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PROPN iajs-2475 3 1 33 33 NUM iajs-2475 3 2 ( ( PUNCT iajs-2475 3 3 3 3 NUM iajs-2475 3 4 ) ) PUNCT iajs-2475 3 5 2020 2020 NUM iajs-2475 3 6 βŠ•-s βŠ•-s ADJ iajs-2475 3 7 - - PUNCT iajs-2475 3 8 extending extend VERB iajs-2475 3 9 modules module NOUN iajs-2475 3 10 department department NOUN iajs-2475 3 11 of of ADP iajs-2475 3 12 mathematics mathematics PROPN iajs-2475 3 13 , , PUNCT iajs-2475 3 14 college college NOUN iajs-2475 3 15 of of ADP iajs-2475 3 16 science science NOUN iajs-2475 3 17 , , PUNCT iajs-2475 3 18 mustansiriyah mustansiriyah NOUN iajs-2475 3 19 university university NOUN iajs-2475 3 20 , , PUNCT iajs-2475 3 21 baghdad baghdad PROPN iajs-2475 3 22 , , PUNCT iajs-2475 3 23 iraq iraq PROPN iajs-2475 3 24 . . PUNCT iajs-2475 4 1 ayaadnan994@gmail.com ayaadnan994@gmail.com X iajs-2475 5 1 saalsaadi@uomustansiriyah.edu.iq saalsaadi@uomustansiriyah.edu.iq PROPN iajs-2475 5 2 π€π›π¬π­π«πšπœπ­ π€π›π¬π­π«πšπœπ­ PROPN iajs-2475 5 3 the the DET iajs-2475 5 4 concept concept NOUN iajs-2475 5 5 βŠ•-s βŠ•-s NOUN iajs-2475 5 6 - - PUNCT iajs-2475 5 7 extending extend VERB iajs-2475 5 8 modules module NOUN iajs-2475 5 9 will will AUX iajs-2475 5 10 be be AUX iajs-2475 5 11 purpose purpose NOUN iajs-2475 5 12 of of ADP iajs-2475 5 13 this this DET iajs-2475 5 14 paper paper NOUN iajs-2475 5 15 , , PUNCT iajs-2475 5 16 a a DET iajs-2475 5 17 module module NOUN iajs-2475 5 18 m m NOUN iajs-2475 5 19 is be AUX iajs-2475 5 20 βŠ•s βŠ•s NOUN iajs-2475 5 21 - - PUNCT iajs-2475 5 22 extending extend VERB iajs-2475 5 23 if if SCONJ iajs-2475 5 24 each each DET iajs-2475 5 25 submodule submodule NOUN iajs-2475 5 26 in in ADP iajs-2475 5 27 m m PROPN iajs-2475 5 28 is be AUX iajs-2475 5 29 essential essential ADJ iajs-2475 5 30 in in ADP iajs-2475 5 31 submodule submodule NOUN iajs-2475 5 32 has have VERB iajs-2475 5 33 a a DET iajs-2475 5 34 supplement supplement NOUN iajs-2475 5 35 that that PRON iajs-2475 5 36 is be AUX iajs-2475 5 37 direct direct ADJ iajs-2475 5 38 summand summand NOUN iajs-2475 5 39 . . 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PUNCT iajs-2475 7 1 in in ADP iajs-2475 7 2 fact fact NOUN iajs-2475 7 3 , , PUNCT iajs-2475 7 4 we we PRON iajs-2475 7 5 gives give VERB iajs-2475 7 6 the the DET iajs-2475 7 7 following follow VERB iajs-2475 7 8 implications implication NOUN iajs-2475 7 9 : : PUNCT iajs-2475 7 10 lifting lift VERB iajs-2475 7 11 modules module NOUN iajs-2475 7 12 ⟹ ⟹ PUNCT iajs-2475 7 13 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 7 14 modules module NOUN iajs-2475 7 15 ⟹ ⟹ ADP iajs-2475 7 16 βŠ•-s βŠ•-s ADJ iajs-2475 7 17 - - PUNCT iajs-2475 7 18 extending extend VERB iajs-2475 7 19 modules module NOUN iajs-2475 7 20 ⟹ ⟹ PRON iajs-2475 7 21 weakly weakly ADV iajs-2475 7 22 supplement supplement VERB iajs-2475 7 23 extending extend VERB iajs-2475 7 24 modules module NOUN iajs-2475 7 25 . . PUNCT iajs-2475 8 1 it it PRON iajs-2475 8 2 is be AUX iajs-2475 8 3 also also ADV iajs-2475 8 4 we we PRON iajs-2475 8 5 give give VERB iajs-2475 8 6 examples example NOUN iajs-2475 8 7 show show VERB iajs-2475 8 8 that that SCONJ iajs-2475 8 9 , , PUNCT iajs-2475 8 10 the the DET iajs-2475 8 11 converse converse NOUN iajs-2475 