1977 33 37 ๐‘“ ๐‘“ PRON iajs-1977 33 38 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 33 39 โ†’ โ†’ SYM iajs-1977 33 40 ๐‘“ ๐‘“ X iajs-1977 33 41 ๐‘Ÿ. ๐‘Ÿ. NOUN iajs-1977 33 42 a a DET iajs-1977 33 43 mapping mapping NOUN iajs-1977 33 44 ๐‘“ ๐‘“ PRON iajs-1977 33 45 โˆถ โˆถ NOUN iajs-1977 33 46 โ„ณ โ„ณ PROPN iajs-1977 33 47 โ†’ โ†’ SYM iajs-1977 33 48 โ„ณ โ„ณ PROPN iajs-1977 33 49 is be AUX iajs-1977 33 50 said say VERB iajs-1977 33 51 to to PART iajs-1977 33 52 be be AUX iajs-1977 33 53 ๐น-contraction ๐น-contraction PROPN iajs-1977 33 54 if if SCONJ iajs-1977 33 55 there there PRON iajs-1977 33 56 exists exist VERB iajs-1977 33 57 ๐œ ๐œ X iajs-1977 33 58 > > X iajs-1977 33 59 0 0 NUM iajs-1977 33 60 such such ADJ iajs-1977 33 61 that that DET iajs-1977 33 62 for for ADP iajs-1977 33 63 all all DET iajs-1977 33 64 ๐‘ž ๐‘ž PROPN iajs-1977 33 65 , , PUNCT iajs-1977 33 66 ๐‘ข ๐‘ข PROPN iajs-1977 33 67 , , PUNCT iajs-1977 33 68 ๐‘ฃ ๐‘ฃ PRON iajs-1977 33 69 โˆˆ โˆˆ PROPN iajs-1977 33 70 โ„ณ โ„ณ PROPN iajs-1977 33 71 , , PUNCT iajs-1977 33 72 ๐›ถ ๐›ถ PROPN iajs-1977 33 73 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 33 74 , , PUNCT iajs-1977 33 75 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 33 76 , , PUNCT iajs-1977 33 77 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 33 78 0 0 NUM iajs-1977 33 79 , , PUNCT iajs-1977 33 80 ๐œ ๐œ X iajs-1977 33 81 ๐น ๐น PROPN iajs-1977 33 82 ๐›ถ ๐›ถ PROPN iajs-1977 33 83 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 33 84 , , PUNCT iajs-1977 33 85 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 33 86 , , PUNCT iajs-1977 33 87 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 33 88 ๐น ๐น PROPN iajs-1977 33 89 ๐›ถ ๐›ถ PROPN iajs-1977 33 90 ๐‘ž ๐‘ž PROPN iajs-1977 33 91 , , PUNCT iajs-1977 33 92 ๐‘ข ๐‘ข PROPN iajs-1977 33 93 , , PUNCT iajs-1977 33 94 ๐‘ฃ ๐‘ฃ X iajs-1977 33 95 . . PUNCT iajs-1977 34 1 for for SCONJ iajs-1977 34 2 all all DET iajs-1977 34 3 ๐‘ž ๐‘ž PROPN iajs-1977 34 4 , , PUNCT iajs-1977 34 5 ๐‘ข ๐‘ข PROPN iajs-1977 34 6 , , PUNCT iajs-1977 34 7 ๐‘ฃ ๐‘ฃ PRON iajs-1977 34 8 โˆˆ โˆˆ PROPN iajs-1977 34 9 โ„ณ โ„ณ PROPN iajs-1977 34 10 ( ( PUNCT iajs-1977 34 11 1 1 X iajs-1977 34 12 ) ) PUNCT iajs-1977 34 13 let let VERB iajs-1977 34 14 ๐ท ๐ท NOUN iajs-1977 34 15 be be AUX iajs-1977 34 16 the the DET iajs-1977 34 17 class class NOUN iajs-1977 34 18 of of ADP iajs-1977 34 19 all all DET iajs-1977 34 20 functions function NOUN iajs-1977 34 21 ๐น ๐น PROPN iajs-1977 34 22 : : PUNCT iajs-1977 34 23 ๐‘… ๐‘… PROPN iajs-1977 34 24 โ†’ โ†’ SYM iajs-1977 34 25 ๐‘… ๐‘… PROPN iajs-1977 34 26 is be AUX iajs-1977 34 27 a a DET iajs-1977 34 28 mapping mapping NOUN iajs-1977 34 29 satisfying satisfy VERB iajs-1977 34 30 the the DET iajs-1977 34 31 following follow VERB iajs-1977 34 32 conditions condition NOUN iajs-1977 34 33 : : PUNCT iajs-1977 34 34 ( ( PUNCT iajs-1977 34 35 d1 d1 NOUN iajs-1977 34 36 ) ) PUNCT iajs-1977 34 37 ๐น ๐น PROPN iajs-1977 34 38 is be AUX iajs-1977 34 39 strictly strictly ADV iajs-1977 34 40 increasing increase VERB iajs-1977 34 41 , , PUNCT iajs-1977 34 42 i.e. i.e. X iajs-1977 34 43 , , PUNCT iajs-1977 34 44 for for ADP iajs-1977 34 45 all all DET iajs-1977 34 46 ๐‘ž ๐‘ž PROPN iajs-1977 34 47 , , PUNCT iajs-1977 34 48 ๐‘ข ๐‘ข PROPN iajs-1977 34 49 , , PUNCT iajs-1977 34 50 ๐‘ฃ ๐‘ฃ PRON iajs-1977 34 51 โˆˆ โˆˆ PROPN iajs-1977 34 52 ๐‘… ๐‘… PROPN iajs-1977 34 53 such such ADJ iajs-1977 34 54 that that SCONJ iajs-1977 34 55 ๐‘ž ๐‘ž PROPN iajs-1977 34 56 ๐‘ข ๐‘ข ADP iajs-1977 34 57 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 34 58 , , PUNCT iajs-1977 34 59 ๐น ๐น PROPN iajs-1977 34 60 ๐‘ž ๐‘ž X iajs-1977 34 61 ๐น ๐น PROPN iajs-1977 34 62 ๐‘ข ๐‘ข PROPN iajs-1977 34 63 ๐น ๐น PROPN iajs-1977 34 64 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 34 65 , , PUNCT iajs-1977 34 66 d2 d2 VERB iajs-1977 34 67 for for ADP iajs-1977 34 68 each each DET iajs-1977 34 69 sequence sequence NOUN iajs-1977 34 70 ฮฑ ฮฑ PROPN iajs-1977 34 71 โŠ‚ โŠ‚ PROPN iajs-1977 34 72 0 0 PROPN iajs-1977 34 73 , , PUNCT iajs-1977 34 74 โˆž โˆž PROPN iajs-1977 34 75 , , PUNCT iajs-1977 35 1 lim lim PROPN iajs-1977 35 2 โ†’ โ†’ SYM iajs-1977 35 3 ฮฑ ฮฑ PROPN iajs-1977 35 4 = = SYM iajs-1977 35 5 0 0 PROPN iajs-1977 35 6 iff iff PROPN iajs-1977 35 7 lim lim PROPN iajs-1977 35 8 โ†’ โ†’ SYM iajs-1977 35 9 ๐น ๐น PROPN iajs-1977 35 10 ฮฑ ฮฑ NOUN iajs-1977 35 11 = = PUNCT iajs-1977 35 12 -โˆž. -โˆž. PUNCT iajs-1977 35 13 ( ( PUNCT iajs-1977 35 14 d3 d3 PROPN iajs-1977 35 15 ) ) PUNCT iajs-1977 35 16 โˆƒ๐‘˜ โˆƒ๐‘˜ PROPN iajs-1977 35 17 โˆˆ โˆˆ PROPN iajs-1977 35 18 0 0 NUM iajs-1977 35 19 , , PUNCT iajs-1977 35 20 1 1 NUM iajs-1977 35 21 such such ADJ iajs-1977 35 22 that that SCONJ iajs-1977 35 23 lim lim PROPN iajs-1977 35 24 โ†’ โ†’ PUNCT iajs-1977 35 25 ๐›ผ ๐›ผ PROPN iajs-1977 35 26 ๐น ๐น PROPN iajs-1977 35 27 ฮฑ ฮฑ NOUN iajs-1977 35 28 = = NOUN iajs-1977 35 29 0 0 NUM iajs-1977 35 30 . . PUNCT iajs-1977 36 1 every every DET iajs-1977 36 2 ๐น-contraction ๐น-contraction PROPN iajs-1977 36 3 is be AUX iajs-1977 36 4 contractive contractive ADJ iajs-1977 36 5 ( ( PUNCT iajs-1977 36 6 byd1 byd1 PROPN iajs-1977 36 7 ) ) PUNCT iajs-1977 36 8 and and CCONJ iajs-1977 36 9 then then ADV iajs-1977 36 10 every every DET iajs-1977 36 11 ๐น ๐น PROPN iajs-1977 36 12 contraction contraction NOUN iajs-1977 36 13 is be AUX iajs-1977 36 14 ๐›ถ-continuous ๐›ถ-continuous PROPN iajs-1977 36 15 . . PUNCT iajs-1977 37 1 clearly clearly ADV iajs-1977 37 2 , , PUNCT iajs-1977 37 3 ( ( PUNCT iajs-1977 37 4 1 1 X iajs-1977 37 5 ) ) PUNCT iajs-1977 37 6 and and CCONJ iajs-1977 37 7 ( ( PUNCT iajs-1977 37 8 d1 d1 NOUN iajs-1977 37 9 ) ) PUNCT iajs-1977 37 10 implies imply VERB iajs-1977 37 11 that that SCONJ iajs-1977 37 12 every every DET iajs-1977 37 13 ๐น-contraction ๐น-contraction PROPN iajs-1977 37 14 mapping mapping NOUN iajs-1977 37 15 is be AUX iajs-1977 37 16 ๐›ถ-continuous ๐›ถ-continuous ADJ iajs-1977 37 17 , , PUNCT iajs-1977 37 18 since since SCONJ iajs-1977 37 19 for for SCONJ iajs-1977 37 20 all all DET iajs-1977 37 21 ๐‘ž ๐‘ž PROPN iajs-1977 37 22 , , PUNCT iajs-1977 37 23 ๐‘ข ๐‘ข PROPN iajs-1977 37 24 , , PUNCT iajs-1977 37 25 ๐‘ฃ ๐‘ฃ PRON iajs-1977 37 26 โˆˆ โˆˆ PROPN iajs-1977 37 27 โ„ณ โ„ณ PROPN iajs-1977 37 28 , , PUNCT iajs-1977 37 29 with๐‘“ with๐‘“ VERB iajs-1977 37 30 ๐‘ž ๐‘ž PROPN iajs-1977 37 31 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 37 32 ๐‘“๐‘ฃ ๐‘“๐‘ฃ PROPN iajs-1977 37 33 , , PUNCT iajs-1977 37 34 ๐น ๐น PROPN iajs-1977 37 35 ๐‘“ ๐‘“ PROPN iajs-1977 37 36 ๐‘ž ๐‘ž PROPN iajs-1977 37 37 , , PUNCT iajs-1977 37 38 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 37 39 , , PUNCT iajs-1977 37 40 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 37 41 ๐น ๐น PROPN iajs-1977 37 42 ๐›ถ ๐›ถ PROPN iajs-1977 37 43 ๐‘ž ๐‘ž PROPN iajs-1977 37 44 , , PUNCT iajs-1977 37 45 ๐‘ข ๐‘ข PROPN iajs-1977 37 46 , , PUNCT iajs-1977 37 47 ๐‘ฃ ๐‘ฃ X iajs-1977 37 48 . . PUNCT iajs-1977 38 1 for for ADP iajs-1977 38 2 illustration illustration NOUN iajs-1977 38 3 , , PUNCT iajs-1977 38 4 we we PRON iajs-1977 38 5 give give VERB iajs-1977 38 6 the the DET iajs-1977 38 7 following follow VERB iajs-1977 38 8 example example NOUN iajs-1977 38 9 . . PUNCT iajs-1977 39 1 example example NOUN iajs-1977 39 2 4 4 NUM iajs-1977 39 3 aconsider aconsider NOUN iajs-1977 39 4 ๐น ๐น PROPN iajs-1977 39 5 : : PUNCT iajs-1977 39 6 0 0 NUM iajs-1977 39 7 , , PUNCT iajs-1977 39 8 โˆž โˆž PROPN iajs-1977 39 9 โ†’ โ†’ SYM iajs-1977 39 10 ๐‘… ๐‘… PROPN iajs-1977 39 11 as as ADP iajs-1977 39 12 ๐น ๐น PROPN iajs-1977 39 13 ( ( PUNCT iajs-1977 39 14 ๐›ผ ๐›ผ X iajs-1977 39 15 ๐‘™๐‘›๐›ผ. ๐‘™๐‘›๐›ผ. NOUN iajs-1977 39 16 it it PRON iajs-1977 39 17 is be AUX iajs-1977 39 18 clear clear ADJ iajs-1977 39 19 that๐น that๐น PROPN iajs-1977 39 20 โˆˆ โˆˆ PROPN iajs-1977 39 21 ๐ท. ๐ท. PROPN iajs-1977 39 22 then then ADV iajs-1977 39 23 each each DET iajs-1977 39 24 self self NOUN iajs-1977 39 25 mappings mapping VERB iajs-1977 39 26 ๐‘“ ๐‘“ PRON iajs-1977 39 27 on on ADP iajs-1977 39 28 a a DET iajs-1977 39 29 ๐บ-metric ๐บ-metric ADJ iajs-1977 39 30 space space NOUN iajs-1977 39 31 ( ( PUNCT iajs-1977 39 32 โ„ณ,๐›ถ โ„ณ,๐›ถ ADJ iajs-1977 39 33 ) ) PUNCT iajs-1977 39 34 is be AUX iajs-1977 39 35 an an DET iajs-1977 39 36 ๐น ๐น PROPN iajs-1977 39 37 -contraction -contraction PROPN iajs-1977 39 38 โˆ‹ โˆ‹ NOUN iajs-1977 39 39 for for ADP iajs-1977 39 40 all all DET iajs-1977 39 41 ๐‘ž ๐‘ž PROPN iajs-1977 39 42 , , PUNCT iajs-1977 39 43 ๐‘ข ๐‘ข PROPN iajs-1977 39 44 , , PUNCT iajs-1977 39 45 ๐‘ฃ ๐‘ฃ PRON iajs-1977 39 46 โˆˆ โˆˆ PROPN iajs-1977 39 47 โ„ณ โ„ณ PROPN iajs-1977 39 48 , , PUNCT iajs-1977 39 49 ๐‘“๐‘ž ๐‘“๐‘ž PRON iajs-1977 39 50 ๐‘“๐‘ข ๐‘“๐‘ข NOUN iajs-1977 39 51 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 39 52 ๐›ถ ๐›ถ PROPN iajs-1977 39 53 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 39 54 , , PUNCT iajs-1977 39 55 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 39 56 , , PUNCT iajs-1977 39 57 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 39 58 ๐‘’ ๐‘’ PROPN iajs-1977 39 59 ๐›ถ ๐›ถ PROPN iajs-1977 39 60 ๐‘ž ๐‘ž PROPN iajs-1977 39 61 , , PUNCT iajs-1977 39 62 ๐‘ข ๐‘ข PROPN iajs-1977 39 63 , , PUNCT iajs-1977 39 64 ๐‘ฃ ๐‘ฃ X iajs-1977 39 65 then then ADV iajs-1977 39 66 for for ADP iajs-1977 39 67 ๐‘ž ๐‘ž PROPN iajs-1977 39 68 , , PUNCT iajs-1977 39 69 ๐‘ข ๐‘ข PROPN iajs-1977 39 70 , , PUNCT iajs-1977 39 71 ๐‘ฃ ๐‘ฃ PRON iajs-1977 39 72 โˆˆ โˆˆ PROPN iajs-1977 39 73 โ„ณ โ„ณ VERB iajs-1977 39 74 such such ADJ iajs-1977 39 75 that that SCONJ iajs-1977 39 76 ๐‘“๐‘ž ๐‘“๐‘ž PRON iajs-1977 39 77 ๐‘“ ๐‘“ PRON iajs-1977 39 78 ๐‘ข ๐‘ข NOUN iajs-1977 39 79 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 39 80 the the DET iajs-1977 39 81 inequality inequality NOUN iajs-1977 39 82 ๐›ถ ๐›ถ PROPN iajs-1977 39 83 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 39 84 , , PUNCT iajs-1977 39 85 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 39 86 , , PUNCT iajs-1977 39 87 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 39 88 ๐‘’ ๐‘’ PROPN iajs-1977 39 89 ๐›ถ ๐›ถ PROPN iajs-1977 39 90 ๐‘ž ๐‘ž PROPN iajs-1977 39 91 , , PUNCT iajs-1977 39 92 ๐‘ข ๐‘ข PROPN iajs-1977 39 93 , , PUNCT iajs-1977 39 94 ๐‘ฃ ๐‘ฃ PRON iajs-1977 39 95 holds hold VERB iajs-1977 39 96 . . PUNCT iajs-1977 40 1 therefore therefore ADV iajs-1977 40 2 , , PUNCT iajs-1977 40 3 ๐‘“ ๐‘“ PRON iajs-1977 40 4 is be AUX iajs-1977 40 5 a a DET iajs-1977 40 6 contraction contraction NOUN iajs-1977 40 7 with with ADP iajs-1977 40 8 โ„Ž โ„Ž PROPN iajs-1977 40 9 ๐‘’ ๐‘’ PROPN iajs-1977 40 10 . . PUNCT iajs-1977 41 1 blet blet NOUN iajs-1977 41 2 ๐น ๐น PROPN iajs-1977 41 3 : : PUNCT iajs-1977 41 4 0 0 NUM iajs-1977 41 5 , , PUNCT iajs-1977 41 6 โˆž โˆž PROPN iajs-1977 41 7 โ†’ โ†’ SYM iajs-1977 41 8 ๐‘… ๐‘… NOUN iajs-1977 41 9 be be AUX iajs-1977 41 10 ๐น ๐น PROPN iajs-1977 41 11 ๐›ผ ๐›ผ VERB iajs-1977 41 12 ๐›ผ ๐›ผ NOUN iajs-1977 41 13 ๐‘™๐‘›๐›ผ. ๐‘™๐‘›๐›ผ. NOUN iajs-1977 41 14 it it PRON iajs-1977 41 15 is be AUX iajs-1977 41 16 clear clear ADJ iajs-1977 41 17 that that SCONJ iajs-1977 41 18 ๐น ๐น PROPN iajs-1977 41 19 โˆˆ โˆˆ PROPN iajs-1977 41 20 ๐ท. ๐ท. PROPN iajs-1977 41 21 then then ADV iajs-1977 41 22 each each DET iajs-1977 41 23 self self NOUN iajs-1977 41 24 mappings mapping VERB iajs-1977 41 25 ๐‘“ ๐‘“ PRON iajs-1977 41 26 on on ADP iajs-1977 41 27 a a DET iajs-1977 41 28 ๐บ-metric ๐บ-metric ADJ iajs-1977 41 29 space space NOUN iajs-1977 41 30 ( ( PUNCT iajs-1977 41 31 โ„ณ โ„ณ PROPN iajs-1977 41 32 , , PUNCT iajs-1977 41 33 ๐›ถ ๐›ถ NOUN iajs-1977 41 34 ) ) PUNCT iajs-1977 41 35 satisfying satisfy VERB iajs-1977 41 36 ( ( PUNCT iajs-1977 41 37 1.1 1.1 NUM iajs-1977 41 38 ) ) PUNCT iajs-1977 41 39 is be AUX iajs-1977 41 40 an an DET iajs-1977 41 41 ๐น ๐น PROPN iajs-1977 41 42 -contraction -contraction NOUN iajs-1977 41 43 such such ADJ iajs-1977 41 44 that that SCONJ iajs-1977 41 45 ย  ย  ย  ย  SPACE iajs-1977 41 46 141 141 NUM iajs-1977 41 47 ibn ibn PROPN iajs-1977 41 48 al al PROPN iajs-1977 41 49 - - PUNCT iajs-1977 41 50 haitham haitham PROPN iajs-1977 41 51 jour jour X iajs-1977 41 52 . . PROPN iajs-1977 42 1 for for ADP iajs-1977 42 2 pure pure ADJ iajs-1977 42 3 & & CCONJ iajs-1977 42 4 appl appl PROPN iajs-1977 42 5 . . PUNCT iajs-1977 43 1 sc sc PROPN iajs-1977 43 2 i. i. PROPN iajs-1977 43 3 ihjpas ihjpas PROPN iajs-1977 43 4 ย  ย  SPACE iajs-1977 43 5 https://doi.org/10.30526/32.1.1977 https://doi.org/10.30526/32.1.1977 NOUN iajs-1977 43 6 ย  ย  SPACE iajs-1977 43 7 vol vol NOUN iajs-1977 43 8 . . PUNCT iajs-1977 44 1 32 32 NUM iajs-1977 44 2 ( ( PUNCT iajs-1977 44 3 1 1 NUM iajs-1977 44 4 ) ) PUNCT iajs-1977 44 5 2019 2019 NUM iajs-1977 44 6 , , PUNCT iajs-1977 44 7 , , PUNCT iajs-1977 44 8 , , PUNCT iajs-1977 44 9 , , PUNCT iajs-1977 44 10 e e NOUN iajs-1977 44 11 , , PUNCT iajs-1977 44 12 , , PUNCT iajs-1977 44 13 , , PUNCT iajs-1977 44 14 , , PUNCT iajs-1977 44 15 ๐‘’ ๐‘’ PROPN iajs-1977 44 16 , , PUNCT iajs-1977 44 17 for for ADP iajs-1977 44 18 all๐‘ž all๐‘ž PROPN iajs-1977 44 19 , , PUNCT iajs-1977 44 20 ๐‘ข ๐‘ข PROPN iajs-1977 44 21 , , PUNCT iajs-1977 44 22 ๐‘ฃ ๐‘ฃ PRON iajs-1977 44 23 โˆˆ โˆˆ PROPN iajs-1977 44 24 โ„ณ โ„ณ PROPN iajs-1977 44 25 , , PUNCT iajs-1977 44 26 ๐‘“๐‘ž ๐‘“๐‘ž PRON iajs-1977 44 27 ๐‘“ ๐‘“ PRON iajs-1977 44 28 ๐‘ข ๐‘ข X iajs-1977 44 29 ๐‘“๐‘ฃ ๐‘“๐‘ฃ ADP iajs-1977 44 30 2 2 NUM iajs-1977 44 31 . . PUNCT iajs-1977 44 32 main main ADJ iajs-1977 44 33 results result NOUN iajs-1977 44 34 throughout throughout ADP iajs-1977 44 35 the the DET iajs-1977 44 36 following follow VERB iajs-1977 44 37 โ„ณ โ„ณ PROPN iajs-1977 44 38 is be AUX iajs-1977 44 39 a a DET iajs-1977 44 40 complete complete ADJ iajs-1977 44 41 ๐บ ๐บ NOUN iajs-1977 44 42 metric metric ADJ iajs-1977 44 43 space space NOUN iajs-1977 44 44 ๐‘ค. ๐‘ค. NOUN iajs-1977 44 45 ๐‘Ÿ. ๐‘Ÿ. NOUN iajs-1977 44 46 ๐‘ก. ๐‘ก. NOUN