8 12 of of ADP iajs-2475 8 13 this this DET iajs-2475 8 14 result result NOUN iajs-2475 8 15 is be AUX iajs-2475 8 16 not not PART iajs-2475 8 17 true true ADJ iajs-2475 8 18 . . PUNCT iajs-2475 9 1 moreover moreover ADV iajs-2475 9 2 , , PUNCT iajs-2475 9 3 we we PRON iajs-2475 9 4 study study VERB iajs-2475 9 5 when when SCONJ iajs-2475 9 6 the the DET iajs-2475 9 7 converse converse NOUN iajs-2475 9 8 of of ADP iajs-2475 9 9 this this DET iajs-2475 9 10 result result NOUN iajs-2475 9 11 is be AUX iajs-2475 9 12 true true ADJ iajs-2475 9 13 . . PUNCT iajs-2475 10 1 keyword keyword NOUN iajs-2475 10 2 : : PUNCT iajs-2475 10 3 extending extend VERB iajs-2475 10 4 modules module NOUN iajs-2475 10 5 , , PUNCT iajs-2475 10 6 weakly weakly ADJ iajs-2475 10 7 supplement supplement NOUN iajs-2475 10 8 extending extend VERB iajs-2475 10 9 modules module NOUN iajs-2475 10 10 , , PUNCT iajs-2475 10 11 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 10 12 modules module NOUN iajs-2475 10 13 , , PUNCT iajs-2475 10 14 βŠ•-s βŠ•-s NOUN iajs-2475 10 15 - - PUNCT iajs-2475 10 16 extending extend VERB iajs-2475 10 17 modules module NOUN iajs-2475 10 18 . . PUNCT iajs-2475 11 1 1 1 X iajs-2475 11 2 . . X iajs-2475 11 3 introduction introduction NOUN iajs-2475 11 4 in in ADP iajs-2475 11 5 this this DET iajs-2475 11 6 paper paper NOUN iajs-2475 11 7 , , PUNCT iajs-2475 11 8 motivated motivate VERB iajs-2475 11 9 by by ADP iajs-2475 11 10 the the DET iajs-2475 11 11 concept concept NOUN iajs-2475 11 12 of of ADP iajs-2475 11 13 closed closed ADJ iajs-2475 11 14 ⨁-supplemented ⨁-supplemente VERB iajs-2475 11 15 that that PRON iajs-2475 11 16 is be AUX iajs-2475 11 17 , , PUNCT iajs-2475 11 18 a a DET iajs-2475 11 19 module module NOUN iajs-2475 11 20 m m VERB iajs-2475 11 21 is be AUX iajs-2475 11 22 called call VERB iajs-2475 11 23 closed closed ADJ iajs-2475 11 24 ⨁-supplemented ⨁-supplemente VERB iajs-2475 11 25 , , PUNCT iajs-2475 11 26 if if SCONJ iajs-2475 11 27 each each DET iajs-2475 11 28 closed closed ADJ iajs-2475 11 29 submodule submodule NOUN iajs-2475 11 30 in in ADP iajs-2475 11 31 m m PROPN iajs-2475 11 32 has have VERB iajs-2475 11 33 a a DET iajs-2475 11 34 supplement supplement NOUN iajs-2475 11 35 which which PRON iajs-2475 11 36 is be AUX iajs-2475 11 37 direct direct ADJ iajs-2475 11 38 summand summand NOUN iajs-2475 11 39 , , PUNCT iajs-2475 11 40 and and CCONJ iajs-2475 11 41 weakly weakly ADJ iajs-2475 11 42 supplement supplement NOUN iajs-2475 11 43 extending extend VERB iajs-2475 11 44 module module NOUN iajs-2475 11 45 we we PRON iajs-2475 11 46 give give VERB iajs-2475 11 47 concept concept NOUN iajs-2475 11 48 that that PRON iajs-2475 11 49 is be AUX iajs-2475 11 50 βŠ•s βŠ•s NOUN iajs-2475 11 51 - - PUNCT iajs-2475 11 52 extending extend VERB iajs-2475 11 53 module module NOUN iajs-2475 11 54 , , PUNCT iajs-2475 11 55 a a DET iajs-2475 11 56 module module NOUN iajs-2475 11 57 m m VERB iajs-2475 11 58 is be AUX iajs-2475 11 59 βŠ•-s βŠ•-s ADV iajs-2475 11 60 - - PUNCT iajs-2475 11 61 extending extend VERB iajs-2475 11 62 if if SCONJ iajs-2475 11 63 each each DET iajs-2475 11 64 submodule submodule NOUN iajs-2475 11 65 in in ADP iajs-2475 11 66 m m PROPN iajs-2475 11 67 is be AUX iajs-2475 11 