iajs-1977 44 47 distance distance NOUN iajs-1977 44 48 function function NOUN iajs-1977 44 49 ๐›ถ. ๐›ถ. NOUN iajs-1977 44 50 we we PRON iajs-1977 44 51 can can AUX iajs-1977 44 52 prove prove VERB iajs-1977 44 53 the the DET iajs-1977 44 54 following follow VERB iajs-1977 44 55 theorem theorem NOUN iajs-1977 44 56 theorem theorem NOUN iajs-1977 44 57 5 5 NUM iajs-1977 44 58 let let VERB iajs-1977 44 59 ๐‘“ ๐‘“ PRON iajs-1977 44 60 : : PUNCT iajs-1977 44 61 โ„ณ โ„ณ PROPN iajs-1977 44 62 โ†’ โ†’ SYM iajs-1977 44 63 โ„ณ โ„ณ NOUN iajs-1977 44 64 be be AUX iajs-1977 44 65 ๐›ถ ๐›ถ PROPN iajs-1977 44 66 continuous continuous ADJ iajs-1977 44 67 self self NOUN iajs-1977 44 68 - - PUNCT iajs-1977 44 69 mapping mapping NOUN iajs-1977 44 70 , , PUNCT iajs-1977 44 71 ๐œ– ๐œ– X iajs-1977 44 72 0 0 PUNCT iajs-1977 44 73 and and CCONJ iajs-1977 44 74 ๐‘ž ๐‘ž PROPN iajs-1977 44 75 โˆˆ โˆˆ PROPN iajs-1977 44 76 โ„ณ. โ„ณ. PROPN iajs-1977 44 77 suppose suppose VERB iajs-1977 44 78 that that SCONJ iajs-1977 44 79 โˆƒ โˆƒ PROPN iajs-1977 44 80 โ„Ž โ„Ž PROPN iajs-1977 44 81 โˆˆ โˆˆ PROPN iajs-1977 44 82 0 0 NUM iajs-1977 44 83 , , PUNCT iajs-1977 44 84 1 1 NUM iajs-1977 44 85 , , PUNCT iajs-1977 44 86 ๐œ ๐œ NOUN iajs-1977 44 87 0 0 NUM iajs-1977 44 88 , , PUNCT iajs-1977 44 89 and and CCONJ iajs-1977 44 90 ๐น ๐น PROPN iajs-1977 44 91 โˆˆ โˆˆ PROPN iajs-1977 44 92 ๐ท.if ๐ท.if PROPN iajs-1977 44 93 for for ADP iajs-1977 44 94 all all DET iajs-1977 44 95 ๐‘ž ๐‘ž PROPN iajs-1977 44 96 , , PUNCT iajs-1977 44 97 ๐‘ข ๐‘ข PROPN iajs-1977 44 98 , , PUNCT iajs-1977 44 99 ๐‘ฃ ๐‘ฃ PRON iajs-1977 44 100 โˆˆ โˆˆ PROPN iajs-1977 44 101 ๐ต ๐ต PROPN iajs-1977 44 102 ๐‘ž ๐‘ž PROPN iajs-1977 44 103 , , PUNCT iajs-1977 44 104 ๐œ– ๐œ– PROPN iajs-1977 44 105 โŠ‚ โŠ‚ PROPN iajs-1977 44 106 โ„ณ โ„ณ PROPN iajs-1977 44 107 with with ADP iajs-1977 44 108 ๐›ถ ๐›ถ PROPN iajs-1977 44 109 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 44 110 , , PUNCT iajs-1977 44 111 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 44 112 , , PUNCT iajs-1977 44 113 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 44 114 0 0 NUM iajs-1977 44 115 โˆ‹ โˆ‹ NOUN iajs-1977 44 116 ๐œ ๐œ X iajs-1977 44 117 ๐น ๐น PROPN iajs-1977 44 118 ๐›ถ ๐›ถ PROPN iajs-1977 44 119 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 44 120 , , PUNCT iajs-1977 44 121 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 44 122 , , PUNCT iajs-1977 44 123 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 44 124 ๐น ๐น PROPN iajs-1977 44 125 โ„Ž โ„Ž PROPN iajs-1977 44 126 ๐›ถ ๐›ถ PROPN iajs-1977 44 127 ๐‘ž ๐‘ž PROPN iajs-1977 44 128 , , PUNCT iajs-1977 44 129 ๐‘ข ๐‘ข PROPN iajs-1977 44 130 , , PUNCT iajs-1977 44 131 ๐‘ฃ ๐‘ฃ X iajs-1977 44 132 , , PUNCT iajs-1977 44 133 ( ( PUNCT iajs-1977 44 134 2 2 NUM iajs-1977 44 135 ) ) PUNCT iajs-1977 44 136 and and CCONJ iajs-1977 44 137 ๐›ถ ๐›ถ PROPN iajs-1977 44 138 ๐‘ž ๐‘ž PROPN iajs-1977 44 139 , , PUNCT iajs-1977 44 140 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 44 141 , , PUNCT iajs-1977 44 142 ๐‘“๐‘ž ๐‘“๐‘ž X iajs-1977 44 143 < < X iajs-1977 44 144 1 1 NUM iajs-1977 44 145 ๐‘˜ ๐‘˜ DET iajs-1977 44 146 ๐œ–. ๐œ–. NOUN iajs-1977 44 147 ( ( PUNCT iajs-1977 44 148 3 3 NUM iajs-1977 44 149 ) ) PUNCT iajs-1977 44 150 then then ADV iajs-1977 44 151 โˆƒ โˆƒ PROPN iajs-1977 44 152 ! ! PROPN iajs-1977 44 153 ๐‘Ÿโˆ— ๐‘Ÿโˆ— PROPN iajs-1977 44 154 in in ADP iajs-1977 44 155 ๐ต ๐ต PROPN iajs-1977 44 156 ๐‘ž ๐‘ž PROPN iajs-1977 44 157 , , PUNCT iajs-1977 44 158 ๐œ– ๐œ– PROPN iajs-1977 44 159 โˆ‹ โˆ‹ NOUN iajs-1977 44 160 ๐‘Ÿโˆ— ๐‘Ÿโˆ— NOUN iajs-1977 44 161 ๐‘“๐‘Ÿโˆ—. ๐‘“๐‘Ÿโˆ—. PROPN iajs-1977 44 162 proof proof NOUN iajs-1977 44 163 : : PUNCT iajs-1977 44 164 suppose suppose VERB iajs-1977 44 165 ๐‘ž ๐‘ž X iajs-1977 44 166 โˆˆ โˆˆ PROPN iajs-1977 44 167 โ„ณ โ„ณ PROPN iajs-1977 44 168 such such ADJ iajs-1977 44 169 that that SCONJ iajs-1977 44 170 ๐‘ž ๐‘ž PROPN iajs-1977 44 171 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 44 172 , , PUNCT iajs-1977 44 173 ๐‘ž ๐‘ž PROPN iajs-1977 44 174 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 44 175 . . PUNCT iajs-1977 45 1 continuing continue VERB iajs-1977 45 2 in in ADP iajs-1977 45 3 this this DET iajs-1977 45 4 way way NOUN iajs-1977 45 5 , , PUNCT iajs-1977 45 6 we we PRON iajs-1977 45 7 get get VERB iajs-1977 45 8 ๐‘ž ๐‘ž X iajs-1977 45 9 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 45 10 , , PUNCT iajs-1977 45 11 โˆ€๐‘› โˆ€๐‘› PROPN iajs-1977 45 12 0 0 NUM iajs-1977 45 13 . . PUNCT iajs-1977 45 14 implies imply VERB iajs-1977 45 15 that that SCONJ iajs-1977 45 16 { { PUNCT iajs-1977 45 17 ๐‘ž ๐‘ž NOUN iajs-1977 45 18 } } PUNCT iajs-1977 45 19 is be AUX iajs-1977 45 20 non non ADJ iajs-1977 45 21 - - ADJ iajs-1977 45 22 increasing increase VERB iajs-1977 45 23 sequence sequence NOUN iajs-1977 45 24 . . PUNCT iajs-1977 46 1 first first ADV iajs-1977 46 2 , , PUNCT iajs-1977 46 3 to to PART iajs-1977 46 4 prove prove VERB iajs-1977 46 5 ๐‘ž ๐‘ž X iajs-1977 46 6 โˆˆ โˆˆ PROPN iajs-1977 46 7 ๐ต ๐ต PROPN iajs-1977 46 8 ๐‘ž ๐‘ž PROPN iajs-1977 46 9 , , PUNCT iajs-1977 46 10 ๐œ– ๐œ– PROPN iajs-1977 46 11 , , PUNCT iajs-1977 46 12 โˆ€๐‘› โˆ€๐‘› PROPN iajs-1977 46 13 โˆˆ โˆˆ PROPN iajs-1977 46 14 โ„• โ„• PROPN iajs-1977 46 15 , , PUNCT iajs-1977 46 16 by by ADP iajs-1977 46 17 using use VERB iajs-1977 46 18 mathematical mathematical ADJ iajs-1977 46 19 induction induction NOUN iajs-1977 46 20 . . PUNCT iajs-1977 47 1 from from ADP iajs-1977 47 2 ( ( PUNCT iajs-1977 47 3 3 3 NUM iajs-1977 47 4 ) ) PUNCT iajs-1977 47 5 , , PUNCT iajs-1977 47 6 we we PRON iajs-1977 47 7 get get VERB iajs-1977 47 8 ๐›ถ ๐›ถ PROPN iajs-1977 47 9 ๐‘ž ๐‘ž PROPN iajs-1977 47 10 , , PUNCT iajs-1977 47 11 ๐‘ž ๐‘ž X iajs-1977 47 12 , , PUNCT iajs-1977 47 13 ๐‘ž ๐‘ž X iajs-1977 47 14 = = PROPN iajs-1977 47 15 ๐›ถ ๐›ถ PROPN iajs-1977 47 16 ๐‘ž ๐‘ž PROPN iajs-1977 47 17 , , PUNCT iajs-1977 47 18 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 47 19 , , PUNCT iajs-1977 47 20 ๐‘“๐‘ž ๐‘“๐‘ž PRON iajs-1977 47 21 1 1 NUM iajs-1977 47 22 ๐‘˜ ๐‘˜ ADP iajs-1977 47 23 ๐œ– ๐œ– X iajs-1977 47 24 ๐œ–. ๐œ–. NOUN iajs-1977 47 25 ( ( PUNCT iajs-1977 47 26 4 4 NUM iajs-1977 47 27 ) ) PUNCT iajs-1977 47 28 hence hence ADV iajs-1977 47 29 , , PUNCT iajs-1977 47 30 ๐‘ž ๐‘ž PROPN iajs-1977 47 31 โˆˆ โˆˆ PROPN iajs-1977 47 32 ๐ต ๐ต PROPN iajs-1977 47 33 ๐‘ž ๐‘ž PROPN iajs-1977 47 34 , , PUNCT iajs-1977 47 35 ๐œ– ๐œ– PROPN iajs-1977 47 36 . . PUNCT iajs-1977 47 37 suppose suppose VERB iajs-1977 47 38 ๐‘ž ๐‘ž X iajs-1977 47 39 , , PUNCT iajs-1977 47 40 โ€ฆ โ€ฆ PUNCT iajs-1977 47 41 .. .. PUNCT iajs-1977 47 42 , , PUNCT iajs-1977 47 43 ๐‘ž ๐‘ž PROPN iajs-1977 47 44 โˆˆ โˆˆ PROPN iajs-1977 47 45 ๐ต ๐ต PROPN iajs-1977 47 46 ๐‘ž ๐‘ž PROPN iajs-1977 47 47 , , PUNCT iajs-1977 47 48 ๐œ– ๐œ– PROPN iajs-1977 47 49 for for ADP iajs-1977 47 50 some๐‘– some๐‘– PROPN iajs-1977 47 51 โˆˆ โˆˆ PROPN iajs-1977 47 52 โ„•. โ„•. PROPN iajs-1977 47 53 then then ADV iajs-1977 47 54 from from ADP iajs-1977 47 55 ( ( PUNCT iajs-1977 47 56 2 2 X iajs-1977 47 57 ) ) PUNCT iajs-1977 47 58 we we PRON iajs-1977 47 59 obtain obtain VERB iajs-1977 47 60 ๐น ๐น PROPN iajs-1977 47 61 ๐›ถ ๐›ถ PROPN iajs-1977 47 62 ๐‘ž ๐‘ž PROPN iajs-1977 47 63 , , PUNCT iajs-1977 47 64 ๐‘ž ๐‘ž PROPN iajs-1977 47 65 , , PUNCT iajs-1977 47 66 ๐‘ž ๐‘ž PROPN iajs-1977 47 67 ๐น ๐น PROPN iajs-1977 47 68 ๐›ถ ๐›ถ PROPN iajs-1977 47 69 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 47 70 , , PUNCT iajs-1977 47 71 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 47 72 , , PUNCT iajs-1977 47 73 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 48 1 ๐น ๐น PROPN iajs-1977 48 2 โ„Ž โ„Ž PROPN iajs-1977 48 3 ๐›ถ ๐›ถ PROPN iajs-1977 48 4 ๐‘ž ๐‘ž PROPN iajs-1977 48 5 , , PUNCT iajs-1977 48 6 ๐‘ž ๐‘ž PROPN iajs-1977 48 7 , , PUNCT iajs-1977 48 8 ๐‘ž ๐‘ž X iajs-1977 48 9 ๐œ ๐œ PROPN iajs-1977 48 10 since since SCONJ iajs-1977 48 11 ๐น ๐น PROPN iajs-1977 48 12 is be AUX iajs-1977 48 13 strictly strictly ADV iajs-1977 48 14 increasing increase VERB iajs-1977 48 15 , , PUNCT iajs-1977 48 16 we we PRON iajs-1977 48 17 get get VERB iajs-1977 48 18 ๐›ถ ๐›ถ PROPN iajs-1977 48 19 ๐‘ž ๐‘ž PROPN iajs-1977 48 20 , , PUNCT iajs-1977 48 21 ๐‘ž ๐‘ž PROPN iajs-1977 48 22 , , PUNCT iajs-1977 48 23 ๐‘ž ๐‘ž X iajs-1977 48 24 โ„Ž โ„Ž PROPN iajs-1977 48 25 ๐›ถ ๐›ถ PROPN iajs-1977 48 26 ๐‘ž ๐‘ž PROPN iajs-1977 48 27 , , PUNCT iajs-1977 48 28 ๐‘ž ๐‘ž PROPN iajs-1977 48 29 , , PUNCT iajs-1977 48 30 ๐‘ž ๐‘ž X iajs-1977 48 31 ( ( PUNCT iajs-1977 48 32 5 5 NUM iajs-1977 48 33 ) ) PUNCT iajs-1977 48 34 now now ADV iajs-1977 48 35 , , PUNCT iajs-1977 48 36 ๐›ถ ๐›ถ PROPN iajs-1977 48 37 ๐‘ž ๐‘ž PROPN iajs-1977 48 38 , , PUNCT iajs-1977 48 39 ๐‘ž ๐‘ž PROPN iajs-1977 48 40 , , PUNCT iajs-1977 48 41 ๐‘ž ๐‘ž PROPN iajs-1977 48 42 ๐›ถ ๐›ถ PROPN iajs-1977 48 43 ๐‘ž ๐‘ž PROPN iajs-1977 48 44 , , PUNCT iajs-1977 48 45 ๐‘ž ๐‘ž X iajs-1977 48 46 , , PUNCT iajs-1977 48 47 ๐‘ž ๐‘ž PROPN iajs-1977 48 48 โ‹ฏ โ‹ฏ PROPN iajs-1977 48 49 ๐›ถ ๐›ถ PROPN iajs-1977 48 50 ๐‘ž ๐‘ž PROPN iajs-1977 48 51 , , PUNCT iajs-1977 48 52 ๐‘ž ๐‘ž PROPN iajs-1977 48 53 , , PUNCT iajs-1977 48 54 ๐‘ž ๐‘ž PROPN iajs-1977 48 55 ๐›ถ ๐›ถ PROPN iajs-1977 48 56 ๐‘ž ๐‘ž PROPN iajs-1977 48 57 , , PUNCT iajs-1977 48 58 ๐‘ž ๐‘ž X iajs-1977 48 59 , , PUNCT iajs-1977 48 60 ๐‘ž ๐‘ž PROPN iajs-1977 48 61 1 1 NUM iajs-1977 48 62 ๐‘˜ ๐‘˜ PRON iajs-1977 48 63 โ‹ฏ โ‹ฏ PROPN iajs-1977 48 64 ๐‘˜ ๐‘˜ X iajs-1977 48 65 ] ] PUNCT iajs-1977 48 66 1 1 NUM iajs-1977 48 67 ๐‘˜ ๐‘˜ PRON iajs-1977 48 68 ๐œ– ๐œ– PROPN iajs-1977 48 69 1 1 NUM iajs-1977 48 70 ๐‘˜ ๐‘˜ DET iajs-1977 48 71 1 1 NUM iajs-1977 48 72 ๐‘˜ ๐‘˜ DET iajs-1977 48 73 ๐œ–. ๐œ–. NOUN iajs-1977 48 74 thus thus ADV iajs-1977 48 75 ๐‘ž ๐‘ž X iajs-1977 48 76 โˆˆ โˆˆ PROPN iajs-1977 48 77 ๐ต ๐ต PROPN iajs-1977 48 78 ๐‘ž ๐‘ž PROPN iajs-1977 48 79 , , PUNCT iajs-1977 48 80 ๐œ– ๐œ– PROPN iajs-1977 48 81 , , PUNCT iajs-1977 48 82 hence hence ADV iajs-1977 48 83 ๐‘Ÿ ๐‘Ÿ X iajs-1977 48 84 โˆˆ โˆˆ PROPN iajs-1977 48 85 ๐ต ๐ต PROPN iajs-1977 48 86 ๐‘ž ๐‘ž PROPN iajs-1977 48 87 , , PUNCT iajs-1977 48 88 ๐œ– ๐œ– X iajs-1977 48 89 โˆ€ โˆ€ X iajs-1977 48 90 ๐‘› ๐‘› DET iajs-1977 48 91 โˆˆ โˆˆ PROPN iajs-1977 48 92 โ„•. โ„•. PROPN iajs-1977 48 93 continuing continue VERB iajs-1977 48 94 , , PUNCT iajs-1977 48 95 we we PRON iajs-1977 48 96 have have VERB iajs-1977 48 97 ๐น ๐น PROPN iajs-1977 48 98 ๐›ถ ๐›ถ PROPN iajs-1977 48 99 ๐‘ž ๐‘ž PROPN iajs-1977 48 100 , , PUNCT iajs-1977 48 101 ๐‘ž ๐‘ž X iajs-1977 48 102 , , PUNCT iajs-1977 48 103 ๐‘Ÿ ๐‘Ÿ X iajs-1977 48 104 ๐น ๐น PROPN iajs-1977 48 105 ๐›ถ ๐›ถ PROPN iajs-1977 48 106 ๐‘ž ๐‘ž PROPN iajs-1977 48 107 , , PUNCT iajs-1977 48 108 ๐‘ž ๐‘ž X iajs-1977 48 109 , , PUNCT iajs-1977 48 110 ๐‘ž ๐‘ž PROPN iajs-1977 48 111 ๐‘›๐œ ๐‘›๐œ PROPN iajs-1977 48 112 this this PRON iajs-1977 48 113 implies imply VERB iajs-1977 48 114 that that SCONJ iajs-1977 48 115 ๐น ๐น PROPN iajs-1977 48 116 ๐›ถ ๐›ถ PROPN iajs-1977 48 117 ๐‘ž ๐‘ž PROPN iajs-1977 48 118 , , PUNCT iajs-1977 48 119 ๐‘ž ๐‘ž PROPN iajs-1977 48 120 , , PUNCT iajs-1977 48 121 ๐‘ž ๐‘ž PROPN iajs-1977 48 122 ๐น ๐น PROPN iajs-1977 48 123 ๐›ถ ๐›ถ PROPN iajs-1977 48 124 ๐‘ž ๐‘ž PROPN iajs-1977 48 125 , , PUNCT iajs-1977 48 126 ๐‘ž ๐‘ž X iajs-1977 48 127 , , PUNCT iajs-1977 48 128 ๐‘ž ๐‘ž PROPN iajs-1977 48 129 ๐‘›๐œ ๐‘›๐œ PROPN iajs-1977 48 130 ( ( PUNCT iajs-1977 48 131 6 6 NUM iajs-1977 48 132 ) ) PUNCT iajs-1977 48 133 from from ADP iajs-1977 48 134 ( ( PUNCT iajs-1977 48 135 6 6 NUM iajs-1977 48 136 ) ) PUNCT iajs-1977 48 137 we we PRON iajs-1977 48 138 get get AUX iajs-1977 48 139 ๐‘™๐‘–๐‘š ๐‘™๐‘–๐‘š VERB iajs-1977 48 140 โ†’ โ†’ PUNCT iajs-1977 48 141 ๐น ๐น PROPN iajs-1977 48 142 ๐›ถ ๐›ถ PROPN iajs-1977 48 143 ๐‘ž ๐‘ž PROPN iajs-1977 48 144 , , PUNCT iajs-1977 48 145 ๐‘ž ๐‘ž PROPN iajs-1977 48 146 , , PUNCT iajs-1977 48 147 ๐‘ž ๐‘ž X iajs-1977 48 148 โˆž โˆž PROPN iajs-1977 48 149 . . PUNCT iajs-1977 49 1 since๐น since๐น PROPN iajs-1977 49 2 โˆˆ โˆˆ PROPN iajs-1977 49 3 ๐ท. ๐ท. NOUN iajs-1977 49 4 we we PRON iajs-1977 49 5 get get AUX iajs-1977 49 6 ๐‘™๐‘–๐‘š ๐‘™๐‘–๐‘š VERB iajs-1977 49 7 โ†’ โ†’ PUNCT iajs-1977 49 8 ๐›ถ ๐›ถ PROPN iajs-1977 49 9 ๐‘ž ๐‘ž PROPN iajs-1977 49 10 , , PUNCT iajs-1977 49 11 ๐‘ž ๐‘ž PROPN iajs-1977 49 12 , , PUNCT iajs-1977 49 13 ๐‘ž ๐‘ž PROPN iajs-1977 49 14 0 0 X iajs-1977 49 15 ( ( PUNCT iajs-1977 49 16 7 7 NUM iajs-1977 49 17 ) ) PUNCT iajs-1977 49 18 from from ADP iajs-1977 49 19 ( ( PUNCT iajs-1977 49 20 d3 d3 PROPN iajs-1977 49 21 ) ) PUNCT iajs-1977 49 22 there there PRON iajs-1977 49 23 exists exist VERB iajs-1977 49 24 ๐‘ ๐‘ PROPN iajs-1977 49 25 โˆˆ โˆˆ NOUN iajs-1977 49 26 0,1 0,1 NUM iajs-1977 49 27 such such ADJ iajs-1977 49 28 that that DET iajs-1977 49 29 ๐‘™๐‘–๐‘š ๐‘™๐‘–๐‘š NOUN iajs-1977 49 30 โ†’ โ†’ PUNCT iajs-1977 49 31 ๐›ถ ๐›ถ PROPN iajs-1977 49 32 ๐‘ž ๐‘ž PROPN iajs-1977 49 33 , , PUNCT iajs-1977 49 34 ๐‘ž ๐‘ž X iajs-1977 49 35 , , PUNCT iajs-1977 49 36 ๐‘ž ๐‘ž PROPN iajs-1977 49 37 ๐น ๐น PROPN iajs-1977 49 38 ๐›ถ ๐›ถ PROPN iajs-1977 49 39 ๐‘ž ๐‘ž PROPN iajs-1977 49 40 , , PUNCT iajs-1977 49 41 ๐‘ž ๐‘ž PROPN iajs-1977 49 42 , , PUNCT iajs-1977 49 43 ๐‘ž ๐‘ž PROPN iajs-1977 49 44 0 0 PUNCT iajs-1977 49 45 ( ( PUNCT iajs-1977 49 46 8) 8) NUM iajs-1977 49 47 ย  ย  ย  ย  SPACE iajs-1977 49 48 142 142 NUM iajs-1977 49 49 ibn ibn PROPN iajs-1977 49 50 al al PROPN iajs-1977 49 51 - - PUNCT iajs-1977 49 52 haitham haitham PROPN iajs-1977 49 53 jour jour X iajs-1977 49 54 . . PROPN iajs-1977 50 1 for for ADP iajs-1977 50 2 pure pure ADJ iajs-1977 50 3 & & CCONJ iajs-1977 50 4 appl appl PROPN iajs-1977 50 5 . . PUNCT iajs-1977 51 1 sc sc PROPN iajs-1977 51 2 i. i. PROPN iajs-1977 51 3 ihjpas ihjpas PROPN iajs-1977 51 4 ย  ย  SPACE iajs-1977 51 5 https://doi.org/10.30526/32.1.1977 https://doi.org/10.30526/32.1.1977 NOUN iajs-1977 51 6 ย  ย  SPACE iajs-1977 51 7 vol vol NOUN iajs-1977 51 8 . . PUNCT iajs-1977 52 1 32 32 NUM iajs-1977 52 2 ( ( PUNCT iajs-1977 52 3 1 1 NUM iajs-1977 52 4 ) ) PUNCT iajs-1977 52 5 2019 2019 NUM iajs-1977 52 6 from from ADP iajs-1977 52 7 ( ( PUNCT iajs-1977 52 8 6 6 NUM iajs-1977 52 9 ) ) PUNCT iajs-1977 52 10 we we PRON iajs-1977 52 11 have have VERB iajs-1977 52 12 ๐›ถ ๐›ถ PROPN iajs-1977 52 13 ๐‘ž ๐‘ž PROPN iajs-1977 52 14 , , PUNCT iajs-1977 52 15 ๐‘ž ๐‘ž X iajs-1977 52 16 , , PUNCT iajs-1977 52 17 ๐‘ž ๐‘ž PROPN iajs-1977 52 18 ๐น ๐น PROPN iajs-1977 52 19 ๐›ถ ๐›ถ PROPN iajs-1977 52 20 ๐‘ž ๐‘ž PROPN iajs-1977 52 21 , , PUNCT iajs-1977 52 22 ๐‘ž ๐‘ž PROPN iajs-1977 52 23 , , PUNCT iajs-1977 52 24 ๐‘ž ๐‘ž PROPN iajs-1977 52 25 ๐น ๐น PROPN iajs-1977 52 26 ๐›ถ ๐›ถ PROPN iajs-1977 52 27 ๐‘ž ๐‘ž PROPN iajs-1977 52 28 , , PUNCT iajs-1977 52 29 ๐‘ž ๐‘ž PROPN iajs-1977 52 30 , , PUNCT iajs-1977 52 31 ๐‘ž ๐‘ž PROPN iajs-1977 52 32 ๐›ถ ๐›ถ PROPN iajs-1977 52 33 ๐‘ž ๐‘ž PROPN iajs-1977 52 34 , , PUNCT iajs-1977 52 35 ๐‘ž ๐‘ž X iajs-1977 52 36 , , PUNCT iajs-1977 52 37 ๐‘ž ๐‘ž X iajs-1977 52 38 ๐‘›๐œ ๐‘›๐œ PROPN iajs-1977 52 39 0 0 NUM iajs-1977 52 40 . . PUNCT iajs-1977 53 1 ( ( PUNCT iajs-1977 53 2 9 9 NUM iajs-1977 53 3 ) ) PUNCT iajs-1977 53 4 by by ADP iajs-1977 53 5 ( ( PUNCT iajs-1977 53 6 7 7 NUM iajs-1977 53 7 ) ) PUNCT iajs-1977 53 8 , , PUNCT iajs-1977 53 9 ( ( PUNCT iajs-1977 53 10 8) 8) NUM iajs-1977 53 11 and and CCONJ iajs-1977 53 12 letting let VERB iajs-1977 53 13 ๐‘› ๐‘› PRON iajs-1977 53 14 โ†’ โ†’ SYM iajs-1977 53 15 โˆž โˆž PROPN iajs-1977 53 16 , , PUNCT iajs-1977 53 17 in in ADP iajs-1977 53 18 ( ( PUNCT iajs-1977 53 19 9 9 X iajs-1977 53 20 ) ) PUNCT iajs-1977 53 21 we we PRON iajs-1977 53 22 get get AUX iajs-1977 53 23 ๐‘™๐‘–๐‘š ๐‘™๐‘–๐‘š VERB iajs-1977 53 24 โ†’ โ†’ SYM iajs-1977 53 25 ๐‘› ๐‘› PROPN iajs-1977 53 26 ๐›ถ ๐›ถ PROPN iajs-1977 53 27 ๐‘ž ๐‘ž PROPN iajs-1977 53 28 , , PUNCT iajs-1977 53 29 ๐‘ž ๐‘ž PROPN iajs-1977 53 30 , , PUNCT iajs-1977 53 31 ๐‘ž ๐‘ž PROPN iajs-1977 53 32 0 0 PROPN iajs-1977 53 33 . . PUNCT iajs-1977 54 1 ( ( PUNCT iajs-1977 54 2 10 10 NUM iajs-1977 54 3 ) ) PUNCT iajs-1977 54 4 we we PRON iajs-1977 54 5 observe observe VERB iajs-1977 54 6 that that SCONJ iajs-1977 54 7 from from ADP iajs-1977 54 8 ( ( PUNCT iajs-1977 54 9 10 10 NUM iajs-1977 54 10 ) ) PUNCT iajs-1977 54 11 , , PUNCT iajs-1977 54 12 then then ADV iajs-1977 54 13 โˆƒ โˆƒ PROPN iajs-1977 54 14 ๐‘› ๐‘› PROPN iajs-1977 54 15 โˆˆ โˆˆ PROPN iajs-1977 54 16 โ„• โ„• PROPN iajs-1977 54 17 โˆ‹ โˆ‹ NOUN iajs-1977 54 18 ๐‘› ๐‘› PRON iajs-1977 54 19 ๐›ถ ๐›ถ PROPN iajs-1977 54 20 ๐‘ž ๐‘ž PROPN iajs-1977 54 21 , , PUNCT iajs-1977 54 22 ๐‘ž ๐‘ž X iajs-1977 54 23 , , PUNCT iajs-1977 54 24 ๐‘ž ๐‘ž PROPN iajs-1977 54 25 1 1 NUM iajs-1977 54 26 , , PUNCT iajs-1977 54 27 โˆ€ โˆ€ NOUN iajs-1977 54 28 ๐‘› ๐‘› PRON iajs-1977 54 29 ๐‘› ๐‘› NOUN iajs-1977 54 30 we we PRON iajs-1977 54 31 have have VERB iajs-1977 54 32 ๐›ถ ๐›ถ PROPN iajs-1977 54 33 ๐‘ž ๐‘ž PROPN iajs-1977 54 34 , , PUNCT iajs-1977 54 35 ๐‘ž ๐‘ž PROPN iajs-1977 54 36 , , PUNCT iajs-1977 54 37 ๐‘ž ๐‘ž X iajs-1977 54 38 , , PUNCT iajs-1977 54 39 โˆ€ โˆ€ X iajs-1977 54 40 ๐‘› ๐‘› PRON iajs-1977 54 41 ๐‘› ๐‘› PROPN iajs-1977 54 42 ( ( PUNCT iajs-1977 54 43 11 11 NUM iajs-1977 54 44 ) ) PUNCT iajs-1977 54 45 now now ADV iajs-1977 54 46 , , PUNCT iajs-1977 54 47 ๐‘š ๐‘š PROPN iajs-1977 54 48 , , PUNCT iajs-1977 54 49 ๐‘› ๐‘› PRON iajs-1977 54 50 โˆˆ โˆˆ PROPN iajs-1977 54 51 โ„• โ„• PROPN iajs-1977 54 52 โˆ‹ โˆ‹ NOUN iajs-1977 54 53 ๐‘š ๐‘š PROPN iajs-1977 54 54 ๐‘› ๐‘› DET iajs-1977 54 55 ๐‘› ๐‘› PROPN iajs-1977 54 56 . . PUNCT iajs-1977 55 1 then then ADV iajs-1977 55 2 , , PUNCT iajs-1977 55 3 by by ADP iajs-1977 55 4 properties property NOUN iajs-1977 55 5 of of ADP iajs-1977 55 6 ๐›ถ ๐›ถ PROPN iajs-1977 55 7 and and CCONJ iajs-1977 55 8 ( ( PUNCT iajs-1977 55 9 11 11 NUM iajs-1977 55 10 ) ) PUNCT iajs-1977 55 11 we we PRON iajs-1977 55 12 obtain obtain VERB iajs-1977 55 13 ๐›ถ ๐›ถ PROPN iajs-1977 55 14 ๐‘ž ๐‘ž PROPN iajs-1977 55 15 , , PUNCT iajs-1977 55 16 ๐‘ž ๐‘ž PROPN iajs-1977 55 17 , , PUNCT iajs-1977 55 18 ๐‘ž ๐‘ž PROPN iajs-1977 55 19 ๐›ถ ๐›ถ PROPN iajs-1977 55 20 ๐‘ž ๐‘ž PROPN iajs-1977 55 21 , , PUNCT iajs-1977 55 22 ๐‘ž ๐‘ž PROPN iajs-1977 55 23 , , PUNCT iajs-1977 55 24 ๐‘ž ๐‘ž PROPN iajs-1977 55 25 ๐›ถ ๐›ถ PROPN iajs-1977 55 26 ๐‘ž ๐‘ž PROPN iajs-1977 55 27 , , PUNCT iajs-1977 55 28 ๐‘ž ๐‘ž X iajs-1977 55 29 , , PUNCT iajs-1977 55 30 ๐‘ž ๐‘ž X iajs-1977 55 31 โ€ฆ โ€ฆ PUNCT iajs-1977 55 32 ๐›ถ ๐›ถ PROPN iajs-1977 55 33 ๐‘ž ๐‘ž PROPN iajs-1977 55 34 , , PUNCT iajs-1977 55 35 ๐‘ž ๐‘ž X iajs-1977 55 36 , , PUNCT iajs-1977 55 37 ๐‘ž ๐‘ž PROPN iajs-1977 55 38 = = PROPN iajs-1977 55 39 โˆ‘ โˆ‘ PROPN iajs-1977 55 40 ๐›ถ ๐›ถ PROPN iajs-1977 55 41 ๐‘ž ๐‘ž PROPN iajs-1977 55 42 , , PUNCT iajs-1977 55 43 ๐‘ž ๐‘ž X iajs-1977 55 44 , , PUNCT iajs-1977 55 45 ๐‘ž ๐‘ž X iajs-1977 55 46 โˆ‘ โˆ‘ ADP iajs-1977 55 47 ๐›ถ ๐›ถ PROPN iajs-1977 55 48 ๐‘ž ๐‘ž PROPN iajs-1977 55 49 , , PUNCT iajs-1977 55 50 ๐‘ž ๐‘ž X iajs-1977 55 51 , , PUNCT iajs-1977 55 52 ๐‘ž ๐‘ž X iajs-1977 55 53 โˆ‘ โˆ‘ PROPN iajs-1977 55 54 ( ( PUNCT iajs-1977 55 55 12 12 NUM iajs-1977 55 56 ) ) PUNCT iajs-1977 55 57 the the DET iajs-1977 55 58 series series NOUN iajs-1977 55 59 โˆ‘ โˆ‘ PUNCT iajs-1977 55 60 is be AUX iajs-1977 55 61 ๐›ถ ๐›ถ PROPN iajs-1977 55 62 convergent convergent NOUN iajs-1977 55 63 . . PUNCT iajs-1977 56 1 as as ADP iajs-1977 56 2 ๐‘› ๐‘› PROPN iajs-1977 56 3 โ†’ โ†’ SYM iajs-1977 56 4 โˆž โˆž PROPN iajs-1977 56 5 , , PUNCT iajs-1977 56 6 from from ADP iajs-1977 56 7 ( ( PUNCT iajs-1977 56 8 12 12 NUM iajs-1977 56 9 ) ) PUNCT iajs-1977 56 10 we we PRON iajs-1977 56 11 get get VERB iajs-1977 56 12 { { PUNCT iajs-1977 56 13 ๐‘ž ๐‘ž NOUN iajs-1977 56 14 is be AUX iajs-1977 56 15 a a DET iajs-1977 56 16 ๐›ถ-cauchy ๐›ถ-cauchy PROPN iajs-1977 56 17 sequence sequence NOUN iajs-1977 56 18 since since SCONJ iajs-1977 56 19 lim lim PROPN iajs-1977 56 20 , , PUNCT iajs-1977 56 21 โ†’ โ†’ PUNCT iajs-1977 56 22 ๐›ถ ๐›ถ PROPN iajs-1977 56 23 ๐‘ž ๐‘ž PROPN iajs-1977 56 24 , , PUNCT iajs-1977 56 25 ๐‘ž ๐‘ž PROPN iajs-1977 56 26 , , PUNCT iajs-1977 56 27 ๐‘ž ๐‘ž PROPN iajs-1977 56 28 0 0 PROPN iajs-1977 56 29 . . PUNCT iajs-1977 57 1 by by ADP iajs-1977 57 2 completeness completeness NOUN iajs-1977 57 3 of of ADP iajs-1977 57 4 โ„ณ โ„ณ PROPN iajs-1977 57 5 , , PUNCT iajs-1977 57 6 โˆƒ๐‘Ÿโˆ— โˆƒ๐‘Ÿโˆ— NOUN iajs-1977 57 7 โˆˆ โˆˆ PROPN iajs-1977 57 8 ๐ต ๐ต PROPN iajs-1977 57 9 ๐‘Ÿ ๐‘Ÿ X iajs-1977 57 10 , , PUNCT iajs-1977 57 11 ๐œ– ๐œ– PROPN iajs-1977 57 12 โˆ‹ โˆ‹ NOUN iajs-1977 57 13 ๐‘ž ๐‘ž X iajs-1977 57 14 โ†’ โ†’ SYM iajs-1977 57 15 ๐‘Ÿโˆ— ๐‘Ÿโˆ— PROPN iajs-1977 57 16 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN iajs-1977 57 17 ๐‘› ๐‘› PROPN iajs-1977 57 18 โ†’ โ†’ PUNCT iajs-1977 57 19 โˆž. โˆž. PROPN iajs-1977 57 20 since since SCONJ iajs-1977 57 21 ๐‘“ ๐‘“ PRON iajs-1977 57 22 is be AUX iajs-1977 57 23 ๐›ถ ๐›ถ PROPN iajs-1977 57 24 continuous continuous ADJ iajs-1977 57 25 . . PUNCT iajs-1977 58 1 then then ADV iajs-1977 58 2 , , PUNCT iajs-1977 58 3 ๐‘ž ๐‘ž X iajs-1977 58 4 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 58 5 โ†’ โ†’ PUNCT iajs-1977 58 6 ๐‘“๐‘Ÿโˆ— ๐‘“๐‘Ÿโˆ— ADJ iajs-1977 58 7 as as ADP iajs-1977 58 8 ๐‘› ๐‘› PROPN iajs-1977 58 9 โ†’ โ†’ SYM iajs-1977 58 10 โˆž โˆž PROPN iajs-1977 58 11 , , PUNCT iajs-1977 58 12 that that ADV iajs-1977 58 13 is is ADV iajs-1977 58 14 , , PUNCT iajs-1977 58 15 ๐‘Ÿโˆ— ๐‘Ÿโˆ— PROPN iajs-1977 58 16 ๐‘“๐‘Ÿโˆ—. ๐‘“๐‘Ÿโˆ—. PROPN iajs-1977 58 17 hence hence ADV iajs-1977 58 18 ๐‘Ÿโˆ— ๐‘Ÿโˆ— PROPN iajs-1977 58 19 is be AUX iajs-1977 58 20 a a DET iajs-1977 58 21 fixed fix VERB iajs-1977 58 22 point point NOUN iajs-1977 58 23 of of ADP iajs-1977 58 24 ๐‘“. ๐‘“. NOUN iajs-1977 58 25 to to PART iajs-1977 58 26 prove prove VERB iajs-1977 58 27 uniqueness uniqueness NOUN iajs-1977 58 28 , , PUNCT iajs-1977 58 29 let let VERB iajs-1977 58 30 ๐‘ž ๐‘ž PRON iajs-1977 58 31 , , PUNCT iajs-1977 58 32 ๐‘ข ๐‘ข PROPN iajs-1977 58 33 โˆˆ โˆˆ PROPN iajs-1977 58 34 ๐ต ๐ต NOUN iajs-1977 58 35 ๐‘ž ๐‘ž PROPN iajs-1977 58 36 , , PUNCT iajs-1977 58 37 ๐œ– ๐œ– PROPN iajs-1977 58 38 and and CCONJ iajs-1977 58 39 ๐‘ž ๐‘ž X iajs-1977 58 40 ๐‘ข ๐‘ข NOUN iajs-1977 58 41 be be VERB iajs-1977 58 42 any any DET iajs-1977 58 43 two two NUM iajs-1977 58 44 fixed fix VERB iajs-1977 58 45 point point NOUN iajs-1977 58 46 of of ADP iajs-1977 58 47 ๐‘“. ๐‘“. NOUN iajs-1977 58 48 then then ADV iajs-1977 58 49 from from ADP iajs-1977 58 50 ( ( PUNCT iajs-1977 58 51 2 2 X iajs-1977 58 52 ) ) PUNCT iajs-1977 58 53 we we PRON iajs-1977 58 54 have have VERB iajs-1977 58 55 ๐œ ๐œ PRON iajs-1977 58 56 ๐น ๐น PROPN iajs-1977 58 57 ๐›ถ ๐›ถ PROPN iajs-1977 58 58 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 58 59 , , PUNCT iajs-1977 58 60 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 58 61 , , PUNCT iajs-1977 58 62 ๐‘“๐‘ข ๐‘“๐‘ข VERB iajs-1977 58 63 ๐น ๐น PROPN iajs-1977 58 64 โ„Ž โ„Ž PROPN iajs-1977 58 65 ๐›ถ ๐›ถ PROPN iajs-1977 58 66 ๐‘ž ๐‘ž PROPN iajs-1977 58 67 , , PUNCT iajs-1977 58 68 ๐‘ข ๐‘ข PROPN iajs-1977 58 69 , , PUNCT iajs-1977 58 70 ๐‘ข ๐‘ข PRON iajs-1977 58 71 , , PUNCT iajs-1977 58 72 we we PRON iajs-1977 58 73 obtain obtain VERB iajs-1977 58 74 , , PUNCT iajs-1977 58 75 ๐œ ๐œ NOUN iajs-1977 58 76 ๐น ๐น PROPN iajs-1977 58 77 ๐›ถ ๐›ถ PROPN iajs-1977 58 78 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 58 79 , , PUNCT iajs-1977 58 80 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 58 81 , , PUNCT iajs-1977 58 82 ๐‘“๐‘ข ๐‘“๐‘ข VERB iajs-1977 58 83 ๐น ๐น PROPN iajs-1977 58 84 ๐›ถ ๐›ถ PROPN iajs-1977 58 85 ๐‘ž ๐‘ž PROPN iajs-1977 58 86 , , PUNCT iajs-1977 58 87 ๐‘ข ๐‘ข PROPN iajs-1977 58 88 , , PUNCT iajs-1977 58 89 ๐‘ข ๐‘ข NOUN iajs-1977 58 90 . . PUNCT iajs-1977 59 1 which which PRON iajs-1977 59 2 is be AUX iajs-1977 59 3 contradiction contradiction NOUN iajs-1977 59 4 , , PUNCT iajs-1977 59 5 so so ADV iajs-1977 59 6 , , PUNCT iajs-1977 59 7 ๐‘ž ๐‘ž X iajs-1977 59 8 ๐‘ข. ๐‘ข. NOUN iajs-1977 59 9 for for ADP iajs-1977 59 10 more more ADJ iajs-1977 59 11 illustration illustration NOUN iajs-1977 59 12 we we PRON iajs-1977 59 13 give give VERB iajs-1977 59 14 the the DET iajs-1977 59 15 following follow VERB iajs-1977 59 16 example example NOUN iajs-1977 59 17 . . PUNCT iajs-1977 60 1 example example NOUN iajs-1977 60 2 6 6 NUM iajs-1977 60 3 let let VERB iajs-1977 60 4 โ„ณ โ„ณ PROPN iajs-1977 60 5 ๐‘… ๐‘… PROPN iajs-1977 60 6 and and CCONJ iajs-1977 60 7 ๐›ถ ๐›ถ PROPN iajs-1977 60 8 ๐‘ž ๐‘ž PROPN iajs-1977 60 9 , , PUNCT iajs-1977 60 10 ๐‘ข ๐‘ข PROPN iajs-1977 60 11 , , PUNCT iajs-1977 60 12 ๐‘ฃ ๐‘ฃ ADP iajs-1977 60 13 |๐‘ž |๐‘ž DET iajs-1977 60 14 ๐‘ข| ๐‘ข| ADJ iajs-1977 60 15 |๐‘ข |๐‘ข PROPN iajs-1977 60 16 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 60 17 |๐‘ž |๐‘ž PROPN iajs-1977 60 18 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 60 19 . . PUNCT iajs-1977 61 1 then then ADV iajs-1977 61 2 ( ( PUNCT iajs-1977 61 3 โ„ณ,๐›ถ โ„ณ,๐›ถ ADJ iajs-1977 61 4 ) ) PUNCT iajs-1977 61 5 is be AUX iajs-1977 61 6 a a DET iajs-1977 61 7 complete complete ADJ iajs-1977 61 8 g g NOUN iajs-1977 61 9 - - PUNCT iajs-1977 61 10 metric metric ADJ iajs-1977 61 11 space space NOUN iajs-1977 61 12 . . PUNCT iajs-1977 62 1 define define VERB iajs-1977 62 2 the the DET iajs-1977 62 3 mapping mapping NOUN iajs-1977 62 4 ๐‘“ ๐‘“ NOUN iajs-1977 62 5 : : PUNCT iajs-1977 62 6 โ„ณโ†’ โ„ณโ†’ PROPN iajs-1977 62 7 โ„ณ โ„ณ PROPN iajs-1977 62 8 by by ADP iajs-1977 62 9 , , PUNCT iajs-1977 62 10 ๐‘“ ๐‘“ PRON iajs-1977 62 11 ๐‘ž ๐‘ž X iajs-1977 62 12 ๐‘ž ๐‘ž PROPN iajs-1977 62 13 4 4 NUM iajs-1977 62 14 , , PUNCT iajs-1977 62 15 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 62 16 โˆˆ โˆˆ NOUN iajs-1977 62 17 0,1 0,1 NUM iajs-1977 62 18 ๐‘ž ๐‘ž PROPN iajs-1977 62 19 1 1 NUM iajs-1977 62 20 2 2 NUM iajs-1977 62 21 , , PUNCT iajs-1977 62 22 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 62 23 โˆˆ โˆˆ PROPN iajs-1977 62 24 1 1 NUM iajs-1977 62 25 , , PUNCT iajs-1977 62 26 โˆž โˆž NUM iajs-1977 62 27 . . PUNCT iajs-1977 63 1 ๐‘Ÿ ๐‘Ÿ X iajs-1977 63 2 1 1 NUM iajs-1977 63 3 , , PUNCT iajs-1977 63 4 ๐œ– ๐œ– X iajs-1977 63 5 3 3 NUM iajs-1977 63 6 , , PUNCT iajs-1977 63 7 ๐ต ๐ต NOUN iajs-1977 63 8 ๐‘Ÿ ๐‘Ÿ X iajs-1977 63 9 , , PUNCT iajs-1977 63 10 ๐œ– ๐œ– PROPN iajs-1977 63 11 , , PUNCT iajs-1977 63 12 . . PUNCT iajs-1977 64 1 if if SCONJ iajs-1977 64 2 ๐น ๐น PROPN iajs-1977 64 3 ๐›ผ ๐›ผ VERB iajs-1977 64 4 ln ln NOUN iajs-1977 64 5 ๐›ผ ๐›ผ PROPN iajs-1977 64 6 , , PUNCT iajs-1977 64 7 ๐›ผ ๐›ผ DET iajs-1977 64 8 0 0 NUM iajs-1977 64 9 and and CCONJ iajs-1977 64 10 ๐œ ๐œ NOUN iajs-1977 64 11 0 0 NUM iajs-1977 64 12 , , PUNCT iajs-1977 64 13 then then ADV iajs-1977 64 14 ๐›ถ ๐›ถ PROPN iajs-1977 64 15 1 1 NUM iajs-1977 64 16 , , PUNCT iajs-1977 64 17 ๐‘“1 ๐‘“1 PROPN iajs-1977 64 18 , , PUNCT iajs-1977 64 19 ๐‘“1 ๐‘“1 ADJ iajs-1977 64 20 ๐œ–. ๐œ–. NOUN iajs-1977 64 21 if if SCONJ iajs-1977 64 22 ๐‘ž ๐‘ž PROPN iajs-1977 64 23 , , PUNCT iajs-1977 64 24 ๐‘  ๐‘  PROPN iajs-1977 64 25 , , PUNCT iajs-1977 64 26 ๐‘ก ๐‘ก PROPN iajs-1977 64 27 โˆˆ โˆˆ PROPN iajs-1977 64 28 ๐ต ๐ต NOUN iajs-1977 64 29 ๐‘Ÿ ๐‘Ÿ X iajs-1977 64 30 , , PUNCT iajs-1977 64 31 ๐œ– ๐œ– PROPN iajs-1977 64 32 then then ADV iajs-1977 64 33 1 1 NUM iajs-1977 64 34 4 4 NUM iajs-1977 64 35 |๐‘ž |๐‘ž PRON iajs-1977 64 36 ๐‘ข| ๐‘ข| ADJ iajs-1977 64 37 |๐‘ข |๐‘ข PROPN iajs-1977 64 38 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 64 39 |๐‘ž |๐‘ž PROPN iajs-1977 64 40 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 64 41 1 1 NUM iajs-1977 64 42 2 2 NUM iajs-1977 65 1 |๐‘ž |๐‘ž DET iajs-1977 65 2 ๐‘ข| ๐‘ข| ADJ iajs-1977 65 3 |๐‘ข |๐‘ข PROPN iajs-1977 65 4 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 65 5 |๐‘ž |๐‘ž PROPN iajs-1977 65 6 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 65 7 so so ADV iajs-1977 65 8 , , PUNCT iajs-1977 65 9 ๐›ถ ๐›ถ PROPN iajs-1977 65 10 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 