68 essential essential ADJ iajs-2475 11 69 in in ADP iajs-2475 11 70 submodule submodule NOUN iajs-2475 11 71 has have VERB iajs-2475 11 72 a a DET iajs-2475 11 73 supplement supplement NOUN iajs-2475 11 74 which which PRON iajs-2475 11 75 is be AUX iajs-2475 11 76 direct direct ADJ iajs-2475 11 77 summand summand NOUN iajs-2475 11 78 . . PUNCT iajs-2475 12 1 we we PRON iajs-2475 12 2 can can AUX iajs-2475 12 3 answer answer VERB iajs-2475 12 4 the the DET iajs-2475 12 5 question question NOUN iajs-2475 12 6 : : PUNCT iajs-2475 12 7 when when SCONJ iajs-2475 12 8 is be AUX iajs-2475 12 9 the the DET iajs-2475 12 10 βŠ•-s βŠ•-s ADJ iajs-2475 12 11 - - PUNCT iajs-2475 12 12 extending extend VERB iajs-2475 12 13 module module NOUN iajs-2475 12 14 inherited inherit VERB iajs-2475 12 15 by by ADP iajs-2475 12 16 direct direct ADJ iajs-2475 12 17 summand summand NOUN iajs-2475 12 18 . . PUNCT iajs-2475 13 1 following follow VERB iajs-2475 13 2 [ [ X iajs-2475 13 3 1 1 NUM iajs-2475 13 4 ] ] PUNCT iajs-2475 13 5 . . PUNCT iajs-2475 14 1 studied study VERB iajs-2475 14 2 a a DET iajs-2475 14 3 weakly weakly ADJ iajs-2475 14 4 supplement supplement NOUN iajs-2475 14 5 extending extend VERB iajs-2475 14 6 modules module NOUN iajs-2475 14 7 , , PUNCT iajs-2475 14 8 a a DET iajs-2475 14 9 module module NOUN iajs-2475 14 10 m m VERB iajs-2475 14 11 is be AUX iajs-2475 14 12 called call VERB iajs-2475 14 13 weakly weakly ADJ iajs-2475 14 14 supplement supplement NOUN iajs-2475 14 15 extending extend VERB iajs-2475 14 16 if if SCONJ iajs-2475 14 17 each each DET iajs-2475 14 18 submodule submodule NOUN iajs-2475 14 19 in in ADP iajs-2475 14 20 m m PROPN iajs-2475 14 21 is be AUX iajs-2475 14 22 essential essential ADJ iajs-2475 14 23 in in ADP iajs-2475 14 24 weakly weakly ADJ iajs-2475 14 25 supplement supplement NOUN iajs-2475 14 26 submodule submodule NOUN iajs-2475 14 27 in in ADP iajs-2475 14 28 m. m. NOUN iajs-2475 14 29 also also ADV iajs-2475 14 30 , , PUNCT iajs-2475 14 31 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 14 32 module module NOUN iajs-2475 14 33 were be AUX iajs-2475 14 34 studied study VERB iajs-2475 14 35 by by ADP iajs-2475 14 36 [ [ X iajs-2475 14 37 2 2 NUM iajs-2475 14 38 ] ] PUNCT iajs-2475 14 39 . . PUNCT iajs-2475 15 1 a a DET iajs-2475 15 2 module module NOUN iajs-2475 15 3 m m NOUN iajs-2475 15 4 is be AUX iajs-2475 15 5 ⨁-supplemented ⨁-supplemente VERB iajs-2475 15 6 , , PUNCT iajs-2475 15 7 if if SCONJ iajs-2475 15 8 for for ADP iajs-2475 15 9 any any DET iajs-2475 15 10 submodule submodule NOUN iajs-2475 15 11 n n PROPN iajs-2475 15 12 in in ADP iajs-2475 15 13 m m PROPN iajs-2475 15 14 has have VERB iajs-2475 15 15 a a DET iajs-2475 15 16 supplement supplement NOUN iajs-2475 15 17 which which PRON iajs-2475 15 18 is be AUX iajs-2475 15 19 article article NOUN iajs-2475 15 20 history history NOUN iajs-2475 15 21 : : PUNCT iajs-2475 15 22 received receive VERB iajs-2475 15 23 31 31 NUM iajs-2475 15 24 october october PROPN iajs-2475 15 25 2019 2019 NUM iajs-2475 15 26 , , PUNCT iajs-2475 15 27 accepted accept VERB iajs-2475 15 28 16 16 NUM iajs-2475 15 29 december december PROPN iajs-2475 15 30 2019 2019 NUM iajs-2475 15 31 , , PUNCT iajs-2475 15 32 published publish VERB iajs-2475 