65 11 , , PUNCT iajs-1977 65 12 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 65 13 , , PUNCT iajs-1977 65 14 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 65 15 โ„Ž โ„Ž ADP iajs-1977 65 16 ๐›ถ ๐›ถ PROPN iajs-1977 65 17 ๐‘ž ๐‘ž PROPN iajs-1977 65 18 , , PUNCT iajs-1977 65 19 ๐‘ข ๐‘ข PROPN iajs-1977 65 20 , , PUNCT iajs-1977 65 21 ๐‘ฃ ๐‘ฃ DET iajs-1977 65 22 ย  ย  ย  ย  SPACE iajs-1977 65 23 143 143 NUM iajs-1977 65 24 ibn ibn PROPN iajs-1977 65 25 al al PROPN iajs-1977 65 26 - - PUNCT iajs-1977 65 27 haitham haitham PROPN iajs-1977 65 28 jour jour X iajs-1977 65 29 . . PROPN iajs-1977 66 1 for for ADP iajs-1977 66 2 pure pure ADJ iajs-1977 66 3 & & CCONJ iajs-1977 66 4 appl appl PROPN iajs-1977 66 5 . . PUNCT iajs-1977 67 1 sc sc PROPN iajs-1977 67 2 i. i. PROPN iajs-1977 67 3 ihjpas ihjpas PROPN iajs-1977 67 4 ย  ย  SPACE iajs-1977 67 5 https://doi.org/10.30526/32.1.1977 https://doi.org/10.30526/32.1.1977 NOUN iajs-1977 67 6 ย  ย  SPACE iajs-1977 67 7 vol vol NOUN iajs-1977 67 8 . . PUNCT iajs-1977 68 1 32 32 NUM iajs-1977 68 2 ( ( PUNCT iajs-1977 68 3 1 1 NUM iajs-1977 68 4 ) ) PUNCT iajs-1977 68 5 2019 2019 NUM iajs-1977 68 6 hence hence ADV iajs-1977 69 1 ๐œ ๐œ X iajs-1977 69 2 ๐น ๐น PROPN iajs-1977 69 3 ๐›ถ ๐›ถ PROPN iajs-1977 69 4 ๐‘“ ๐‘“ PROPN iajs-1977 69 5 ๐‘ž ๐‘ž PROPN iajs-1977 69 6 , , PUNCT iajs-1977 69 7 ๐‘“ ๐‘“ DET iajs-1977 69 8 ๐‘ข ๐‘ข X iajs-1977 69 9 , , PUNCT iajs-1977 69 10 ๐‘“ ๐‘“ PRON iajs-1977 69 11 ๐‘ฃ ๐‘ฃ X iajs-1977 69 12 ๐œ ๐œ X iajs-1977 69 13 ln ln ADV iajs-1977 69 14 ๐›ถ ๐›ถ PROPN iajs-1977 69 15 ๐‘“ ๐‘“ PROPN iajs-1977 69 16 ๐‘ž ๐‘ž PROPN iajs-1977 69 17 , , PUNCT iajs-1977 69 18 ๐‘“ ๐‘“ DET iajs-1977 69 19 ๐‘ข ๐‘ข X iajs-1977 69 20 , , PUNCT iajs-1977 69 21 ๐‘“ ๐‘“ PRON iajs-1977 69 22 ๐‘ฃ ๐‘ฃ X iajs-1977 69 23 ln ln NOUN iajs-1977 69 24 โ„Ž โ„Ž ADP iajs-1977 69 25 ๐›ถ ๐›ถ PROPN iajs-1977 69 26 ๐‘ž ๐‘ž PROPN iajs-1977 69 27 , , PUNCT iajs-1977 69 28 ๐‘ข ๐‘ข PROPN iajs-1977 69 29 , , PUNCT iajs-1977 69 30 ๐‘ฃ ๐‘ฃ X iajs-1977 69 31 = = PUNCT iajs-1977 69 32 ๐น ๐น PROPN iajs-1977 69 33 โ„Ž โ„Ž PROPN iajs-1977 69 34 ๐›ถ ๐›ถ PROPN iajs-1977 69 35 ๐‘ž ๐‘ž PROPN iajs-1977 69 36 , , PUNCT iajs-1977 69 37 ๐‘ข ๐‘ข PROPN iajs-1977 69 38 , , PUNCT iajs-1977 69 39 ๐‘ฃ ๐‘ฃ X iajs-1977 69 40 if if SCONJ iajs-1977 69 41 ๐‘ž ๐‘ž PROPN iajs-1977 69 42 , , PUNCT iajs-1977 69 43 ๐‘ข ๐‘ข PROPN iajs-1977 69 44 , , PUNCT iajs-1977 69 45 ๐‘ฃ ๐‘ฃ PRON iajs-1977 69 46 โˆˆ โˆˆ PROPN iajs-1977 69 47 1 1 NUM iajs-1977 69 48 , , PUNCT iajs-1977 69 49 โˆž โˆž PROPN iajs-1977 69 50 then then ADV iajs-1977 69 51 ๐‘ž ๐‘ž X iajs-1977 69 52 ๐‘ข ๐‘ข X iajs-1977 69 53 ๐‘ข ๐‘ข X iajs-1977 69 54 ๐‘ฃ ๐‘ฃ ADP iajs-1977 69 55 ๐‘ฃ ๐‘ฃ ADP iajs-1977 69 56 ๐‘ž ๐‘ž X iajs-1977 69 57 |๐‘Ÿ |๐‘Ÿ X iajs-1977 69 58 ๐‘ข| ๐‘ข| ADJ iajs-1977 69 59 |๐‘ข |๐‘ข PROPN iajs-1977 69 60 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 69 61 |๐‘Ÿ |๐‘Ÿ ADP iajs-1977 69 62 ๐‘ฃ| ๐‘ฃ| VERB iajs-1977 69 63 ๐œ ๐œ ADP iajs-1977 69 64 |๐‘“ |๐‘“ NOUN iajs-1977 69 65 ๐‘ž ๐‘ž PROPN iajs-1977 69 66 ๐‘“ ๐‘“ DET iajs-1977 69 67 ๐‘ข| ๐‘ข| ADJ iajs-1977 69 68 |๐‘“ |๐‘“ NOUN iajs-1977 69 69 ๐‘ข ๐‘ข PROPN iajs-1977 69 70 ๐‘“ ๐‘“ DET iajs-1977 69 71 ๐‘ฃ| ๐‘ฃ| NOUN iajs-1977 69 72 |๐‘“ |๐‘“ PROPN iajs-1977 69 73 ๐‘ž ๐‘ž PROPN iajs-1977 69 74 ๐‘“ ๐‘“ PROPN iajs-1977 69 75 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 69 76 |๐‘ž |๐‘ž DET iajs-1977 69 77 ๐‘ข| ๐‘ข| ADJ iajs-1977 69 78 |๐‘ข |๐‘ข PROPN iajs-1977 69 79 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 69 80 |๐‘ž |๐‘ž PROPN iajs-1977 69 81 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 69 82 so so ADV iajs-1977 69 83 , , PUNCT iajs-1977 69 84 ๐œ ๐œ NOUN iajs-1977 69 85 ๐น ๐น PROPN iajs-1977 69 86 ๐›ถ ๐›ถ PROPN iajs-1977 69 87 ๐‘“ ๐‘“ PROPN iajs-1977 69 88 ๐‘ž ๐‘ž PROPN iajs-1977 69 89 , , PUNCT iajs-1977 69 90 ๐‘“ ๐‘“ DET iajs-1977 69 91 ๐‘ข ๐‘ข X iajs-1977 69 92 , , PUNCT iajs-1977 69 93 ๐‘“ ๐‘“ PRON iajs-1977 69 94 ๐‘ฃ ๐‘ฃ ADP iajs-1977 69 95 ๐น ๐น PROPN iajs-1977 69 96 ๐›ถ ๐›ถ PROPN iajs-1977 69 97 ๐‘ž ๐‘ž PROPN iajs-1977 69 98 , , PUNCT iajs-1977 69 99 ๐‘ข ๐‘ข PROPN iajs-1977 69 100 , , PUNCT iajs-1977 69 101 ๐‘ฃ ๐‘ฃ X iajs-1977 69 102 . . PUNCT iajs-1977 70 1 then then ADV iajs-1977 70 2 the the DET iajs-1977 70 3 contraction contraction NOUN iajs-1977 70 4 does do AUX iajs-1977 70 5 not not PART iajs-1977 70 6 hold hold VERB iajs-1977 70 7 on on ADP iajs-1977 70 8 โ„ณ. โ„ณ. PROPN iajs-1977 70 9 now now ADV iajs-1977 70 10 , , PUNCT iajs-1977 70 11 we we PRON iajs-1977 70 12 present present VERB iajs-1977 70 13 two two NUM iajs-1977 70 14 properties property NOUN iajs-1977 70 15 of๐‘“ of๐‘“ NOUN iajs-1977 70 16 , , PUNCT iajs-1977 70 17 ๐‘“ ๐‘“ X iajs-1977 70 18 : : PUNCT iajs-1977 70 19 โ„ณ โ„ณ PROPN iajs-1977 70 20 โ†’ โ†’ SYM iajs-1977 70 21 โ„ณ. โ„ณ. PROPN iajs-1977 70 22 we we PRON iajs-1977 70 23 say say VERB iajs-1977 70 24 that that SCONJ iajs-1977 70 25 ๐‘“ ๐‘“ PRON iajs-1977 70 26 satisfies satisfy VERB iajs-1977 70 27 the the DET iajs-1977 70 28 condition condition NOUN iajs-1977 70 29 : : PUNCT iajs-1977 71 1 i i PROPN iajs-1977 71 2 ฯ‰ ฯ‰ NUM iajs-1977 71 3 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 71 4 , , PUNCT iajs-1977 71 5 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 71 6 , , PUNCT iajs-1977 71 7 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 71 8 1 1 NUM iajs-1977 71 9 , , PUNCT iajs-1977 71 10 โˆ€ โˆ€ X iajs-1977 71 11 ๐‘ž ๐‘ž NOUN iajs-1977 71 12 , , PUNCT iajs-1977 71 13 ๐‘ข ๐‘ข PROPN iajs-1977 71 14 , , PUNCT iajs-1977 71 15 ๐‘ฃ ๐‘ฃ PRON iajs-1977 71 16 โˆˆ โˆˆ PROPN iajs-1977 71 17 โ„ณ โ„ณ PROPN iajs-1977 71 18 whenevere whenevere NOUN iajs-1977 71 19 ฯ‰ ฯ‰ NUM iajs-1977 71 20 โˆถ โˆถ NOUN iajs-1977 71 21 โ„ณ โ„ณ PROPN iajs-1977 71 22 โ†’ โ†’ SYM iajs-1977 71 23 ๐‘… ๐‘… PROPN iajs-1977 71 24 , , PUNCT iajs-1977 71 25 ฯ‰ ฯ‰ PROPN iajs-1977 71 26 ๐‘ž ๐‘ž PROPN iajs-1977 71 27 , , PUNCT iajs-1977 71 28 ๐‘ข ๐‘ข PROPN iajs-1977 71 29 , , PUNCT iajs-1977 71 30 ๐‘ฃ ๐‘ฃ DET iajs-1977 71 31 1 1 NUM iajs-1977 71 32 . . PUNCT iajs-1977 72 1 iifor iifor ADP iajs-1977 72 2 given give VERB iajs-1977 72 3 a a DET iajs-1977 72 4 sequence sequence NOUN iajs-1977 72 5 { { PUNCT iajs-1977 72 6 ๐‘ž ๐‘ž X iajs-1977 72 7 โŠ‚ โŠ‚ X iajs-1977 72 8 โ„ณ โ„ณ PROPN iajs-1977 72 9 with with ADP iajs-1977 72 10 ๐‘ž ๐‘ž PROPN iajs-1977 72 11 โ†’ โ†’ SYM iajs-1977 72 12 ๐‘ž ๐‘ž PROPN iajs-1977 72 13 โˆˆ โˆˆ PROPN iajs-1977 72 14 โ„ณ โ„ณ PROPN iajs-1977 72 15 as as ADP iajs-1977 72 16 ๐‘› ๐‘› PROPN iajs-1977 72 17 โ†’ โ†’ SYM iajs-1977 72 18 โˆž โˆž PROPN iajs-1977 72 19 , , PUNCT iajs-1977 72 20 if if SCONJ iajs-1977 72 21 ฯ‰ ฯ‰ PROPN iajs-1977 72 22 ๐‘ž ๐‘ž PROPN iajs-1977 72 23 , , PUNCT iajs-1977 72 24 ๐‘ž ๐‘ž PROPN iajs-1977 72 25 , , PUNCT iajs-1977 72 26 ๐‘ž ๐‘ž PROPN iajs-1977 72 27 ฯ† ฯ† PROPN iajs-1977 72 28 ๐‘ž ๐‘ž PROPN iajs-1977 72 29 , , PUNCT iajs-1977 72 30 ๐‘ž ๐‘ž PROPN iajs-1977 72 31 , , PUNCT iajs-1977 72 32 ๐‘ž ๐‘ž X iajs-1977 72 33 , , PUNCT iajs-1977 72 34 โˆ€๐‘› โˆ€๐‘› NOUN iajs-1977 72 35 โˆˆ โˆˆ NOUN iajs-1977 73 1 ๐‘ ๐‘ PROPN iajs-1977 73 2 โ‡’ โ‡’ VERB iajs-1977 73 3 ๐‘“ ๐‘“ PRON iajs-1977 73 4 ๐‘ž ๐‘ž X iajs-1977 73 5 โ†’ โ†’ SYM iajs-1977 73 6 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 73 7 , , PUNCT iajs-1977 73 8 where where SCONJ iajs-1977 73 9 ฯ‰ ฯ‰ NOUN iajs-1977 73 10 , , PUNCT iajs-1977 73 11 ฯ† ฯ† NOUN iajs-1977 73 12 : : PUNCT iajs-1977 73 13 โ„ณ โ„ณ PROPN iajs-1977 73 14 โ†’ โ†’ SYM iajs-1977 73 15 ๐‘… ๐‘… PROPN iajs-1977 73 16 are be AUX iajs-1977 73 17 two two NUM iajs-1977 73 18 functions function NOUN iajs-1977 73 19 . . PUNCT iajs-1977 74 1 if if SCONJ iajs-1977 74 2 ฯ† ฯ† PROPN iajs-1977 74 3 ๐‘ž ๐‘ž PROPN iajs-1977 74 4 , , PUNCT iajs-1977 74 5 ๐‘ข ๐‘ข PROPN iajs-1977 74 6 , , PUNCT iajs-1977 74 7 ๐‘ฃ ๐‘ฃ PRON iajs-1977 74 8 1 1 NUM iajs-1977 74 9 then then ADV iajs-1977 74 10 ( ( PUNCT iajs-1977 74 11 ii ii NOUN iajs-1977 74 12 ) ) PUNCT iajs-1977 74 13 reduces reduce VERB iajs-1977 74 14 ( ( PUNCT iajs-1977 74 15 i i NOUN iajs-1977 74 16 ) ) PUNCT iajs-1977 74 17 . . PUNCT iajs-1977 75 1 let let VERB iajs-1977 75 2 โˆ†๐œ“ โˆ†๐œ“ PROPN iajs-1977 75 3 ๐œ“ ๐œ“ PROPN iajs-1977 75 4 : : PUNCT iajs-1977 75 5 ๐‘… ๐‘… PROPN iajs-1977 75 6 โ†’ โ†’ SYM iajs-1977 75 7 ๐‘… ๐‘… PROPN iajs-1977 75 8 โˆถ โˆถ NOUN iajs-1977 75 9 โˆ€ โˆ€ X iajs-1977 75 10 ๐‘ก ๐‘ก PROPN iajs-1977 75 11 , , PUNCT iajs-1977 75 12 ๐‘ก ๐‘ก PROPN iajs-1977 75 13 , , PUNCT iajs-1977 75 14 ๐‘ก ๐‘ก PROPN iajs-1977 75 15 , , PUNCT iajs-1977 75 16 ๐‘ก ๐‘ก PROPN iajs-1977 75 17 โˆˆ โˆˆ PROPN iajs-1977 75 18 ๐‘… ๐‘… PROPN iajs-1977 75 19 , , PUNCT iajs-1977 75 20 ๐‘ก ๐‘ก PROPN iajs-1977 75 21 ๐‘ก ๐‘ก PROPN iajs-1977 75 22 ๐‘ก ๐‘ก VERB iajs-1977 75 23 ๐‘ก ๐‘ก PROPN iajs-1977 75 24 0 0 NUM iajs-1977 75 25 , , PUNCT iajs-1977 75 26 โˆƒ๐œ โˆƒ๐œ NUM iajs-1977 75 27 0 0 NUM iajs-1977 75 28 โˆ‹ โˆ‹ NOUN iajs-1977 75 29 ๐œ“ ๐œ“ PROPN iajs-1977 75 30 ๐‘ก ๐‘ก PROPN iajs-1977 75 31 , , PUNCT iajs-1977 75 32 ๐‘ก ๐‘ก PROPN iajs-1977 75 33 , , PUNCT iajs-1977 75 34 ๐‘ก ๐‘ก PROPN iajs-1977 75 35 , , PUNCT iajs-1977 75 36 ๐‘ก ๐‘ก PROPN iajs-1977 75 37 ๐œ ๐œ NOUN iajs-1977 75 38 . . PUNCT iajs-1977 76 1 definition definition NOUN iajs-1977 76 2 7 7 NUM iajs-1977 76 3 let let VERB iajs-1977 76 4 ๐‘“ ๐‘“ PRON iajs-1977 76 5 be be AUX iajs-1977 76 6 a a DET iajs-1977 76 7 selfmapping selfmapping NOUN iajs-1977 76 8 on on ADP iajs-1977 76 9 a a DET iajs-1977 76 10 ๐บ-metric ๐บ-metric ADJ iajs-1977 76 11 space space NOUN iajs-1977 76 12 ( ( PUNCT iajs-1977 76 13 โ„ณ โ„ณ PROPN iajs-1977 76 14 , , PUNCT iajs-1977 76 15 ๐›ถ ๐›ถ PROPN iajs-1977 76 16 ) ) PUNCT iajs-1977 76 17 and and CCONJ iajs-1977 76 18 ๐‘Ÿ ๐‘Ÿ DET iajs-1977 76 19 โˆˆโ„ณ โˆˆโ„ณ NOUN iajs-1977 76 20 with with ADP iajs-1977 76 21 ๐œ– ๐œ– PROPN iajs-1977 76 22 0 0 NUM iajs-1977 76 23 . . PUNCT iajs-1977 76 24 suppose suppose VERB iajs-1977 76 25 that that SCONJ iajs-1977 76 26 ๐œ” ๐œ” ADP iajs-1977 76 27 : : PUNCT iajs-1977 76 28 โ„ณ โ„ณ NOUN iajs-1977 76 29 โ†’ โ†’ SYM iajs-1977 76 30 0 0 NUM iajs-1977 76 31 , , PUNCT iajs-1977 76 32 โˆž โˆž PROPN iajs-1977 76 33 , , PUNCT iajs-1977 76 34 ฯ† ฯ† PROPN iajs-1977 76 35 : : PUNCT iajs-1977 76 36 โ„ณ โ„ณ PROPN iajs-1977 76 37 โ†’ โ†’ SYM iajs-1977 76 38 ๐‘… ๐‘… PROPN iajs-1977 76 39 two two NUM iajs-1977 76 40 functions function NOUN iajs-1977 76 41 . . PUNCT iajs-1977 77 1 we we PRON iajs-1977 77 2 say say VERB iajs-1977 77 3 that that SCONJ iajs-1977 77 4 ๐‘“ ๐‘“ PRON iajs-1977 77 5 is be AUX iajs-1977 77 6 called call VERB iajs-1977 77 7 ๐œ”-ฯ†๐œ“๐น ๐œ”-ฯ†๐œ“๐น ADJ iajs-1977 77 8 contraction contraction NOUN iajs-1977 77 9 on on ADP iajs-1977 77 10 a a DET iajs-1977 77 11 closed closed ADJ iajs-1977 77 12 ball ball NOUN iajs-1977 77 13 if if SCONJ iajs-1977 77 14 for for ADP iajs-1977 77 15 all all DET iajs-1977 77 16 ๐‘ž ๐‘ž PROPN iajs-1977 77 17 , , PUNCT iajs-1977 77 18 ๐‘ข ๐‘ข PROPN iajs-1977 77 19 , , PUNCT iajs-1977 77 20 ๐‘ฃ ๐‘ฃ PRON iajs-1977 77 21 โˆˆ โˆˆ PROPN iajs-1977 77 22 ๐ต ๐ต NOUN iajs-1977 77 23 ๐‘ž ๐‘ž PROPN iajs-1977 77 24 , , PUNCT iajs-1977 77 25 ๐œ– ๐œ– PROPN iajs-1977 77 26 โŠ† โŠ† NUM iajs-1977 77 27 โ„ณ โ„ณ PROPN iajs-1977 77 28 , , PUNCT iajs-1977 77 29 with with ADP iajs-1977 77 30 ๐œ” ๐œ” X iajs-1977 77 31 ๐‘ž ๐‘ž PROPN iajs-1977 77 32 , , PUNCT iajs-1977 77 33 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 77 34 , , PUNCT iajs-1977 77 35 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 77 36 , , PUNCT iajs-1977 77 37 ๐‘ข ๐‘ข PROPN iajs-1977 77 38 , , PUNCT iajs-1977 77 39 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 77 40 , , PUNCT iajs-1977 77 41 ๐‘“๐‘ข ๐‘“๐‘ข INTJ iajs-1977 77 42 , , PUNCT iajs-1977 77 43 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 77 44 , , PUNCT iajs-1977 77 45 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 77 46 , , PUNCT iajs-1977 77 47 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 77 48 ฯ† ฯ† PROPN iajs-1977 77 49 ๐‘ž ๐‘ž PROPN iajs-1977 77 50 , , PUNCT iajs-1977 77 51 ๐‘ข ๐‘ข PROPN iajs-1977 77 52 , , PUNCT iajs-1977 77 53 ๐‘ฃ ๐‘ฃ X iajs-1977 77 54 and and CCONJ iajs-1977 77 55 ๐›ถ ๐›ถ PROPN iajs-1977 77 56 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 77 57 , , PUNCT iajs-1977 77 58 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 77 59 , , PUNCT iajs-1977 77 60 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 77 61 0 0 NUM iajs-1977 77 62 , , PUNCT iajs-1977 77 63 we we PRON iajs-1977 77 64 have have VERB iajs-1977 77 65 ๐œ“ ๐œ“ ADP iajs-1977 77 66 ๐›ถ ๐›ถ PROPN iajs-1977 77 67 ๐‘ž ๐‘ž PROPN iajs-1977 77 68 , , PUNCT iajs-1977 77 69 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 77 70 , , PUNCT iajs-1977 77 71 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 77 72 , , PUNCT iajs-1977 77 73 ๐›ถ ๐›ถ PROPN iajs-1977 77 74 ๐‘ข ๐‘ข PROPN iajs-1977 77 75 , , PUNCT iajs-1977 77 76 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 77 77 , , PUNCT iajs-1977 77 78 ๐‘“๐‘ข ๐‘“๐‘ข INTJ iajs-1977 77 79 , , PUNCT iajs-1977 77 80 ๐›ถ ๐›ถ PROPN iajs-1977 77 81 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 77 82 , , PUNCT iajs-1977 77 83 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 77 84 , , PUNCT iajs-1977 77 85 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 77 86 ๐น ๐น PROPN iajs-1977 77 87 ๐›ถ ๐›ถ PROPN iajs-1977 77 88 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 77 89 , , PUNCT iajs-1977 77 90 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 77 91 , , PUNCT iajs-1977 77 92 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 77 93 ๐น ๐น PROPN iajs-1977 77 94 โ„Ž โ„Ž PROPN iajs-1977 77 95 ๐›ถ ๐›ถ PROPN iajs-1977 77 96 ๐‘ž ๐‘ž PROPN iajs-1977 77 97 , , PUNCT iajs-1977 77 98 ๐‘ข ๐‘ข PROPN iajs-1977 77 99 , , PUNCT iajs-1977 77 100 ๐‘ฃ ๐‘ฃ X iajs-1977 77 101 , , PUNCT iajs-1977 77 102 13 13 NUM iajs-1977 77 103 and and CCONJ iajs-1977 77 104 ๐›ถ ๐›ถ PROPN iajs-1977 77 105 ๐‘ž ๐‘ž PROPN iajs-1977 77 