15 33 in in ADP iajs-2475 15 34 july july PROPN iajs-2475 15 35 2020 2020 NUM iajs-2475 15 36 . . PUNCT iajs-2475 16 1 doi doi NOUN iajs-2475 16 2 : : PUNCT iajs-2475 16 3 10.30526/33.3.2475 10.30526/33.3.2475 PROPN iajs-2475 16 4 aya aya PROPN iajs-2475 16 5 a. a. PROPN iajs-2475 16 6 al al PROPN iajs-2475 16 7 - - PUNCT iajs-2475 16 8 rubaye rubaye VERB iajs-2475 16 9 ibn ibn PROPN iajs-2475 16 10 al al PROPN iajs-2475 16 11 haitham haitham PROPN iajs-2475 16 12 journal journal PROPN iajs-2475 16 13 for for ADP iajs-2475 16 14 pure pure ADJ iajs-2475 16 15 and and CCONJ iajs-2475 16 16 applied apply VERB iajs-2475 16 17 science science NOUN iajs-2475 16 18 journal journal PROPN iajs-2475 16 19 homepage homepage NOUN iajs-2475 16 20 : : PUNCT iajs-2475 16 21 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index PROPN iajs-2475 16 22 saad saad PROPN iajs-2475 16 23 a. a. PROPN iajs-2475 16 24 al al PROPN iajs-2475 16 25 - - PUNCT iajs-2475 16 26 saadi saadi PROPN iajs-2475 16 27 mailto mailto PROPN iajs-2475 16 28 : : PUNCT iajs-2475 16 29 ayaadnan994@gmail.com2 ayaadnan994@gmail.com2 PROPN iajs-2475 16 30 mailto:saalsaadi@uomustansiriyah.edu.iq mailto:saalsaadi@uomustansiriyah.edu.iq PROPN iajs-2475 16 31 81 81 NUM iajs-2475 16 32 ibn ibn PROPN iajs-2475 16 33 al al PROPN iajs-2475 16 34 - - PUNCT iajs-2475 16 35 haitham haitham PROPN iajs-2475 16 36 jour jour X iajs-2475 16 37 . . PROPN iajs-2475 17 1 for for ADP iajs-2475 17 2 pure pure ADJ iajs-2475 17 3 & & CCONJ iajs-2475 17 4 appl appl PROPN iajs-2475 17 5 . . PUNCT iajs-2475 18 1 sci sci PROPN iajs-2475 18 2 . . PROPN iajs-2475 19 1 33 33 NUM iajs-2475 19 2 ( ( PUNCT iajs-2475 19 3 3 3 NUM iajs-2475 19 4 ) ) PUNCT iajs-2475 19 5 2020 2020 NUM iajs-2475 19 6 direct direct ADJ iajs-2475 19 7 summand summand NOUN iajs-2475 19 8 . . PUNCT iajs-2475 20 1 in in ADP iajs-2475 20 2 this this DET iajs-2475 20 3 work work NOUN iajs-2475 20 4 , , PUNCT iajs-2475 20 5 r r NOUN iajs-2475 20 6 is be AUX iajs-2475 20 7 associative associative ADJ iajs-2475 20 8 rings ring NOUN iajs-2475 20 9 with with ADP iajs-2475 20 10 identity identity NOUN iajs-2475 20 11 and and CCONJ iajs-2475 20 12 m m NOUN iajs-2475 20 13 is be AUX iajs-2475 20 14 left leave VERB iajs-2475 20 15 rmodule rmodule NOUN iajs-2475 20 16 . . PUNCT iajs-2475 21 1 a a DET iajs-2475 21 2 submodule submodule PROPN iajs-2475 21 3 w w PROPN iajs-2475 21 4 of of ADP iajs-2475 21 5 m m PROPN iajs-2475 21 6 is be AUX iajs-2475 21 7 said say VERB iajs-2475 21 8 to to PART iajs-2475 21 9 be be AUX iajs-2475 21 10 essential essential ADJ iajs-2475 21 11 , , PUNCT iajs-2475 21 12 if if SCONJ iajs-2475 21 13 w∩v=0 w∩v=0 NOUN iajs-2475 21 14 then then ADV iajs-2475 21 15 v=0[2 v=0[2 VERB iajs-2475 21 16 ] ] PUNCT iajs-2475 21 17 . . PUNCT iajs-2475 22 1 a a DET iajs-2475 22 2 submodule submodule PROPN iajs-2475 22 3 w w PROPN iajs-2475 22 4 of of ADP iajs-2475 22 5 m m PROPN iajs-2475 22 6 is be AUX iajs-2475 22 7 called call VERB iajs-2475 22 8 small small ADJ iajs-2475 22 9 ( ( PUNCT iajs-2475 22 10 denoted denote VERB iajs-2475 22 11 by by ADP iajs-2475 22 12 wβ‰ͺm wβ‰ͺm NOUN iajs-2475 22 13 ) ) PUNCT iajs-2475 22 14 if if SCONJ iajs-2475 22 15 for for ADP iajs-2475 22 16 each each DET iajs-2475 22 17 submodule submodule NOUN iajs-2475 22 18 l l PROPN iajs-2475 22 19 in in ADP iajs-2475 22 20 m m PRON iajs-2475 22 21 such such ADJ iajs-2475 22 22 that that SCONJ iajs-2475 22 23 mβ‰ w+l mβ‰ w+l PROPN iajs-2475 22 24 implies imply VERB iajs-2475 22 25 lβ‰ m lβ‰ m PUNCT iajs-2475 23 1 [ [ X iajs-2475 23 2 2 2 NUM iajs-2475 23 3 ] ] PUNCT iajs-2475 23 4 . . PUNCT iajs-2475 24 1 a a DET iajs-2475 24 2 submodule submodule PROPN iajs-2475 24 3 n n PROPN iajs-2475 24 4 of of ADP iajs-2475 24 5 m m PROPN iajs-2475 24 6 is be AUX iajs-2475 24 7 called call VERB iajs-2475 24 8 closed closed ADJ iajs-2475 24 9 , , PUNCT iajs-2475 24 10 if if SCONJ iajs-2475 24 11 n n PRON iajs-2475 24 12 has have VERB iajs-2475 24 13 no no DET iajs-2475 24 14 proper proper ADJ iajs-2475 24 15 essential essential ADJ iajs-2475 24 16 extension extension NOUN iajs-2475 24 17 in in ADP iajs-2475 24 18 m m PROPN iajs-2475 24 19 [ [ X iajs-2475 24 20 2 2 NUM iajs-2475 24 21 ] ] PUNCT iajs-2475 24 22 . . PUNCT iajs-2475 25 1 a a DET iajs-2475 25 2 submodule submodule PROPN iajs-2475 25 3 n n PROPN iajs-2475 25 4 of of ADP iajs-2475 25 5 m m PROPN iajs-2475 25 6 is be AUX iajs-2475 25 7 weakly weakly ADJ iajs-2475 25 8 supplement supplement NOUN iajs-2475 25 9 , , PUNCT iajs-2475 25 10 if if SCONJ iajs-2475 25 11 there there PRON iajs-2475 25 12 exists exist VERB iajs-2475 25 13 a a DET iajs-2475 25 14 submodule submodule NOUN iajs-2475 25 15 w w NOUN iajs-2475 25 16 in in ADP iajs-2475 25 17 m m PRON iajs-2475 25 18 such such ADJ iajs-2475 25 19 that that SCONJ iajs-2475 25 20 m m NOUN iajs-2475 25 21 = = NOUN iajs-2475 25 22 w+n w+n NOUN iajs-2475 25 23 and and CCONJ iajs-2475 25 24 n∩ n∩ PROPN iajs-2475 25 25 wβ‰ͺm wβ‰ͺm PROPN iajs-2475 25 26 . . PUNCT iajs-2475 26 1 a a DET iajs-2475 26 2 module module NOUN iajs-2475 26 3 m m VERB iajs-2475 26 4 is be AUX iajs-2475 26 5 called call VERB iajs-2475 26 6 weakly weakly ADV iajs-2475 26 7 supplemented supplement VERB iajs-2475 26 8 if if SCONJ iajs-2475 26 9 each each DET iajs-2475 26 10 submodule submodule NOUN iajs-2475 26 11 w w PROPN iajs-2475 26 12 in in ADP iajs-2475 26 13 m m PROPN iajs-2475 26 14 is be AUX iajs-2475 26 15 weakly weakly ADJ iajs-2475 26 16 supplement supplement NOUN iajs-2475 26 17 of of ADP iajs-2475 26 18 m m PROPN iajs-2475 26 19 [ [ X iajs-2475 26 20 2 2 NUM iajs-2475 26 21 ] ] PUNCT iajs-2475 26 22 . . PUNCT iajs-2475 27 1 a a DET iajs-2475 27 2 submodule submodule NOUN iajs-2475 27 3 v v NOUN iajs-2475 27 4 of of ADP iajs-2475 27 5 m m PROPN iajs-2475 27 6 is be AUX iajs-2475 27 7 supplement supplement ADJ iajs-2475 27 8 , , PUNCT iajs-2475 27 9 if if SCONJ iajs-2475 27 10 there there PRON iajs-2475 27 11 is be VERB iajs-2475 27 12 a a DET iajs-2475 27 13 submodule submodule NOUN iajs-2475 27 14 w w NOUN iajs-2475 27 15 of of ADP iajs-2475 27 16 m m PRON iajs-2475 27 17 such such ADJ iajs-2475 27 18 that that SCONJ iajs-2475 27 19 m m NOUN iajs-2475 27 20 = = NOUN iajs-2475 27 21 v+w v+w NUM iajs-2475 27 22 and and CCONJ iajs-2475 27 23 v∩ v∩ PROPN iajs-2475 27 24 wβ‰ͺv wβ‰ͺv PUNCT iajs-2475 28 1 [ [ X iajs-2475 28 2 2 2 NUM iajs-2475 28 3 ] ] PUNCT iajs-2475 28 4 . . PUNCT iajs-2475 29 1 a a DET iajs-2475 29 2 module module NOUN iajs-2475 29 3 m m VERB iajs-2475 29 4 is