106 , , PUNCT iajs-1977 77 107 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 77 108 , , PUNCT iajs-1977 77 109 ๐‘“๐‘ž ๐‘“๐‘ž PRON iajs-1977 77 110 1 1 NUM iajs-1977 77 111 ๐‘˜ ๐‘˜ ADP iajs-1977 77 112 ๐œ– ๐œ– PROPN iajs-1977 77 113 , , PUNCT iajs-1977 77 114 14 14 NUM iajs-1977 77 115 where where SCONJ iajs-1977 77 116 0 0 NUM iajs-1977 77 117 ๐‘˜ ๐‘˜ DET iajs-1977 77 118 1 1 NUM iajs-1977 77 119 , , PUNCT iajs-1977 77 120 ๐œ“ ๐œ“ PROPN iajs-1977 77 121 โˆˆ โˆˆ PROPN iajs-1977 77 122 โˆ†๐œ“ โˆ†๐œ“ PROPN iajs-1977 77 123 and and CCONJ iajs-1977 77 124 ๐น ๐น PROPN iajs-1977 77 125 โˆˆ โˆˆ PROPN iajs-1977 77 126 ๐ท. ๐ท. PROPN iajs-1977 77 127 definition definition NOUN iajs-1977 77 128 8 8 NUM iajs-1977 77 129 let let VERB iajs-1977 77 130 ๐‘“ ๐‘“ PRON iajs-1977 77 131 : : PUNCT iajs-1977 77 132 โ„ณ โ„ณ PROPN iajs-1977 77 133 โ†’ โ†’ SYM iajs-1977 77 134 โ„ณ โ„ณ PROPN iajs-1977 77 135 be be VERB iajs-1977 77 136 a a DET iajs-1977 77 137 self self NOUN iajs-1977 77 138 โ€“ โ€“ PUNCT iajs-1977 77 139 mapping mapping NOUN iajs-1977 77 140 and๐œ” and๐œ” NOUN iajs-1977 77 141 , , PUNCT iajs-1977 77 142 ฯ† ฯ† NOUN iajs-1977 77 143 : : PUNCT iajs-1977 77 144 โ„ณ โ„ณ PROPN iajs-1977 77 145 ร— ร— NOUN iajs-1977 77 146 โ„ณร— โ„ณร— PROPN iajs-1977 77 147 โ„ณ โ„ณ PROPN iajs-1977 77 148 โ†’ โ†’ SYM iajs-1977 77 149 [ [ X iajs-1977 77 150 0 0 NUM iajs-1977 77 151 , , PUNCT iajs-1977 77 152 + + NOUN iajs-1977 77 153 โˆž โˆž NOUN iajs-1977 77 154 ) ) PUNCT iajs-1977 77 155 be be VERB iajs-1977 77 156 two two NUM iajs-1977 77 157 functions function NOUN iajs-1977 77 158 . . PUNCT iajs-1977 78 1 ๐‘“ ๐‘“ PRON iajs-1977 78 2 is be AUX iajs-1977 78 3 called call VERB iajs-1977 78 4 that that PRON iajs-1977 78 5 is be AUX iajs-1977 78 6 ๐œ”admissible ๐œ”admissible ADJ iajs-1977 78 7 mapping mapping NOUN iajs-1977 78 8 with with ADP iajs-1977 78 9 respect respect NOUN iajs-1977 78 10 to to ADP iajs-1977 78 11 ฯ† ฯ† NUM iajs-1977 79 1 if if SCONJ iajs-1977 79 2 ๐‘ž ๐‘ž PROPN iajs-1977 79 3 , , PUNCT iajs-1977 79 4 ๐‘ข ๐‘ข PROPN iajs-1977 79 5 , , PUNCT iajs-1977 79 6 ๐‘ฃ ๐‘ฃ PRON iajs-1977 79 7 โˆˆ โˆˆ PROPN iajs-1977 79 8 โ„ณ โ„ณ PROPN iajs-1977 79 9 , , PUNCT iajs-1977 79 10 ฯ† ฯ† PROPN iajs-1977 79 11 ( ( PUNCT iajs-1977 79 12 ๐‘ž ๐‘ž PROPN iajs-1977 79 13 , , PUNCT iajs-1977 79 14 ๐‘ข ๐‘ข PROPN iajs-1977 79 15 , , PUNCT iajs-1977 79 16 ๐‘ฃ ๐‘ฃ NOUN iajs-1977 79 17 ) ) PUNCT iajs-1977 79 18 ๐œ” ๐œ” PRON iajs-1977 79 19 ๐‘ž ๐‘ž PROPN iajs-1977 79 20 , , PUNCT iajs-1977 79 21 ๐‘ข ๐‘ข PROPN iajs-1977 79 22 , , PUNCT iajs-1977 80 1 ๐‘ฃ ๐‘ฃ PRON iajs-1977 80 2 implies imply VERB iajs-1977 80 3 that that SCONJ iajs-1977 80 4 ฯ† ฯ† PROPN iajs-1977 80 5 ( ( PUNCT iajs-1977 80 6 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 80 7 , , PUNCT iajs-1977 80 8 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 80 9 , , PUNCT iajs-1977 80 10 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 80 11 ) ) PUNCT iajs-1977 80 12 ๐œ” ๐œ” PRON iajs-1977 80 13 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 80 14 , , PUNCT iajs-1977 80 15 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 80 16 , , PUNCT iajs-1977 80 17 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 80 18 ) ) PUNCT iajs-1977 80 19 and and CCONJ iajs-1977 80 20 ๐›ถ ๐›ถ PROPN iajs-1977 80 21 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 80 22 , , PUNCT iajs-1977 80 23 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 80 24 , , PUNCT iajs-1977 80 25 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 80 26 0 0 NUM iajs-1977 80 27 , , PUNCT iajs-1977 80 28 we we PRON iajs-1977 80 29 have have VERB iajs-1977 80 30 ๐œ“ ๐œ“ ADP iajs-1977 80 31 ๐›ถ ๐›ถ PROPN iajs-1977 80 32 ๐‘ž ๐‘ž PROPN iajs-1977 80 33 , , PUNCT iajs-1977 80 34 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 80 35 , , PUNCT iajs-1977 80 36 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 80 37 , , PUNCT iajs-1977 80 38 ๐›ถ ๐›ถ PROPN iajs-1977 80 39 ๐‘ข ๐‘ข PROPN iajs-1977 80 40 , , PUNCT iajs-1977 80 41 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 80 42 , , PUNCT iajs-1977 80 43 ๐‘“๐‘ข ๐‘“๐‘ข INTJ iajs-1977 80 44 , , PUNCT iajs-1977 80 45 ๐›ถ ๐›ถ PROPN iajs-1977 80 46 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 80 47 , , PUNCT iajs-1977 80 48 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 80 49 , , PUNCT iajs-1977 80 50 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 80 51 , , PUNCT iajs-1977 80 52 ๐›ถ ๐›ถ PROPN iajs-1977 80 53 ๐‘ž ๐‘ž PROPN iajs-1977 80 54 , , PUNCT iajs-1977 80 55 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 80 56 , , PUNCT iajs-1977 80 57 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 80 58 , , PUNCT iajs-1977 80 59 ๐›ถ ๐›ถ PROPN iajs-1977 80 60 ๐‘ข ๐‘ข PROPN iajs-1977 80 61 , , PUNCT iajs-1977 80 62 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 80 63 , , PUNCT iajs-1977 80 64 ๐‘“๐‘ฃ ๐‘“๐‘ฃ INTJ iajs-1977 80 65 , , PUNCT iajs-1977 80 66 ๐›ถ ๐›ถ PROPN iajs-1977 80 67 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 80 68 , , PUNCT iajs-1977 80 69 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 80 70 , , PUNCT iajs-1977 80 71 ๐‘“๐‘ข ๐‘“๐‘ข VERB iajs-1977 80 72 ๐น ๐น PROPN iajs-1977 80 73 ๐›ถ ๐›ถ PROPN iajs-1977 80 74 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 80 75 , , PUNCT iajs-1977 80 76 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 80 77 , , PUNCT iajs-1977 80 78 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 80 79 ๐น ๐น PROPN iajs-1977 80 80 ๐‘€ ๐‘€ PROPN iajs-1977 80 81 ๐‘ž ๐‘ž PROPN iajs-1977 80 82 , , PUNCT iajs-1977 80 83 ๐‘ข ๐‘ข PROPN iajs-1977 80 84 , , PUNCT iajs-1977 80 85 ๐‘ฃ ๐‘ฃ X iajs-1977 80 86 ( ( PUNCT iajs-1977 80 87 15 15 NUM iajs-1977 80 88 ) ) PUNCT iajs-1977 80 89 where where SCONJ iajs-1977 80 90 ๐‘€ ๐‘€ PROPN iajs-1977 80 91 ๐‘ž ๐‘ž PROPN iajs-1977 80 92 , , PUNCT iajs-1977 80 93 ๐‘ข ๐‘ข PROPN iajs-1977 80 94 , , PUNCT iajs-1977 80 95 ๐‘ฃ ๐‘ฃ ADP iajs-1977 80 96 ๐‘š๐‘Ž๐‘ฅ ๐‘š๐‘Ž๐‘ฅ ADJ iajs-1977 80 97 ๐›ถ ๐›ถ PROPN iajs-1977 80 98 ๐‘ž ๐‘ž PROPN iajs-1977 80 99 , , PUNCT iajs-1977 80 100 ๐‘ข ๐‘ข PROPN iajs-1977 80 101 , , PUNCT iajs-1977 80 102 ๐‘ฃ ๐‘ฃ X iajs-1977 80 103 , , PUNCT iajs-1977 80 104 ๐›ถ ๐›ถ PROPN iajs-1977 80 105 ๐‘ž ๐‘ž PROPN iajs-1977 80 106 , , PUNCT iajs-1977 80 107 ๐‘“๐‘ž ๐‘“๐‘ž PROPN iajs-1977 80 108 , , PUNCT iajs-1977 80 109 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 80 110 , , PUNCT iajs-1977 80 111 ๐›ถ ๐›ถ PROPN iajs-1977 80 112 ๐‘ข ๐‘ข PROPN iajs-1977 80 113 , , PUNCT iajs-1977 80 114 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 80 115 , , PUNCT iajs-1977 80 116 ๐‘“๐‘ข ๐‘“๐‘ข INTJ iajs-1977 80 117 , , PUNCT iajs-1977 80 118 ๐›ถ ๐›ถ PROPN iajs-1977 80 119 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 80 120 , , PUNCT iajs-1977 80 121 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 80 122 , , PUNCT iajs-1977 80 123 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 80 124 , , PUNCT iajs-1977 80 125 ๐›ถ ๐›ถ PROPN iajs-1977 80 126 ๐‘ž ๐‘ž PROPN iajs-1977 80 127 , , PUNCT iajs-1977 80 128 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 80 129 , , PUNCT iajs-1977 80 130 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 80 131 ๐›ถ ๐›ถ PROPN iajs-1977 80 132 ๐‘ข ๐‘ข PROPN iajs-1977 80 133 , , PUNCT iajs-1977 80 134 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 80 135 , , PUNCT iajs-1977 80 136 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 80 137 ๐›ถ ๐›ถ PROPN iajs-1977 80 138 ๐‘ฃ ๐‘ฃ PROPN iajs-1977 80 139 , , PUNCT iajs-1977 80 140 ๐‘“๐‘ž ๐‘“๐‘ž INTJ iajs-1977 80 141 , , PUNCT iajs-1977 80 142 ๐‘“๐‘ข ๐‘“๐‘ข ADP iajs-1977 80 143 3 3 NUM iajs-1977 80 144 and and CCONJ iajs-1977 80 145 ย  ย  ย  ย  SPACE iajs-1977 80 146 144 144 NUM iajs-1977 80 147 ibn ibn PROPN iajs-1977 80 148 al al PROPN iajs-1977 80 149 - - PUNCT iajs-1977 80 150 haitham haitham PROPN iajs-1977 80 151 jour jour X iajs-1977 80 152 . . PROPN iajs-1977 81 1 for for ADP iajs-1977 81 2 pure pure ADJ iajs-1977 81 3 & & CCONJ iajs-1977 81 4 appl appl PROPN iajs-1977 81 5 . . PUNCT iajs-1977 82 1 sc sc PROPN iajs-1977 82 2 i. i. PROPN iajs-1977 82 3 ihjpas ihjpas PROPN iajs-1977 82 4 ย  ย  SPACE iajs-1977 82 5 https://doi.org/10.30526/32.1.1977 https://doi.org/10.30526/32.1.1977 NOUN iajs-1977 82 6 ย  ย  SPACE iajs-1977 82 7 vol vol NOUN iajs-1977 82 8 . . PUNCT iajs-1977 83 1 32 32 NUM iajs-1977 83 2 ( ( PUNCT iajs-1977 83 3 1 1 NUM iajs-1977 83 4 ) ) PUNCT iajs-1977 83 5 2019 2019 NUM iajs-1977 83 6 โˆ‘ โˆ‘ PUNCT iajs-1977 83 7 ๐›ถ ๐›ถ PROPN iajs-1977 83 8 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 83 9 , , PUNCT iajs-1977 83 10 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 83 11 , , PUNCT iajs-1977 83 12 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 83 13 โ‰ค โ‰ค NUM iajs-1977 83 14 , , PUNCT iajs-1977 83 15 โˆ€๐‘— โˆ€๐‘— NOUN iajs-1977 83 16 โˆˆ โˆˆ PROPN iajs-1977 83 17 ๐‘ ๐‘ PROPN iajs-1977 83 18 and and CCONJ iajs-1977 83 19 ๐œ– ๐œ– PROPN iajs-1977 83 20 0 0 PROPN iajs-1977 83 21 . . PUNCT iajs-1977 84 1 ( ( PUNCT iajs-1977 84 2 16 16 NUM iajs-1977 84 3 ) ) PUNCT iajs-1977 84 4 ๐œ“ ๐œ“ PROPN iajs-1977 84 5 โˆˆ โˆˆ PROPN iajs-1977 84 6 โˆ†๐œ“ โˆ†๐œ“ PROPN iajs-1977 84 7 and๐น and๐น PROPN iajs-1977 84 8 โˆˆ โˆˆ PROPN iajs-1977 84 9 ๐ท. ๐ท. PROPN iajs-1977 84 10 theorem theorem VERB iajs-1977 84 11 9 9 NUM iajs-1977 84 12 let let VERB iajs-1977 84 13 ๐‘“ ๐‘“ PRON iajs-1977 84 14 : : PUNCT iajs-1977 84 15 โ„ณ โ„ณ PROPN iajs-1977 84 16 โ†’ โ†’ SYM iajs-1977 84 17 โ„ณ โ„ณ NOUN iajs-1977 84 18 be be AUX iajs-1977 84 19 ๐œ”-ฯ†๐œ“๐น ๐œ”-ฯ†๐œ“๐น ADJ iajs-1977 84 20 contraction contraction NOUN iajs-1977 84 21 mapping mapping NOUN iajs-1977 84 22 on on ADP iajs-1977 84 23 a a DET iajs-1977 84 24 closed closed ADJ iajs-1977 84 25 ball ball NOUN iajs-1977 84 26 where where SCONJ iajs-1977 84 27 ( ( PUNCT iajs-1977 84 28 i i NOUN iajs-1977 84 29 ) ) PUNCT iajs-1977 84 30 ๐‘“ ๐‘“ PRON iajs-1977 84 31 is be AUX iajs-1977 84 32 an an DET iajs-1977 84 33 ๐œ”admissible ๐œ”admissible ADJ iajs-1977 84 34 mapping mapping NOUN iajs-1977 84 35 with with ADP iajs-1977 84 36 respect respect NOUN iajs-1977 84 37 to to ADP iajs-1977 84 38 ฯ† ฯ† PROPN iajs-1977 84 39 , , PUNCT iajs-1977 84 40 ii ii PROPN iajs-1977 84 41 there there ADV iajs-1977 84 42 exists exist VERB iajs-1977 84 43 ๐‘Ÿ ๐‘Ÿ X iajs-1977 84 44 โˆˆ โˆˆ PROPN iajs-1977 84 45 โ„ณ โ„ณ VERB iajs-1977 84 46 such such ADJ iajs-1977 84 47 that that SCONJ iajs-1977 84 48 ๐œ” ๐œ” ADP iajs-1977 84 49 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 84 50 , , PUNCT iajs-1977 84 51 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 84 52 , , PUNCT iajs-1977 84 53 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 84 54 ฯ† ฯ† NUM iajs-1977 84 55 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 84 56 , , PUNCT iajs-1977 84 57 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 84 58 , , PUNCT iajs-1977 84 59 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 84 60 , , PUNCT iajs-1977 84 61 ( ( PUNCT iajs-1977 84 62 iii iii X iajs-1977 84 63 ) ) PUNCT iajs-1977 84 64 ๐‘“ ๐‘“ PRON iajs-1977 84 65 is be AUX iajs-1977 84 66 an an DET iajs-1977 84 67 ๐œ”-ฯ† ๐œ”-ฯ† ADV iajs-1977 84 68 continuous continuous ADJ iajs-1977 84 69 . . PUNCT iajs-1977 85 1 then then ADV iajs-1977 85 2 there there PRON iajs-1977 85 3 exists exist VERB iajs-1977 85 4 a a DET iajs-1977 85 5 point point NOUN iajs-1977 85 6 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 85 7 in in ADP iajs-1977 85 8 ๐ต ๐ต PROPN iajs-1977 85 9 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 85 10 , , PUNCT iajs-1977 85 11 ๐œ– ๐œ– PROPN iajs-1977 85 12 such such ADJ iajs-1977 85 13 that that SCONJ iajs-1977 85 14 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 85 15 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 85 16 proof proof NOUN iajs-1977 85 17 : : PUNCT iajs-1977 85 18 let let VERB iajs-1977 85 19 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 85 20 in in ADP iajs-1977 85 21 โ„ณ โ„ณ PROPN iajs-1977 85 22 such such ADJ iajs-1977 85 23 that that SCONJ iajs-1977 85 24 ๐œ” ๐œ” ADP iajs-1977 85 25 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 85 26 , , PUNCT iajs-1977 85 27 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 85 28 , , PUNCT iajs-1977 85 29 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 85 30 ฯ† ฯ† NUM iajs-1977 85 31 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 85 32 , , PUNCT iajs-1977 85 33 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 85 34 , , PUNCT iajs-1977 85 35 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 85 36 . . PUNCT iajs-1977 86 1 for for ADP iajs-1977 86 2 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 86 3 โˆˆโ„ณ. โˆˆโ„ณ. NOUN iajs-1977 86 4 let let VERB iajs-1977 86 5 us we PRON iajs-1977 86 6 construct construct VERB iajs-1977 86 7 a a DET iajs-1977 86 8 sequence sequence NOUN iajs-1977 86 9 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 86 10 such such ADJ iajs-1977 86 11 that that SCONJ iajs-1977 86 12 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 86 13 ๐‘“ ๐‘“ DET iajs-1977 86 14 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 86 15 , , PUNCT iajs-1977 86 16 ๐‘Ÿ ๐‘Ÿ X iajs-1977 86 17 ๐‘“ ๐‘“ PRON iajs-1977 86 18 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 86 19 ๐‘“ ๐‘“ DET iajs-1977 86 20 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 86 21 continuing continue VERB iajs-1977 86 22 this this DET iajs-1977 86 23 way way NOUN iajs-1977 86 24 , , PUNCT iajs-1977 86 25 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 86 26 ๐‘“ ๐‘“ PRON iajs-1977 86 27 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 86 28 ๐‘“ ๐‘“ PRON iajs-1977 86 29 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 86 30 , , PUNCT iajs-1977 86 31 โˆ€ โˆ€ X iajs-1977 86 32 ๐‘› ๐‘› DET iajs-1977 86 33 โˆˆ โˆˆ NOUN iajs-1977 86 34 ๐‘. ๐‘. PROPN iajs-1977 86 35 since since