be AUX iajs-2475 29 5 called call VERB iajs-2475 29 6 supplemented supplement VERB iajs-2475 29 7 if if SCONJ iajs-2475 29 8 each each DET iajs-2475 29 9 submodule submodule NOUN iajs-2475 29 10 v v NOUN iajs-2475 29 11 of of ADP iajs-2475 29 12 m m PROPN iajs-2475 29 13 has have VERB iajs-2475 29 14 a a DET iajs-2475 29 15 supplement supplement NOUN iajs-2475 29 16 submodule submodule NOUN iajs-2475 29 17 in in ADP iajs-2475 29 18 m m PROPN iajs-2475 29 19 [ [ X iajs-2475 29 20 2 2 NUM iajs-2475 29 21 ] ] PUNCT iajs-2475 29 22 . . PUNCT iajs-2475 30 1 a a DET iajs-2475 30 2 module module NOUN iajs-2475 30 3 m m NOUN iajs-2475 30 4 is be AUX iajs-2475 30 5 said say VERB iajs-2475 30 6 to to PART iajs-2475 30 7 be be AUX iajs-2475 30 8 uniform uniform ADJ iajs-2475 30 9 if if SCONJ iajs-2475 30 10 each each DET iajs-2475 30 11 nonzero nonzero PROPN iajs-2475 30 12 submodule submodule PROPN iajs-2475 30 13 in in ADP iajs-2475 30 14 m m PROPN iajs-2475 30 15 is be AUX iajs-2475 30 16 essential essential ADJ iajs-2475 30 17 submodule submodule NOUN iajs-2475 30 18 in in ADP iajs-2475 30 19 m m PROPN iajs-2475 30 20 [ [ X iajs-2475 30 21 3 3 NUM iajs-2475 30 22 ] ] PUNCT iajs-2475 30 23 . . PUNCT iajs-2475 31 1 a a DET iajs-2475 31 2 module module NOUN iajs-2475 31 3 m m VERB iajs-2475 31 4 is be AUX iajs-2475 31 5 called call VERB iajs-2475 31 6 non non ADJ iajs-2475 31 7 - - ADJ iajs-2475 31 8 singular singular ADJ iajs-2475 31 9 if if SCONJ iajs-2475 31 10 z(m)=0 z(m)=0 NUM iajs-2475 31 11 where where SCONJ iajs-2475 31 12 z(m)={m∈m z(m)={m∈m PROPN iajs-2475 32 1 | | ADV iajs-2475 32 2 xm=0 xm=0 PROPN iajs-2475 32 3 for for ADP iajs-2475 32 4 some some DET iajs-2475 32 5 essential essential ADJ iajs-2475 32 6 left leave VERB iajs-2475 32 7 ideal ideal NOUN iajs-2475 32 8 x x PUNCT iajs-2475 32 9 of of ADP iajs-2475 32 10 r r NOUN iajs-2475 32 11 } } PUNCT iajs-2475 32 12 , , PUNCT iajs-2475 32 13 and and CCONJ iajs-2475 32 14 m m PROPN iajs-2475 32 15 is be AUX iajs-2475 32 16 singular singular ADJ iajs-2475 32 17 if if SCONJ iajs-2475 32 18 z(m)=m z(m)=m PROPN iajs-2475 32 19 [ [ X iajs-2475 32 20 3 3 NUM iajs-2475 32 21 ] ] PUNCT iajs-2475 32 22 . . PUNCT iajs-2475 33 1 a a DET iajs-2475 33 2 module module NOUN iajs-2475 33 3 𝑀 𝑀 PROPN iajs-2475 33 4 is be AUX iajs-2475 33 5 called call VERB iajs-2475 33 6 lifting lift VERB iajs-2475 33 7 , , PUNCT iajs-2475 33 8 if if SCONJ iajs-2475 33 9 each each DET iajs-2475 33 10 submodule submodule NOUN iajs-2475 33 11 v v NOUN iajs-2475 33 12 in in ADP iajs-2475 33 13 m m NOUN iajs-2475 33 14 there there PRON iajs-2475 33 15 is be VERB iajs-2475 33 16 a a DET iajs-2475 33 17 direct direct ADJ iajs-2475 33 18 summand summand NOUN iajs-2475 33 19 w w NOUN iajs-2475 33 20 in in ADP iajs-2475 33 21 m m NOUN iajs-2475 33 22 with with ADP iajs-2475 33 23 wβŠ† wβŠ† NOUN iajs-2475 33 24 v v ADP iajs-2475 33 25 such such ADJ iajs-2475 33 26 that that SCONJ iajs-2475 33 27 m m PROPN iajs-2475 33 28 = = NOUN iajs-2475 33 29 wβŠ•w wβŠ•w NOUN iajs-2475 33 30 ' ' PUNCT iajs-2475 33 31 and and CCONJ iajs-2475 33 32 wβ€² wβ€² PROPN iajs-2475 33 33 ∩ ∩ PROPN iajs-2475 33 34 𝑉 𝑉 PROPN iajs-2475 33 35 β‰ͺw β‰ͺw NOUN iajs-2475 33 36 ' ' PART iajs-2475 33 37 [ [ X iajs-2475 33 38 2 2 NUM