SCONJ iajs-1977 86 36 , , PUNCT iajs-1977 86 37 ๐‘“ ๐‘“ PRON iajs-1977 86 38 is be AUX iajs-1977 86 39 an an DET iajs-1977 86 40 ๐œ”admissible ๐œ”admissible ADJ iajs-1977 86 41 mapping mapping NOUN iajs-1977 86 42 with with ADP iajs-1977 86 43 respect respect NOUN iajs-1977 86 44 to to ADP iajs-1977 86 45 ฯ† ฯ† NUM iajs-1977 86 46 , , PUNCT iajs-1977 86 47 then then ADV iajs-1977 86 48 ๐œ” ๐œ” VERB iajs-1977 86 49 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 86 50 , , PUNCT iajs-1977 86 51 ๐‘Ÿ ๐‘Ÿ X iajs-1977 86 52 , , PUNCT iajs-1977 86 53 ๐‘Ÿ ๐‘Ÿ X iajs-1977 86 54 ๐œ” ๐œ” ADP iajs-1977 86 55 ๐‘Ÿ ๐‘Ÿ X iajs-1977 86 56 , , PUNCT iajs-1977 86 57 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 86 58 , , PUNCT iajs-1977 86 59 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 86 60 ฯ† ฯ† NUM iajs-1977 86 61 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 86 62 , , PUNCT iajs-1977 86 63 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 86 64 , , PUNCT iajs-1977 86 65 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 86 66 ฯ† ฯ† NUM iajs-1977 86 67 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 86 68 , , PUNCT iajs-1977 86 69 ๐‘Ÿ ๐‘Ÿ X iajs-1977 86 70 , , PUNCT iajs-1977 86 71 ๐‘Ÿ ๐‘Ÿ X iajs-1977 86 72 . . PUNCT iajs-1977 87 1 continuous continuous ADJ iajs-1977 87 2 we we PRON iajs-1977 87 3 get get VERB iajs-1977 87 4 ฯ† ฯ† NUM iajs-1977 87 5 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 87 6 , , PUNCT iajs-1977 87 7 ๐‘“ ๐‘“ DET iajs-1977 87 8 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 87 9 , , PUNCT iajs-1977 87 10 ๐‘“ ๐‘“ PRON iajs-1977 87 11 ๐‘Ÿ ๐‘Ÿ X iajs-1977 87 12 = = SYM iajs-1977 87 13 ฯ† ฯ† NUM iajs-1977 87 14 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 87 15 , , PUNCT iajs-1977 87 16 ๐‘Ÿ ๐‘Ÿ X iajs-1977 87 17 , , PUNCT iajs-1977 87 18 ๐‘Ÿ ๐‘Ÿ X iajs-1977 87 19 ๐œ” ๐œ” ADP iajs-1977 87 20 ๐‘Ÿ ๐‘Ÿ X iajs-1977 87 21 , , PUNCT iajs-1977 87 22 ๐‘Ÿ ๐‘Ÿ X iajs-1977 87 23 , , PUNCT iajs-1977 87 24 ๐‘Ÿ ๐‘Ÿ X iajs-1977 87 25 , , PUNCT iajs-1977 87 26 โˆ€ โˆ€ X iajs-1977 87 27 ๐‘› ๐‘› PRON iajs-1977 87 28 โˆˆ โˆˆ NOUN iajs-1977 87 29 ๐‘ ๐‘ PROPN iajs-1977 87 30 ( ( PUNCT iajs-1977 87 31 17 17 NUM iajs-1977 87 32 if if SCONJ iajs-1977 87 33 โˆƒ โˆƒ PROPN iajs-1977 87 34 ๐‘› ๐‘› PRON iajs-1977 87 35 โˆˆ โˆˆ PROPN iajs-1977 87 36 ๐‘ ๐‘ PROPN iajs-1977 87 37 โˆ‹ โˆ‹ NOUN iajs-1977 87 38 ๐›ถ ๐›ถ PROPN iajs-1977 87 39 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 87 40 , , PUNCT iajs-1977 87 41 ๐‘“ ๐‘“ DET iajs-1977 87 42 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 87 43 , , PUNCT iajs-1977 87 44 ๐‘“ ๐‘“ DET iajs-1977 87 45 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 87 46 0 0 NUM iajs-1977 87 47 , , PUNCT iajs-1977 87 48 there there PRON iajs-1977 87 49 is be VERB iajs-1977 87 50 nothing nothing PRON iajs-1977 87 51 to to PART iajs-1977 87 52 prove prove VERB iajs-1977 87 53 . . PUNCT iajs-1977 88 1 so so ADV iajs-1977 88 2 , , PUNCT iajs-1977 88 3 suppose suppose VERB iajs-1977 88 4 that that SCONJ iajs-1977 88 5 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 88 6 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 7 with with ADP iajs-1977 88 8 ๐›ถ ๐›ถ PROPN iajs-1977 88 9 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 10 , , PUNCT iajs-1977 88 11 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 12 , , PUNCT iajs-1977 88 13 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 14 ๐›ถ ๐›ถ PROPN iajs-1977 88 15 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 16 , , PUNCT iajs-1977 88 17 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 18 , , PUNCT iajs-1977 88 19 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 20 0 0 NUM iajs-1977 88 21 , , PUNCT iajs-1977 88 22 โˆ€ โˆ€ X iajs-1977 88 23 ๐‘› ๐‘› PRON iajs-1977 88 24 โˆˆ โˆˆ PROPN iajs-1977 88 25 ๐‘. ๐‘. PROPN iajs-1977 88 26 first first ADV iajs-1977 88 27 , , PUNCT iajs-1977 88 28 we we PRON iajs-1977 88 29 see see VERB iajs-1977 88 30 that that SCONJ iajs-1977 88 31 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 32 โˆˆ โˆˆ PROPN iajs-1977 88 33 ๐ต ๐ต NOUN iajs-1977 88 34 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 35 , , PUNCT iajs-1977 88 36 ๐œ– ๐œ– PROPN iajs-1977 88 37 , , PUNCT iajs-1977 88 38 โˆ€ โˆ€ VERB iajs-1977 88 39 ๐‘› ๐‘› DET iajs-1977 88 40 โˆˆ โˆˆ NOUN iajs-1977 88 41 ๐‘. ๐‘. PROPN iajs-1977 88 42 since since SCONJ iajs-1977 88 43 ๐‘“ ๐‘“ PRON iajs-1977 88 44 be be AUX iajs-1977 88 45 a a DET iajs-1977 88 46 ๐œ”-ฯ†๐œ“๐น ๐œ”-ฯ†๐œ“๐น ADJ iajs-1977 88 47 contraction contraction NOUN iajs-1977 88 48 mapping mapping NOUN iajs-1977 88 49 on on ADP iajs-1977 88 50 a a DET iajs-1977 88 51 closed closed ADJ iajs-1977 88 52 ball ball NOUN iajs-1977 88 53 , , PUNCT iajs-1977 88 54 we we PRON iajs-1977 88 55 get get VERB iajs-1977 88 56 ๐›ถ ๐›ถ PROPN iajs-1977 88 57 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 88 58 , , PUNCT iajs-1977 88 59 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 60 , , PUNCT iajs-1977 88 61 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 62 ๐›ถ ๐›ถ PROPN iajs-1977 88 63 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 88 64 , , PUNCT iajs-1977 88 65 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 66 , , PUNCT iajs-1977 88 67 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 68 1 1 NUM iajs-1977 88 69 ๐‘˜ ๐‘˜ ADP iajs-1977 88 70 ๐œ– ๐œ– X iajs-1977 88 71 ๐œ– ๐œ– X iajs-1977 88 72 ( ( PUNCT iajs-1977 88 73 18 18 NUM iajs-1977 88 74 ) ) PUNCT iajs-1977 88 75 thus thus ADV iajs-1977 88 76 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 77 โˆˆ โˆˆ PROPN iajs-1977 88 78 ๐ต ๐ต NOUN iajs-1977 88 79 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 80 , , PUNCT iajs-1977 88 81 ๐œ– ๐œ– PROPN iajs-1977 88 82 . . PUNCT iajs-1977 88 83 suppose suppose VERB iajs-1977 88 84 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 88 85 , , PUNCT iajs-1977 88 86 โ€ฆ โ€ฆ PUNCT iajs-1977 88 87 , , PUNCT iajs-1977 88 88 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 89 โˆˆ โˆˆ PROPN iajs-1977 88 90 ๐ต ๐ต NOUN iajs-1977 88 91 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 92 , , PUNCT iajs-1977 88 93 ๐œ– ๐œ– PROPN iajs-1977 88 94 for for ADP iajs-1977 88 95 some some DET iajs-1977 88 96 ๐‘— ๐‘— PRON iajs-1977 88 97 โˆˆ โˆˆ PROPN iajs-1977 88 98 ๐‘ ๐‘ PROPN iajs-1977 88 99 , , PUNCT iajs-1977 88 100 such such ADJ iajs-1977 88 101 that that SCONJ iajs-1977 88 102 ๐œ“ ๐œ“ NOUN iajs-1977 88 103 ๐›ถ ๐›ถ PROPN iajs-1977 88 104 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 105 , , PUNCT iajs-1977 88 106 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 107 , , PUNCT iajs-1977 88 108 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 109 , , PUNCT iajs-1977 88 110 ๐›ถ ๐›ถ PROPN iajs-1977 88 111 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 88 112 , , PUNCT iajs-1977 88 113 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 114 , , PUNCT iajs-1977 88 115 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 116 , , PUNCT iajs-1977 88 117 ๐›ถ ๐›ถ PROPN iajs-1977 88 118 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 88 119 , , PUNCT iajs-1977 88 120 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 121 , , PUNCT iajs-1977 88 122 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 123 , , PUNCT iajs-1977 88 124 ๐›ถ ๐›ถ PROPN iajs-1977 88 125 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 88 126 , , PUNCT iajs-1977 88 127 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 128 , , PUNCT iajs-1977 88 129 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 130 + + PROPN iajs-1977 88 131 ๐น ๐น PROPN iajs-1977 88 132 ๐›ถ ๐›ถ PROPN iajs-1977 88 133 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 134 , , PUNCT iajs-1977 88 135 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 88 136 , , PUNCT iajs-1977 88 137 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ VERB iajs-1977 88 138 ๐น ๐น PROPN iajs-1977 88 139 โ„Ž โ„Ž ADP iajs-1977 88 140 ๐›ถ ๐›ถ PROPN iajs-1977 88 141 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 88 142 , , PUNCT iajs-1977 88 143 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 144 , , PUNCT iajs-1977 88 145 ๐‘Ÿ ๐‘Ÿ X iajs-1977 88 146 . . PUNCT iajs-1977 89 1 this this PRON iajs-1977 89 2 implies imply VERB iajs-1977 89 3 , , PUNCT iajs-1977 89 4 ๐œ“ ๐œ“ PROPN iajs-1977 89 5 ๐›ถ ๐›ถ PROPN iajs-1977 89 6 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 89 7 , , PUNCT iajs-1977 89 8 ๐‘Ÿ ๐‘Ÿ X iajs-1977 89 9 , , PUNCT iajs-1977 89 10 ๐‘Ÿ ๐‘Ÿ X iajs-1977 89 11 , , PUNCT iajs-1977 89 12 ๐›ถ ๐›ถ PROPN iajs-1977 89 13 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 89 14 , , PUNCT iajs-1977 89 15 ๐‘Ÿ ๐‘Ÿ X iajs-1977 89 16 , , PUNCT iajs-1977 89 17 ๐‘Ÿ ๐‘Ÿ X iajs-1977 89 18 , , PUNCT iajs-1977 89 19 ๐›ถ ๐›ถ PROPN iajs-1977 89 20 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 89 21 , , PUNCT iajs-1977 89 22 ๐‘Ÿ ๐‘Ÿ X iajs-1977 89 23 , , PUNCT iajs-1977 89 24 ๐‘Ÿ ๐‘Ÿ X iajs-1977 89 25 , , PUNCT iajs-1977 89 26 0 0 NUM iajs-1977 89 27 ๐น ๐น PROPN iajs-1977 89 28 ๐›ถ ๐›ถ PROPN iajs-1977 89 29 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 89 30 , , PUNCT iajs-1977 89 31 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 89 32 , , PUNCT iajs-1977 89 33 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ VERB iajs-1977 89 34 ๐น ๐น PROPN iajs-1977 89 35 โ„Ž โ„Ž ADP iajs-1977 89 36 ๐›ถ ๐›ถ PROPN iajs-1977 89 37 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 89 38 , , PUNCT iajs-1977 89 39 ๐‘Ÿ ๐‘Ÿ X iajs-1977 89 40 , , PUNCT iajs-1977 89 41 ๐‘Ÿ ๐‘Ÿ X iajs-1977 89 42 . . PUNCT iajs-1977 90 1 by by ADP iajs-1977 90 2 definition definition NOUN iajs-1977 90 3 of of ADP iajs-1977 90 4 ๐œ“ ๐œ“ PROPN iajs-1977 90 5 , , PUNCT iajs-1977 90 6 ( ( PUNCT iajs-1977 90 7 ๐›ถ ๐›ถ PROPN iajs-1977 90 8 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 9 , , PUNCT iajs-1977 90 10 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 11 , , PUNCT iajs-1977 90 12 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 13 , , PUNCT iajs-1977 90 14 ๐›ถ ๐›ถ PROPN iajs-1977 90 15 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 16 , , PUNCT iajs-1977 90 17 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 18 , , PUNCT iajs-1977 90 19 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 20 , , PUNCT iajs-1977 90 21 ๐›ถ ๐›ถ PROPN iajs-1977 90 22 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 23 , , PUNCT iajs-1977 90 24 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 25 , , PUNCT iajs-1977 90 26 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 27 , , PUNCT iajs-1977 90 28 0 0 NUM iajs-1977 90 29 0 0 NUM iajs-1977 90 30 , , PUNCT iajs-1977 90 31 so so ADV iajs-1977 90 32 , , PUNCT iajs-1977 90 33 โˆƒ๐œ โˆƒ๐œ NUM iajs-1977 90 34 0 0 NUM iajs-1977 90 35 such such ADJ iajs-1977 90 36 that that PRON iajs-1977 90 37 , , PUNCT iajs-1977 90 38 ๐œ“ ๐œ“ PROPN iajs-1977 90 39 ๐›ถ ๐›ถ PROPN iajs-1977 90 40 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 41 , , PUNCT iajs-1977 90 42 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 43 , , PUNCT iajs-1977 90 44 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 45 , , PUNCT iajs-1977 90 46 ๐›ถ ๐›ถ PROPN iajs-1977 90 47 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 48 , , PUNCT iajs-1977 90 49 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 50 , , PUNCT iajs-1977 90 51 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 52 , , PUNCT iajs-1977 90 53 ๐›ถ ๐›ถ PROPN iajs-1977 90 54 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 55 , , PUNCT iajs-1977 90 56 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 57 , , PUNCT iajs-1977 90 58 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 59 . . PUNCT iajs-1977 90 60 0 0 PUNCT iajs-1977 90 61 ๐œ. ๐œ. PROPN iajs-1977 90 62 therefore therefore ADV iajs-1977 90 63 , , PUNCT iajs-1977 90 64 ๐น ๐น PROPN iajs-1977 90 65 ๐›ถ ๐›ถ PROPN iajs-1977 90 66 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 67 , , PUNCT iajs-1977 90 68 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 69 , , PUNCT iajs-1977 90 70 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 71 ๐น ๐น PROPN iajs-1977 90 72 ๐›ถ ๐›ถ PROPN iajs-1977 90 73 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 74 , , PUNCT iajs-1977 90 75 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 76 , , PUNCT iajs-1977 90 77 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 78 ๐น ๐น PROPN iajs-1977 90 79 โ„Ž โ„Ž VERB iajs-1977 90 80 ๐›ถ ๐›ถ PROPN iajs-1977 90 81 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 82 , , PUNCT iajs-1977 90 83 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 84 , , PUNCT iajs-1977 90 85 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 90 86 ๐œ ๐œ X iajs-1977 90 87 19 19 NUM iajs-1977 90 88 to to PART iajs-1977 90 89 complete complete ADJ iajs-1977 90 90 , , PUNCT iajs-1977 90 91 we we PRON iajs-1977 90 92 follow follow VERB iajs-1977 90 93 the the DET iajs-1977 90 94 same same ADJ iajs-1977 90 95 steps step NOUN iajs-1977 90 96 of of ADP iajs-1977 90 97 the the DET iajs-1977 90 98 theorem theorem NOUN iajs-1977 90 99 ( ( PUNCT iajs-1977 90 100 9),since 9),since NUM iajs-1977 90 101 โ„ณ โ„ณ NOUN iajs-1977 90 102 is be AUX iajs-1977 90 103 complete complete ADJ iajs-1977 90 104 ๐บ ๐บ PROPN iajs-1977 90 105 metric metric ADJ iajs-1977 90 106 space space NOUN iajs-1977 90 107 there there PRON iajs-1977 90 108 exists exist VERB iajs-1977 90 109 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 110 โˆˆ โˆˆ PROPN iajs-1977 90 111 ๐ต ๐ต NOUN iajs-1977 90 112 ๐‘Ÿ ๐‘Ÿ X iajs-1977 90 113 , , PUNCT iajs-1977 90 114 ๐œ– ๐œ– PROPN iajs-1977 90 115 such such ADJ iajs-1977 90 116 that that SCONJ iajs-1977 90 117 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 118 โ†’ โ†’ SYM iajs-1977 90 119 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 90 120 as as ADP iajs-1977 90 121 ๐‘› ๐‘› PROPN iajs-1977 90 122 โ†’ โ†’ SYM iajs-1977 90 123 โˆž โˆž NUM iajs-1977 90 124 . . PUNCT iajs-1977 91 1 ๐‘“ ๐‘“ PRON iajs-1977 91 2 is be AUX iajs-1977 91 3 an an DET iajs-1977 91 4 ๐œ”-ฯ† ๐œ”-ฯ† ADV iajs-1977 91 5 continuous continuous ADJ iajs-1977 91 6 and and CCONJ iajs-1977 91 7 ฯ† ฯ† NUM iajs-1977 91 8 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 91 9 , , PUNCT iajs-1977 91 10 ๐‘Ÿ ๐‘Ÿ X iajs-1977 91 11 , , PUNCT iajs-1977 91 12 ๐‘Ÿ ๐‘Ÿ X iajs-1977 91 13 ๐œ” ๐œ” ADP iajs-1977 91 14 ๐‘Ÿ ๐‘Ÿ X iajs-1977 91 15 , , PUNCT iajs-1977 91 16 ๐‘Ÿ ๐‘Ÿ X iajs-1977 91 17 , , PUNCT iajs-1977 91 18 ๐‘Ÿ ๐‘Ÿ X iajs-1977 91 19 , , PUNCT iajs-1977 91 20 โˆ€ โˆ€ X iajs-1977 91 21 ๐‘› ๐‘› PRON iajs-1977 91 22 โˆˆ โˆˆ PROPN iajs-1977 91 23 ๐‘. ๐‘. PROPN iajs-1977 91 24 then then ADV iajs-1977 91 25 ย  ย  ย  ย  SPACE iajs-1977 91 26 145 145 NUM iajs-1977 91 27 ibn ibn PROPN iajs-1977 91 28 al al PROPN iajs-1977 91 29 - - PUNCT iajs-1977 91 30 haitham haitham PROPN iajs-1977 91 31 jour jour X iajs-1977 91 32 . . PROPN iajs-1977 92 1 for for ADP iajs-1977 92 2 pure pure ADJ iajs-1977 92 3 & & CCONJ iajs-1977 92 4 appl appl PROPN iajs-1977 92 5 . . PUNCT iajs-1977 93 1 sc sc PROPN iajs-1977 93 2 i. i. PROPN iajs-1977 93 3 ihjpas ihjpas PROPN iajs-1977 93 4 ย  ย  SPACE iajs-1977 93 5 https://doi.org/10.30526/32.1.1977 https://doi.org/10.30526/32.1.1977 NOUN iajs-1977 93 6 ย  ย  SPACE iajs-1977 93 7 vol vol NOUN iajs-1977 93 8 . . PUNCT iajs-1977 94 1 32 32 NUM iajs-1977 94 2 ( ( PUNCT iajs-1977 94 3 1 1 NUM iajs-1977 94 4 ) ) PUNCT iajs-1977 94 5 2019 2019 NUM iajs-1977 94 6 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 94 7 ๐‘“ ๐‘“ PRON iajs-1977 94 8 ๐‘Ÿ ๐‘Ÿ X iajs-1977 94 9 โ†’ โ†’ SYM iajs-1977 94 10 ๐‘“ ๐‘“ PRON iajs-1977 94 11 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 94 12 as as ADP iajs-1977 94 13 ๐‘› ๐‘› PROPN iajs-1977 94 14 โ†’ โ†’ SYM iajs-1977 94 15 โˆž. โˆž. PROPN iajs-1977 94 16 that that PRON iajs-1977 94 17 is be AUX iajs-1977 94 18 , , PUNCT iajs-1977 94 19 ๐‘Ÿ ๐‘Ÿ X iajs-1977 94 20 ๐‘“ ๐‘“ PRON iajs-1977 94 21 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 94 22 hence hence ADV iajs-1977 94 23 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 94 24 is be AUX iajs-1977 94 25 a a DET iajs-1977 94 26 fixed fix VERB iajs-1977 94 27 point point NOUN iajs-1977 94 28 of of ADP iajs-1977 94 29 ๐‘“. ๐‘“. NOUN iajs-1977 94 30 to to PART iajs-1977 94 31 illustrate illustrate VERB iajs-1977 94 32 theorem theorem NOUN iajs-1977 94 33 9 9 NUM iajs-1977 94 34 , , PUNCT iajs-1977 94 35 we we PRON iajs-1977 94 36 give give VERB iajs-1977 94 37 the the DET iajs-1977 94 38 following follow VERB iajs-1977 94 39 example example NOUN iajs-1977 94 40 example example NOUN iajs-1977 94 41 10 10 NUM iajs-1977 94 42 let let VERB iajs-1977 94 43 โ„ณ โ„ณ PROPN iajs-1977 94 44 ๐‘… ๐‘… PROPN iajs-1977 94 45 and and CCONJ iajs-1977 94 46 ๐›ถ ๐›ถ PROPN iajs-1977 94 47 be be VERB iajs-1977 94 48 ๐บ ๐บ NOUN iajs-1977 94 49 metric metric NOUN iajs-1977 94 50 on on ADP iajs-1977 94 51 โ„ณ โ„ณ PROPN iajs-1977 94 52 as as ADP iajs-1977 94 53 in in ADP iajs-1977 94 54 example example NOUN iajs-1977 94 55 6 6 NUM iajs-1977 94 56 define define VERB iajs-1977 94 57 ๐‘“ ๐‘“ PRON iajs-1977 94 58 : : PUNCT iajs-1977 94 59 โ„ณ โ„ณ PROPN iajs-1977 94 60 โ†’ โ†’ SYM iajs-1977 94 61 โ„ณ โ„ณ PROPN iajs-1977 94 62 , , PUNCT iajs-1977 94 63 ๐œ” ๐œ” PART iajs-1977 94 64 : : PUNCT iajs-1977 94 65 โ„ณ โ„ณ NOUN iajs-1977 94 66 โ„ณ โ„ณ NOUN iajs-1977 94 67 โ„ณ โ„ณ PROPN iajs-1977 94 68 โ†’ โ†’ SYM iajs-1977 94 69 โˆž โˆž PROPN iajs-1977 94 70 โˆช โˆช ADP iajs-1977 94 71 0 0 NUM iajs-1977 94 72 , , PUNCT iajs-1977 94 73 โˆž โˆž PROPN iajs-1977 94 74 , , PUNCT iajs-1977 94 75 ฯ† ฯ† PROPN iajs-1977 94 76 : : PUNCT iajs-1977 94 77 โ„ณ โ„ณ PROPN iajs-1977 94 78 โ„ณ โ„ณ NOUN iajs-1977 94 79 โ„ณ โ„ณ PROPN iajs-1977 94 80 โ†’ โ†’ SYM iajs-1977 94 81 ๐‘… ๐‘… PROPN iajs-1977 94 82 , , PUNCT iajs-1977 94 83 ๐œ“ ๐œ“ PROPN iajs-1977 94 84 : : PUNCT iajs-1977 94 85 ๐‘… ๐‘… PROPN iajs-1977 94 86 โ†’ โ†’ SYM iajs-1977 94 87 ๐‘… ๐‘… PROPN iajs-1977 94 88 , , PUNCT iajs-1977 94 89 and and CCONJ iajs-1977 94 90 ๐น ๐น PROPN iajs-1977 94 91 : : PUNCT iajs-1977 94 92 ๐‘… ๐‘… PROPN iajs-1977 94 93 โ†’ โ†’ SYM iajs-1977 94 94 ๐‘… ๐‘… PROPN iajs-1977 94 95 by by ADP iajs-1977 94 96 ๐‘“ ๐‘“ PRON iajs-1977 94 97 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 94 98 โˆš๐‘Ÿ โˆš๐‘Ÿ NOUN iajs-1977 94 99 , , PUNCT iajs-1977 94 100 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 94 101 โˆˆ โˆˆ NOUN iajs-1977 94 102 0,1 0,1 NUM iajs-1977 94 103 , , PUNCT iajs-1977 94 104 2๐‘Ÿ 2๐‘Ÿ NUM iajs-1977 94 105 , , PUNCT iajs-1977 94 106 ๐‘Ÿ ๐‘Ÿ X iajs-1977 94 107 โˆˆ โˆˆ PROPN iajs-1977 94 108 1 1 NUM iajs-1977 94 109 , , PUNCT iajs-1977 94 110 โˆž โˆž NUM iajs-1977 94 111 , , PUNCT iajs-1977 94 112 ๐œ” ๐œ” PUNCT iajs-1977 94 113 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 94 114 , , PUNCT iajs-1977 94 115 ๐‘ข ๐‘ข PROPN iajs-1977 94 116 , , PUNCT iajs-1977 94 117 ๐‘ฃ ๐‘ฃ PRON iajs-1977 94 118 ๐‘’ ๐‘’ ADV iajs-1977 94 119 , , PUNCT iajs-1977 94 120 ๐‘Ÿ ๐‘Ÿ X iajs-1977 94 121 โˆˆ โˆˆ NOUN iajs-1977 94 122 0,1 0,1 NUM iajs-1977 94 123 , , PUNCT iajs-1977 94 124 otherwise otherwise ADV iajs-1977 94 125 ฯ† ฯ† PROPN iajs-1977 94 126 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 94 127 , , PUNCT iajs-1977 94 128 ๐‘ข ๐‘ข PROPN iajs-1977 94 129 , , PUNCT iajs-1977 94 130 ๐‘ฃ ๐‘ฃ X iajs-1977 94 131 for for ADP iajs-1977 94 132 all all DET iajs-1977 94 133 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 94 134 , , PUNCT iajs-1977 94 135 ๐‘ข ๐‘ข PROPN iajs-1977 94 136 , , PUNCT iajs-1977 94 137 ๐‘ฃ ๐‘ฃ PRON iajs-1977 94 138 โˆˆ โˆˆ PROPN iajs-1977 94 139 โ„ณ โ„ณ PROPN iajs-1977 94 140 , , PUNCT iajs-1977 94 141 ๐œ“ ๐œ“ PROPN iajs-1977 94 142 ( ( PUNCT iajs-1977 94 143 ๐‘ก ๐‘ก PROPN iajs-1977 94 144 , , PUNCT iajs-1977 94 145 ๐‘ก ๐‘ก PROPN iajs-1977 94 146 , , PUNCT iajs-1977 94 147 ๐‘ก ๐‘ก PROPN iajs-1977 94 148 , , PUNCT iajs-1977 94 149 ๐‘ก ๐‘ก X iajs-1977 94 150 ) ) PUNCT iajs-1977 94 151 = = SYM iajs-1977 94 152 ๐œ ๐œ NOUN iajs-1977 94 153 0 0 NUM iajs-1977 95 1 and and CCONJ iajs-1977 95 2 ๐น ๐น PRON iajs-1977 95 3 ๐‘ž ๐‘ž PROPN iajs-1977 95 4 ln ln ADJ iajs-1977 95 5 ๐‘ž ๐‘ž PROPN iajs-1977 95 6 with with ADP iajs-1977 95 7 ๐‘ž ๐‘ž PROPN iajs-1977 95 8 0 0 NUM iajs-1977 95 9 . . PUNCT iajs-1977 96 1 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 96 2 1 1 NUM iajs-1977 96 3 3 3 NUM iajs-1977 96 4 , , PUNCT iajs-1977 96 5 ๐œ– ๐œ– X iajs-1977 96 6 1 1 NUM iajs-1977 96 7 , , PUNCT iajs-1977 96 8 ๐ต ๐ต NOUN iajs-1977 96 9 ๐‘Ÿ ๐‘Ÿ X iajs-1977 96 10 , , PUNCT iajs-1977 96 11 ๐œ– ๐œ– PROPN iajs-1977 96 12 1 1 NUM iajs-1977 96 13 6 6 NUM iajs-1977 96 14 , , PUNCT iajs-1977 96 15 5 5 NUM iajs-1977 96 16 6 6 NUM iajs-1977 96 17 then then ADV iajs-1977 96 18 ๐›ถ ๐›ถ PROPN iajs-1977 96 19 ( ( PUNCT iajs-1977 96 20 , , PUNCT iajs-1977 96 21 ๐‘“ ๐‘“ X iajs-1977 96 22 , , PUNCT iajs-1977 96 23 ๐‘“ ๐‘“ DET iajs-1977 96 24 0.732 0.732 NUM iajs-1977 96 25 ๐œ–. ๐œ–. NOUN iajs-1977 96 26 if if SCONJ iajs-1977 96 27 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 96 28 , , PUNCT iajs-1977 96 29 ๐‘ข ๐‘ข X iajs-1977 96 30 , , PUNCT iajs-1977 96 31 ๐‘ฃ ๐‘ฃ PRON iajs-1977 96 32 โˆˆ โˆˆ PROPN iajs-1977 96 33 ๐ต ๐ต NOUN iajs-1977 96 34 ๐‘Ÿ ๐‘Ÿ X iajs-1977 96 35 , , PUNCT iajs-1977 96 36 ๐œ– ๐œ– PROPN iajs-1977 96 37 then then ADV iajs-1977 96 38 ๐œ” ๐œ” VERB iajs-1977 96 39 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 96 40 , , PUNCT iajs-1977 96 41 ๐‘ข ๐‘ข PROPN iajs-1977 96 42 , , PUNCT iajs-1977 96 43 ๐‘ฃ ๐‘ฃ X iajs-1977 96 44 ๐‘’ ๐‘’ PROPN iajs-1977 96 45 ฯ† ฯ† NUM iajs-1977 96 46 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 96 47 , , PUNCT iajs-1977 96 48 ๐‘ข ๐‘ข PROPN iajs-1977 96 49 , , PUNCT iajs-1977 96 50 ๐‘ฃ ๐‘ฃ X iajs-1977 96 51 . . PUNCT iajs-1977 97 1 on on ADP iajs-1977 97 2 the the DET iajs-1977 97 3 other other ADJ iajs-1977 97 4 hand hand NOUN iajs-1977 97 5 , , PUNCT iajs-1977 97 6 ๐‘“ ๐‘“ DET iajs-1977 97 7 ๐‘Ÿ ๐‘Ÿ DET iajs-1977 97 8 โˆˆ โˆˆ PROPN iajs-1977 97 9 ๐ต ๐ต PROPN iajs-1977 97 10 ๐‘Ÿ ๐‘Ÿ X iajs-1977 97 11 , , PUNCT iajs-1977 97 12 ๐œ– ๐œ– PROPN iajs-1977 97 13 , , PUNCT iajs-1977 97 14 โˆ€ โˆ€ VERB iajs-1977 97 15 ๐‘Ÿ ๐‘Ÿ X iajs-1977 97 16 โˆˆ โˆˆ PROPN iajs-1977 97 17 ๐ต ๐ต NOUN iajs-1977 97 18 ๐‘Ÿ ๐‘Ÿ X iajs-1977 97 19 , , PUNCT iajs-1977 97 20 ๐œ– ๐œ– PROPN iajs-1977 97 21 . . PUNCT iajs-1977 98 1 then then ADV iajs-1977 98 2 , , PUNCT iajs-1977 98 3 ๐œ” ๐œ” X iajs-1977 98 4 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 98 5 , , PUNCT iajs-1977 98 6 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 98 7 , , PUNCT iajs-1977 98 8 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 98 9 ฯ† ฯ† PROPN iajs-1977 98 10 ๐‘Ÿ ๐‘Ÿ PROPN iajs-1977 98 11 , , PUNCT iajs-1977 98 12 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 98 13 , , PUNCT iajs-1977 98 14 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 98 15 with with ADP iajs-1977 98 16 ๐›ถ ๐›ถ PROPN iajs-1977 98 17 ๐‘“ ๐‘“ PROPN iajs-1977 98 18 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 98 19 , , PUNCT iajs-1977 98 20 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 98 21 , , PUNCT iajs-1977 98 22 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 98 23 โˆš๐‘Ÿ โˆš๐‘Ÿ PROPN iajs-1977 98 24 โˆš๐‘ข โˆš๐‘ข PROPN iajs-1977 98 25 โˆš๐‘ข โˆš๐‘ข PROPN iajs-1977 98 26 โˆš๐‘ฃ โˆš๐‘ฃ NOUN iajs-1977 98 27 โˆš๐‘Ÿ โˆš๐‘Ÿ NOUN iajs-1977 98 28 โˆš๐‘ฃ โˆš๐‘ฃ X iajs-1977 98 29 0 0 X iajs-1977 98 30 . . PUNCT iajs-1977 99 1 clearly clearly ADV iajs-1977 99 2 ๐œ” ๐œ” SYM iajs-1977 99 3 0 0 NUM iajs-1977 99 4 , , PUNCT iajs-1977 99 5 ๐‘“1 ๐‘“1 PROPN iajs-1977 99 6 , , PUNCT iajs-1977 99 7 ๐‘“1 ๐‘“1 PROPN iajs-1977 99 8 ฯ† ฯ† PROPN iajs-1977 99 9 0 0 PROPN iajs-1977 99 10 , , PUNCT iajs-1977 99 11 ๐‘“1 ๐‘“1 PROPN iajs-1977 99 12 , , PUNCT iajs-1977 99 13 ๐‘“1 ๐‘“1 PROPN iajs-1977 99 14 , , PUNCT iajs-1977 99 15 then then ADV iajs-1977 99 16 we we PRON iajs-1977 99 17 have have VERB iajs-1977 99 18 ๐›ถ ๐›ถ PROPN iajs-1977 99 19 ๐‘“๐‘Ÿ ๐‘“๐‘Ÿ PROPN iajs-1977 99 20 , , PUNCT iajs-1977 99 21 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 99 22 , , PUNCT iajs-1977 99 23 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 99 24 โˆš๐‘Ÿ โˆš๐‘Ÿ PROPN iajs-1977 99 25 โˆš๐‘ข โˆš๐‘ข PROPN iajs-1977 99 26 โˆš๐‘Ÿ โˆš๐‘Ÿ NOUN iajs-1977 100 1 โˆš๐‘ข โˆš๐‘ข PROPN iajs-1977 101 1 โˆš๐‘Ÿ โˆš๐‘Ÿ NOUN iajs-1977 102 1 โˆš๐‘ข โˆš๐‘ข PROPN iajs-1977 103 1 โˆš๐‘ข โˆš๐‘ข PROPN iajs-1977 103 2 โˆš๐‘ฃ โˆš๐‘ฃ VERB iajs-1977 103 3 โˆš๐‘ข โˆš๐‘ข PROPN iajs-1977 103 4 โˆš๐‘ฃ โˆš๐‘ฃ PROPN iajs-1977 103 5 โˆš๐‘ข โˆš๐‘ข PROPN iajs-1977 103 6 โˆš๐‘ฃ โˆš๐‘ฃ ADP iajs-1977 103 7 โˆš๐‘Ÿ โˆš๐‘Ÿ NOUN iajs-1977 103 8 โˆš๐‘ฃ โˆš๐‘ฃ X iajs-1977 103 9 โˆš๐‘Ÿ โˆš๐‘Ÿ PROPN iajs-1977 103 10 โˆš๐‘ฃ โˆš๐‘ฃ PUNCT iajs-1977 103 11 โˆš๐‘Ÿ โˆš๐‘Ÿ NOUN iajs-1977 103 12 โˆš๐‘ฃ โˆš๐‘ฃ PUNCT iajs-1977 104 1 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 104 2 ๐‘ข ๐‘ข X iajs-1977 104 3 โˆš๐‘Ÿ โˆš๐‘Ÿ NOUN iajs-1977 104 4 โˆš๐‘ข โˆš๐‘ข ADP iajs-1977 104 5 ๐‘ข ๐‘ข PROPN iajs-1977 104 6 ๐‘ฃ ๐‘ฃ X iajs-1977 104 7 โˆš๐‘ข โˆš๐‘ข NOUN iajs-1977 104 8 โˆš๐‘ฃ โˆš๐‘ฃ PUNCT iajs-1977 104 9 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 104 10 ๐‘ฃ ๐‘ฃ X iajs-1977 104 11 โˆš๐‘Ÿ โˆš๐‘Ÿ NOUN iajs-1977 104 12 โˆš๐‘ฃ โˆš๐‘ฃ PUNCT iajs-1977 105 1 โ„Ž โ„Ž X iajs-1977 105 2 |๐‘Ÿ |๐‘Ÿ ADP iajs-1977 105 3 ๐‘ข| ๐‘ข| ADJ iajs-1977 105 4 |๐‘ข |๐‘ข PROPN iajs-1977 105 5 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 105 6 |๐‘Ÿ |๐‘Ÿ ADP iajs-1977 105 7 ๐‘ฃ| ๐‘ฃ| VERB iajs-1977 105 8 consequently consequently ADV iajs-1977 105 9 ๐œ ๐œ VERB iajs-1977 105 10 ๐น ๐น PROPN iajs-1977 105 11 ๐›ถ ๐›ถ PROPN iajs-1977 105 12 ๐‘“ ๐‘“ PROPN iajs-1977 105 13 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 105 14 , , PUNCT iajs-1977 105 15 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 105 16 , , PUNCT iajs-1977 105 17 ๐‘“๐‘ฃ ๐‘“๐‘ฃ NOUN iajs-1977 105 18 ฯ„ ฯ„ PROPN iajs-1977 105 19 ln ln PROPN iajs-1977 105 20 ๐›ถ ๐›ถ PROPN iajs-1977 105 21 ๐‘“ ๐‘“ PROPN iajs-1977 105 22 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 105 23 , , PUNCT iajs-1977 105 24 ๐‘“๐‘ข ๐‘“๐‘ข PROPN iajs-1977 105 25 , , PUNCT iajs-1977 105 26 ๐‘“๐‘ฃ ๐‘“๐‘ฃ VERB iajs-1977 105 27 ln ln ADJ iajs-1977 105 28 โ„Ž โ„Ž PROPN iajs-1977 105 29 ๐›ถ ๐›ถ PROPN iajs-1977 105 30 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 105 31 , , PUNCT iajs-1977 105 32 ๐‘ข ๐‘ข PROPN iajs-1977 105 33 , , PUNCT iajs-1977 105 34 ๐‘ฃ ๐‘ฃ ADP iajs-1977 105 35 ๐น ๐น PROPN iajs-1977 105 36 โ„Ž โ„Ž PROPN iajs-1977 105 37 ๐›ถ ๐›ถ PROPN iajs-1977 105 38 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 105 39 , , PUNCT iajs-1977 105 40 ๐‘ข ๐‘ข X iajs-1977 105 41 , , PUNCT iajs-1977 105 42 ๐‘ฃ ๐‘ฃ X iajs-1977 105 43 . . PUNCT iajs-1977 106 1 if if SCONJ iajs-1977 106 2 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 106 3 โˆ‰ โˆ‰ PROPN iajs-1977 106 4 ๐ต ๐ต PROPN iajs-1977 106 5 ๐‘Ÿ ๐‘Ÿ X iajs-1977 106 6 , , PUNCT iajs-1977 106 7 ๐œ– ๐œ– PROPN iajs-1977 106 8 or or CCONJ iajs-1977 106 9 ๐‘ข ๐‘ข X iajs-1977 106 10 โˆ‰ โˆ‰ PROPN iajs-1977 106 11 ๐ต ๐ต PROPN iajs-1977 106 12 ๐‘Ÿ ๐‘Ÿ X iajs-1977 106 13 , , PUNCT iajs-1977 106 14 ๐œ– ๐œ– PROPN iajs-1977 106 15 or or CCONJ iajs-1977 106 16 ๐‘ฃ ๐‘ฃ ADP iajs-1977 106 17 โˆ‰ โˆ‰ ADJ iajs-1977 106 18 ๐ต ๐ต PROPN iajs-1977 106 19 ๐‘Ÿ ๐‘Ÿ X iajs-1977 106 20 , , PUNCT iajs-1977 106 21 ๐œ– ๐œ– PROPN iajs-1977 106 22 then then ADV iajs-1977 106 23 ๐œ” ๐œ” ADP iajs-1977 106 24 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 106 25 , , PUNCT iajs-1977 106 26 ๐‘ข ๐‘ข X iajs-1977 106 27 , , PUNCT iajs-1977 106 28 ๐‘ฃ ๐‘ฃ DET iajs-1977 106 29 1 1 NUM iajs-1977 106 30 5 5 NUM iajs-1977 106 31 โ‰ฑ โ‰ฑ PROPN iajs-1977 106 32 1 1 NUM iajs-1977 106 33 3 3 NUM iajs-1977 106 34 ฯ† ฯ† NUM iajs-1977 106 35 