iajs-2475 33 39 ] ] PUNCT iajs-2475 33 40 . . PUNCT iajs-2475 34 1 if if SCONJ iajs-2475 34 2 f(n)βŠ†(n f(n)βŠ†(n PROPN iajs-2475 34 3 ) ) PUNCT iajs-2475 34 4 for for ADP iajs-2475 34 5 each each DET iajs-2475 34 6 r r NOUN iajs-2475 34 7 - - PUNCT iajs-2475 34 8 endomorphism endomorphism PROPN iajs-2475 34 9 f f PROPN iajs-2475 34 10 of of ADP iajs-2475 34 11 m m PROPN iajs-2475 34 12 , , PUNCT iajs-2475 34 13 then then ADV iajs-2475 34 14 a a DET iajs-2475 34 15 submodule submodule NOUN iajs-2475 34 16 n n PROPN iajs-2475 34 17 of of ADP iajs-2475 34 18 a a DET iajs-2475 34 19 module module NOUN iajs-2475 34 20 m m NOUN iajs-2475 34 21 is be AUX iajs-2475 34 22 said say VERB iajs-2475 34 23 to to PART iajs-2475 34 24 be be AUX iajs-2475 34 25 fully fully ADV iajs-2475 34 26 invariant invariant ADJ iajs-2475 34 27 [ [ X iajs-2475 34 28 4 4 NUM iajs-2475 34 29 ] ] PUNCT iajs-2475 34 30 . . PUNCT iajs-2475 35 1 a a DET iajs-2475 35 2 module module NOUN iajs-2475 35 3 m m NOUN iajs-2475 35 4 is be AUX iajs-2475 35 5 semi semi ADJ iajs-2475 35 6 - - ADJ iajs-2475 35 7 simple simple ADJ iajs-2475 35 8 , , PUNCT iajs-2475 35 9 if if SCONJ iajs-2475 35 10 every every DET iajs-2475 35 11 submodule submodule NOUN iajs-2475 35 12 is be AUX iajs-2475 35 13 direct direct ADJ iajs-2475 35 14 summand summand NOUN iajs-2475 36 1 [ [ X iajs-2475 36 2 3 3 NUM iajs-2475 36 3 ] ] PUNCT iajs-2475 36 4 . . PUNCT iajs-2475 37 1 a a DET iajs-2475 37 2 module module NOUN iajs-2475 37 3 m m VERB iajs-2475 37 4 is be AUX iajs-2475 37 5 called call VERB iajs-2475 37 6 extending extend VERB iajs-2475 37 7 if if SCONJ iajs-2475 37 8 each each DET iajs-2475 37 9 submodule submodule NOUN iajs-2475 37 10 in in ADP iajs-2475 37 11 m m PROPN iajs-2475 37 12 is be AUX iajs-2475 37 13 essential essential ADJ iajs-2475 37 14 in in ADP iajs-2475 37 15 direct direct ADJ iajs-2475 37 16 summand summand NOUN iajs-2475 37 17 of of ADP iajs-2475 37 18 m m PROPN iajs-2475 37 19 [ [ X iajs-2475 37 20 3 3 NUM iajs-2475 37 21 ] ] PUNCT iajs-2475 37 22 . . PUNCT iajs-2475 38 1 the the DET iajs-2475 38 2 extending extend VERB iajs-2475 38 3 property property NOUN iajs-2475 38 4 and and CCONJ iajs-2475 38 5 their their PRON iajs-2475 38 6 generalization generalization NOUN iajs-2475 38 7 are be AUX iajs-2475 38 8 studied study VERB iajs-2475 38 9 by by ADP iajs-2475 38 10 different different ADJ iajs-2475 38 11 authors author NOUN iajs-2475 38 12 such such ADJ iajs-2475 38 13 as as ADP iajs-2475 38 14 [ [ X iajs-2475 38 15 1 1 NUM iajs-2475 38 16 , , PUNCT iajs-2475 38 17 5,6 5,6 NUM iajs-2475 38 18 ] ] PUNCT iajs-2475 38 19 . . PUNCT iajs-2475 39 1 2 2 X iajs-2475 39 2 . . X iajs-2475 39 3 βŠ•-s βŠ•-s NOUN iajs-2475 39 4 - - PUNCT iajs-2475 39 5 extending extend VERB iajs-2475 39 6 modules module NOUN iajs-2475 39 7 in in ADP iajs-2475 39 8 this this DET iajs-2475 39 9 work work NOUN iajs-2475 39 10 , , PUNCT iajs-2475 39 11 we we PRON iajs-2475 39 12 will will AUX iajs-2475 39 13 present present VERB iajs-2475 39 14 the the DET iajs-2475 39 15 following follow VERB iajs-2475 39 16 concept concept NOUN iajs-2475 39 17 that that PRON iajs-2475 39 18 is be AUX iajs-2475 39 19 stronger strong ADJ iajs-2475 39 20 than than ADP iajs-2475 39 21 of of ADP iajs-2475 