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 106 36 , , PUNCT iajs-1977 106 37 ๐‘ข ๐‘ข PROPN iajs-1977 106 38 , , PUNCT iajs-1977 106 39 ๐‘ฃ ๐‘ฃ PRON iajs-1977 106 40 2(|๐‘Ÿ 2(|๐‘Ÿ NUM iajs-1977 106 41 ๐‘ข| ๐‘ข| ADJ iajs-1977 106 42 |๐‘ข |๐‘ข PROPN iajs-1977 106 43 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 106 44 |๐‘Ÿ |๐‘Ÿ ADP iajs-1977 106 45 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 106 46 |๐‘Ÿ |๐‘Ÿ ADP iajs-1977 106 47 ๐‘ข| ๐‘ข| ADJ iajs-1977 106 48 |๐‘ข |๐‘ข PROPN iajs-1977 106 49 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 106 50 |๐‘Ÿ |๐‘Ÿ ADP iajs-1977 106 51 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 106 52 |๐‘“ |๐‘“ NOUN iajs-1977 106 53 ๐‘Ÿ ๐‘Ÿ PRON iajs-1977 106 54 ๐‘“ ๐‘“ DET iajs-1977 106 55 ๐‘ข| ๐‘ข| ADJ iajs-1977 106 56 |๐‘“ |๐‘“ NOUN iajs-1977 106 57 ๐‘ข ๐‘ข PROPN iajs-1977 106 58 ๐‘“ ๐‘“ DET iajs-1977 106 59 ๐‘ฃ| ๐‘ฃ| NOUN iajs-1977 106 60 |๐‘“ |๐‘“ NOUN iajs-1977 106 61 ๐‘Ÿ ๐‘Ÿ PROPN iajs-1977 106 62 ๐‘“ ๐‘“ DET iajs-1977 106 63 ๐‘ฃ| ๐‘ฃ| NOUN iajs-1977 106 64 |๐‘Ÿ |๐‘Ÿ ADP iajs-1977 106 65 ๐‘ข| ๐‘ข| ADJ iajs-1977 106 66 |๐‘ข |๐‘ข PROPN iajs-1977 106 67 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 106 68 |๐‘Ÿ |๐‘Ÿ ADP iajs-1977 106 69 ๐‘ฃ| ๐‘ฃ| PROPN iajs-1977 106 70 ๐œ ๐œ X iajs-1977 106 71 ๐น ๐น PROPN iajs-1977 106 72 ๐›ถ ๐›ถ PROPN iajs-1977 106 73 ๐‘“ ๐‘“ PROPN iajs-1977 106 74 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 106 75 , , PUNCT iajs-1977 106 76 ๐‘“ ๐‘“ DET iajs-1977 106 77 ๐‘ข ๐‘ข X iajs-1977 106 78 , , PUNCT iajs-1977 106 79 ๐‘“ ๐‘“ PRON iajs-1977 106 80 ๐‘ฃ ๐‘ฃ ADP iajs-1977 106 81 ๐น ๐น PROPN iajs-1977 106 82 ๐›ถ ๐›ถ PROPN iajs-1977 106 83 ๐‘Ÿ ๐‘Ÿ NOUN iajs-1977 106 84 , , PUNCT iajs-1977 106 85 ๐‘ข ๐‘ข PROPN iajs-1977 106 86 , , PUNCT iajs-1977 106 87 ๐‘ฃ ๐‘ฃ ADP iajs-1977 106 88 the the DET iajs-1977 106 89 contraction contraction NOUN iajs-1977 106 90 does do AUX iajs-1977 106 91 not not PART iajs-1977 106 92 hold hold VERB iajs-1977 106 93 . . PUNCT iajs-1977 107 1 3.conclusion 3.conclusion NUM iajs-1977 107 2 and and CCONJ iajs-1977 107 3 open open ADJ iajs-1977 107 4 problem problem NOUN iajs-1977 107 5 this this DET iajs-1977 107 6 research research NOUN iajs-1977 107 7 focus focus NOUN iajs-1977 107 8 on on ADP iajs-1977 107 9 introducing introduce VERB iajs-1977 107 10 new new ADJ iajs-1977 107 11 idea idea NOUN iajs-1977 107 12 of of ADP iajs-1977 107 13 f f NOUN iajs-1977 107 14 - - PUNCT iajs-1977 107 15 contraction contraction NOUN iajs-1977 107 16 on on ADP iajs-1977 107 17 a a DET iajs-1977 107 18 closed closed ADJ iajs-1977 107 19 ball ball NOUN iajs-1977 107 20 which which PRON iajs-1977 107 21 is be AUX iajs-1977 107 22 different different ADJ iajs-1977 107 23 from from ADP iajs-1977 107 24 f f NOUN iajs-1977 107 25 - - PUNCT iajs-1977 107 26 contraction contraction NOUN iajs-1977 107 27 given give VERB iajs-1977 107 28 in in ADP iajs-1977 107 29 [ [ X iajs-1977 107 30 4 4 NUM iajs-1977 107 31 ] ] PUNCT iajs-1977 107 32 . . PUNCT iajs-1977 108 1 therefore therefore ADV iajs-1977 108 2 a a DET iajs-1977 108 3 generalization generalization NOUN iajs-1977 108 4 of of ADP iajs-1977 108 5 results result NOUN iajs-1977 108 6 is be AUX iajs-1977 108 7 very very ADV iajs-1977 108 8 useful useful ADJ iajs-1977 108 9 so so ADV iajs-1977 108 10 far far ADV iajs-1977 108 11 as as SCONJ iajs-1977 108 12 it it PRON iajs-1977 108 13 requires require VERB iajs-1977 108 14 the the DET iajs-1977 108 15 f f NUM iajs-1977 108 16 - - PUNCT iajs-1977 108 17 contraction contraction NOUN iajs-1977 108 18 mapping mapping NOUN iajs-1977 108 19 only only ADV iajs-1977 108 20 on on ADP iajs-1977 108 21 a a DET iajs-1977 108 22 closed closed ADJ iajs-1977 108 23 ball ball NOUN iajs-1977 108 24 rather rather ADV iajs-1977 108 25 the the DET iajs-1977 108 26 whole whole ADJ iajs-1977 108 27 space space NOUN iajs-1977 108 28 . . PUNCT iajs-1977 109 1 this this DET iajs-1977 109 2 new new ADJ iajs-1977 109 3 idea idea NOUN iajs-1977 109 4 however however ADV iajs-1977 109 5 guides guide VERB iajs-1977 109 6 the the DET iajs-1977 109 7 researcher researcher NOUN iajs-1977 109 8 towards towards ADP iajs-1977 109 9 futher futher NOUN iajs-1977 109 10 investigations investigation NOUN iajs-1977 109 11 and and CCONJ iajs-1977 109 12 applications.at applications.at NUM iajs-1977 109 13 the the DET iajs-1977 109 14 ย  ย  ย  ย  SPACE iajs-1977 109 15 146 146 NUM iajs-1977 109 16 ibn ibn PROPN iajs-1977 109 17 al al PROPN iajs-1977 109 18 - - PUNCT iajs-1977 109 19 haitham haitham PROPN iajs-1977 109 20 jour jour X iajs-1977 109 21 . . PROPN iajs-1977 110 1 for for ADP iajs-1977 110 2 pure pure ADJ iajs-1977 110 3 & & CCONJ iajs-1977 110 4 appl appl PROPN iajs-1977 110 5 . . PUNCT iajs-1977 111 1 sc sc PROPN iajs-1977 111 2 i. i. PROPN iajs-1977 111 3 ihjpas ihjpas PROPN iajs-1977 111 4 ย  ย  SPACE iajs-1977 111 5 https://doi.org/10.30526/32.1.1977 https://doi.org/10.30526/32.1.1977 NOUN iajs-1977 111 6 ย  ย  SPACE iajs-1977 111 7 vol vol NOUN iajs-1977 111 8 . . PUNCT iajs-1977 112 1 32 32 NUM iajs-1977 112 2 ( ( PUNCT iajs-1977 112 3 1 1 NUM iajs-1977 112 4 ) ) PUNCT iajs-1977 112 5 2019 2019 NUM iajs-1977 112 6 same same ADJ iajs-1977 112 7 time time NOUN iajs-1977 112 8 , , PUNCT iajs-1977 112 9 it it PRON iajs-1977 112 10 will will AUX iajs-1977 112 11 be be AUX iajs-1977 112 12 interesting interesting ADJ iajs-1977 112 13 to to PART iajs-1977 112 14 apply apply VERB iajs-1977 112 15 these these DET iajs-1977 112 16 concepts concept NOUN iajs-1977 112 17 in in ADP iajs-1977 112 18 a a DET iajs-1977 112 19 various various ADJ iajs-1977 112 20 spaces space NOUN iajs-1977 112 21 . . PUNCT iajs-1977 113 1 in in ADP iajs-1977 113 2 future future NOUN iajs-1977 113 3 , , PUNCT iajs-1977 113 4 we we PRON iajs-1977 113 5 suggest suggest AUX iajs-1977 113 6 study study VERB iajs-1977 113 7 the the DET iajs-1977 113 8 results result NOUN iajs-1977 113 9 in in ADP iajs-1977 113 10 [ [ X iajs-1977 113 11 8 8 NUM iajs-1977 113 12 ] ] PUNCT iajs-1977 113 13 to to PART iajs-1977 113 14 verify verify VERB iajs-1977 113 15 the the DET iajs-1977 113 16 extent extent NOUN iajs-1977 113 17 achieved achieve VERB iajs-1977 113 18 in in ADP iajs-1977 113 19 the the DET iajs-1977 113 20 setting setting NOUN iajs-1977 113 21 of of ADP iajs-1977 113 22 ๐น ๐น PROPN iajs-1977 113 23 contraction contraction NOUN iajs-1977 113 24 mappings mapping NOUN iajs-1977 113 25 in in ADP iajs-1977 113 26 modular modular ADJ iajs-1977 113 27 spaces space NOUN iajs-1977 113 28 . . PUNCT iajs-1977 114 1 references reference NOUN iajs-1977 114 2 1 1 NUM iajs-1977 114 3 . . NUM iajs-1977 114 4 1.mustafa 1.mustafa NUM iajs-1977 114 5 , , PUNCT iajs-1977 114 6 z. z. PROPN iajs-1977 114 7 ; ; PUNCT iajs-1977 114 8 sims sim NOUN iajs-1977 114 9 , , PUNCT iajs-1977 114 10 b.a b.a PROPN iajs-1977 114 11 . . PROPN iajs-1977 114 12 new new ADJ iajs-1977 114 13 approach approach NOUN iajs-1977 114 14 to to ADP iajs-1977 114 15 generalized generalize VERB iajs-1977 114 16 metric metric ADJ iajs-1977 114 17 space space NOUN iajs-1977 114 18 . . PUNCT iajs-1977 115 1 j. j. PROPN iajs-1977 115 2 of of ADP iajs-1977 115 3 nonlinear nonlinear PROPN iajs-1977 115 4 and and CCONJ iajs-1977 115 5 convex convex ADJ iajs-1977 115 6 analysis analysis NOUN iajs-1977 115 7 . . PUNCT iajs-1977 116 1 2006 2006 NUM iajs-1977 116 2 . . PUNCT iajs-1977 117 1 7 7 NUM iajs-1977 117 2 , , PUNCT iajs-1977 117 3 2 2 NUM iajs-1977 117 4 , , PUNCT iajs-1977 117 5 289 289 NUM iajs-1977 117 6 - - SYM iajs-1977 117 7 297 297 NUM iajs-1977 117 8 . . PUNCT iajs-1977 118 1 2 2 NUM iajs-1977 118 2 . . NUM iajs-1977 118 3 abed abe VERB iajs-1977 118 4 , , PUNCT iajs-1977 118 5 s.s s.s PROPN iajs-1977 118 6 . . PROPN iajs-1977 118 7 ; ; PUNCT iajs-1977 119 1 luaibi luaibi PROPN iajs-1977 119 2 , , PUNCT iajs-1977 119 3 h. h. PROPN iajs-1977 119 4 h. h. PROPN iajs-1977 120 1 two two NUM iajs-1977 120 2 fixed fix VERB iajs-1977 120 3 point point NOUN iajs-1977 120 4 theorems theorem NOUN iajs-1977 120 5 in in ADP iajs-1977 120 6 generalized generalized ADJ iajs-1977 120 7 metric metric ADJ iajs-1977 120 8 spaces space NOUN iajs-1977 120 9 . . PUNCT iajs-1977 121 1 international international ADJ iajs-1977 121 2 journal journal NOUN iajs-1977 121 3 of of ADP iajs-1977 121 4 advanced advanced ADJ iajs-1977 121 5 statistics statistic NOUN iajs-1977 121 6 and and CCONJ iajs-1977 121 7 probability probability NOUN iajs-1977 121 8 . . PUNCT iajs-1977 122 1 2016,4,1 2016,4,1 NUM iajs-1977 122 2 , , PUNCT iajs-1977 122 3 16 16 NUM iajs-1977 122 4 - - SYM iajs-1977 122 5 19 19 NUM iajs-1977 122 6 . . NOUN iajs-1977 122 7 3 3 NUM iajs-1977 122 8 . . X iajs-1977 122 9 gajic gajic PROPN iajs-1977 122 10 , , PUNCT iajs-1977 122 11 l. l. PROPN iajs-1977 122 12 ; ; PUNCT iajs-1977 122 13 stojakovic stojakovic ADJ iajs-1977 122 14 , , PUNCT iajs-1977 122 15 m. m. NOUN iajs-1977 122 16 ; ; PUNCT iajs-1977 122 17 thomas thomas PROPN iajs-1977 122 18 , , PUNCT iajs-1977 122 19 s. s. PROPN iajs-1977 122 20 type type PROPN iajs-1977 122 21 fixed fix VERB iajs-1977 122 22 point point NOUN iajs-1977 122 23 theorems theorem NOUN iajs-1977 122 24 in in ADP iajs-1977 122 25 generalized generalized ADJ iajs-1977 122 26 metric metric ADJ iajs-1977 122 27 spaces space NOUN iajs-1977 122 28 . . PUNCT iajs-1977 123 1 filomat filomat PROPN iajs-1977 123 2 . . PROPN iajs-1977 124 1 2017 2017 NUM iajs-1977 124 2 , , PUNCT iajs-1977 124 3 31,11 31,11 NUM iajs-1977 124 4 , , PUNCT iajs-1977 124 5 3347โ€“3356 3347โ€“3356 NUM iajs-1977 124 6 4 4 NUM iajs-1977 124 7 . . PUNCT iajs-1977 125 1 tahat tahat PROPN iajs-1977 125 2 , , PUNCT iajs-1977 125 3 n. n. PROPN iajs-1977 125 4 ; ; PUNCT iajs-1977 125 5 et et PROPN iajs-1977 125 6 al al PROPN iajs-1977 125 7 . . PROPN iajs-1977 125 8 common common ADJ iajs-1977 125 9 fixed fix VERB iajs-1977 125 10 points point NOUN iajs-1977 125 11 for for ADP iajs-1977 125 12 single single ADJ iajs-1977 125 13 valued value VERB iajs-1977 125 14 and and CCONJ iajs-1977 125 15 multi multi ADJ iajs-1977 125 16 - - ADJ iajs-1977 125 17 valued value VERB iajs-1977 125 18 maps map NOUN iajs-1977 125 19 satisfying satisfy VERB iajs-1977 125 20 generalized generalized ADJ iajs-1977 125 21 contraction contraction NOUN iajs-1977 125 22 in in ADP iajs-1977 125 23 g g NOUN iajs-1977 125 24 - - PUNCT iajs-1977 125 25 metric metric ADJ iajs-1977 125 26 spaces space NOUN iajs-1977 125 27 . . PUNCT iajs-1977 126 1 fixed fix VERB iajs-1977 126 2 point point NOUN iajs-1977 126 3 theory theory NOUN iajs-1977 126 4 and and CCONJ iajs-1977 126 5 applications application NOUN iajs-1977 126 6 . . PUNCT iajs-1977 127 1 2012 2012 NUM iajs-1977 127 2 , , PUNCT iajs-1977 127 3 7 7 NUM iajs-1977 127 4 , , PUNCT iajs-1977 127 5 31 31 NUM iajs-1977 127 6 - - SYM iajs-1977 127 7 39 39 NUM iajs-1977 127 8 . . PUNCT iajs-1977 128 1 5 5 NUM iajs-1977 128 2 . . NUM iajs-1977 128 3 abed abe VERB iajs-1977 128 4 , , PUNCT iajs-1977 128 5 s.s s.s PROPN iajs-1977 128 6 . . PROPN iajs-1977 128 7 ; ; PUNCT iajs-1977 128 8 faraj faraj ADJ iajs-1977 128 9 , , PUNCT iajs-1977 128 10 a.n.topological a.n.topological ADJ iajs-1977 128 11 properties property NOUN iajs-1977 128 12 of of ADP iajs-1977 128 13 ๐บ-hausdorff ๐บ-hausdorff PROPN iajs-1977 128 14 metric metric NOUN iajs-1977 128 15 . . PUNCT iajs-1977 129 1 international international ADJ iajs-1977 129 2 journal journal PROPN iajs-1977 129 3 of of ADP iajs-1977 129 4 applied apply VERB iajs-1977 129 5 mathematics mathematic NOUN iajs-1977 129 6 and and CCONJ iajs-1977 129 7 statistical statistical ADJ iajs-1977 129 8 sciences science NOUN iajs-1977 129 9 ( ( PUNCT iajs-1977 129 10 ijamss ijamss NOUN iajs-1977 129 11 ) ) PUNCT iajs-1977 129 12 . . PUNCT iajs-1977 130 1 2018 2018 NUM iajs-1977 130 2 , , PUNCT iajs-1977 130 3 7 7 NUM iajs-1977 130 4 , , PUNCT iajs-1977 130 5 5 5 NUM iajs-1977 130 6 , , PUNCT iajs-1977 130 7 1 1 NUM iajs-1977 130 8 - - SYM iajs-1977 130 9 18 18 NUM iajs-1977 130 10 . . NOUN iajs-1977 130 11 6 6 NUM iajs-1977 130 12 . . NUM iajs-1977 130 13 abed abe VERB iajs-1977 130 14 , , PUNCT iajs-1977 130 15 s.s s.s PROPN iajs-1977 130 16 . . PROPN iajs-1977 130 17 fixed fix VERB iajs-1977 130 18 point point NOUN iajs-1977 130 19 principles principle NOUN iajs-1977 130 20 in in ADP iajs-1977 130 21 general general ADJ iajs-1977 130 22 ๐‘ ๐‘ DET iajs-1977 130 23 metric metric ADJ iajs-1977 130 24 spaces space NOUN iajs-1977 130 25 and and CCONJ iajs-1977 130 26 ๐‘ ๐‘ DET iajs-1977 130 27 menger menger PROPN iajs-1977 130 28 probabilistic probabilistic ADJ iajs-1977 130 29 spaces space NOUN iajs-1977 130 30 . . PUNCT iajs-1977 131 1 journal journal PROPN iajs-1977 131 2 of of ADP iajs-1977 131 3 al al PROPN iajs-1977 131 4 - - PUNCT iajs-1977 131 5 qadisiyah qadisiyah NOUN iajs-1977 131 6 for for ADP iajs-1977 131 7 computer computer NOUN iajs-1977 131 8 science science NOUN iajs-1977 131 9 and and CCONJ iajs-1977 131 10 mathematics mathematic NOUN iajs-1977 131 11 . . PUNCT iajs-1977 132 1 2018 2018 NUM iajs-1977 132 2 , , PUNCT iajs-1977 132 3 10 10 NUM iajs-1977 132 4 , , PUNCT iajs-1977 132 5 2 2 NUM iajs-1977 132 6 , , PUNCT iajs-1977 132 7 2521 2521 NUM iajs-1977 132 8 - - SYM iajs-1977 132 9 0204 0204 NUM iajs-1977 132 10 . . PUNCT iajs-1977 133 1 7 7 X iajs-1977 133 2 . . NUM iajs-1977 133 3 abed abe VERB iajs-1977 133 4 , , PUNCT iajs-1977 133 5 s.s s.s PROPN iajs-1977 133 6 . . PROPN iajs-1977 133 7 ; ; PUNCT iajs-1977 133 8 luaibi luaibi PROPN iajs-1977 133 9 , , PUNCT iajs-1977 133 10 h.h h.h PROPN iajs-1977 133 11 . . PROPN iajs-1977 133 12 implicit implicit ADJ iajs-1977 133 13 fixed fix VERB iajs-1977 133 14 points point NOUN iajs-1977 133 15 in in ADP iajs-1977 133 16 manger manger NOUN iajs-1977 133 17 gmetric gmetric ADJ iajs-1977 133 18 spaces space VERB iajs-1977 133 19 . . 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