39 22 weakly weakly ADJ iajs-2475 39 23 supplement supplement NOUN iajs-2475 39 24 extending extend VERB iajs-2475 39 25 modules module NOUN iajs-2475 39 26 : : PUNCT iajs-2475 39 27 definition definition NOUN iajs-2475 39 28 ( ( PUNCT iajs-2475 39 29 1 1 X iajs-2475 39 30 ) ) PUNCT iajs-2475 39 31 a a DET iajs-2475 39 32 module module NOUN iajs-2475 39 33 m m VERB iajs-2475 39 34 is be AUX iajs-2475 39 35 called call VERB iajs-2475 39 36 βŠ•-s βŠ•-s NOUN iajs-2475 39 37 - - PUNCT iajs-2475 39 38 extending extend VERB iajs-2475 39 39 if if SCONJ iajs-2475 39 40 each each DET iajs-2475 39 41 submodule submodule NOUN iajs-2475 39 42 in in ADP iajs-2475 39 43 m m PROPN iajs-2475 39 44 is be AUX iajs-2475 39 45 essential essential ADJ iajs-2475 39 46 in in ADP iajs-2475 39 47 submodule submodule NOUN iajs-2475 39 48 has have VERB iajs-2475 39 49 a a DET iajs-2475 39 50 supplement supplement NOUN iajs-2475 39 51 which which PRON iajs-2475 39 52 is be AUX iajs-2475 39 53 direct direct ADJ iajs-2475 39 54 summand summand NOUN iajs-2475 39 55 . . PUNCT iajs-2475 40 1 proposition proposition NOUN iajs-2475 40 2 ( ( PUNCT iajs-2475 40 3 2 2 X iajs-2475 40 4 ) ) PUNCT iajs-2475 40 5 a a DET iajs-2475 40 6 module module NOUN iajs-2475 40 7 m m VERB iajs-2475 40 8 is be AUX iajs-2475 40 9 called call VERB iajs-2475 40 10 βŠ•-s βŠ•-s NOUN iajs-2475 40 11 - - PUNCT iajs-2475 40 12 extending extend VERB iajs-2475 40 13 if if SCONJ iajs-2475 41 1 and and CCONJ iajs-2475 41 2 only only ADV iajs-2475 41 3 if if SCONJ iajs-2475 41 4 each each DET iajs-2475 41 5 submodule submodule NOUN iajs-2475 41 6 in in ADP iajs-2475 41 7 m m PROPN iajs-2475 41 8 is be AUX iajs-2475 41 9 essential essential ADJ iajs-2475 41 10 in in ADP iajs-2475 41 11 submodule submodule NOUN iajs-2475 41 12 has have VERB iajs-2475 41 13 a a DET iajs-2475 41 14 weakly weakly ADJ iajs-2475 41 15 supplement supplement NOUN iajs-2475 41 16 which which PRON iajs-2475 41 17 is be AUX iajs-2475 41 18 direct direct ADJ iajs-2475 41 19 summand summand NOUN iajs-2475 41 20 . . PUNCT iajs-2475 42 1 proof proof NOUN iajs-2475 42 2 let let VERB iajs-2475 42 3 n n PRON iajs-2475 42 4 be be AUX iajs-2475 42 5 a a DET iajs-2475 42 6 submodule submodule NOUN iajs-2475 42 7 of of ADP iajs-2475 42 8 βŠ•-s βŠ•-s ADJ iajs-2475 42 9 - - PUNCT iajs-2475 42 10 extending extend VERB iajs-2475 42 11 module module NOUN iajs-2475 42 12 , , PUNCT iajs-2475 42 13 then then ADV iajs-2475 42 14 n n PROPN iajs-2475 42 15 is be AUX iajs-2475 42 16 essential essential ADJ iajs-2475 42 17 in in ADP iajs-2475 42 18 submodule submodule PROPN iajs-2475 42 19 k k PROPN iajs-2475 42 20 in in ADP iajs-2475 42 21 m m PROPN iajs-2475 42 22 has have VERB iajs-2475 42 23 a a DET iajs-2475 42 24 supplement supplement NOUN iajs-2475 42 25 which which PRON iajs-2475 42 26 is be AUX iajs-2475 42 27 direct direct ADJ iajs-2475 42 28 summand summand NOUN iajs-2475 42 29 . . PUNCT iajs-2475 43 1 since since SCONJ iajs-2475 43 2 every every DET iajs-2475 43 3 supplement supplement NOUN iajs-2475 43 4 submodule submodule NOUN iajs-2475 43 5 is be AUX iajs-2475 43 6 weakly weakly ADJ iajs-2475 43 7 supplement supplement NOUN iajs-2475 43 8 , , PUNCT iajs-2475 43 9 then then ADV iajs-2475 43 10 we we PRON iajs-2475 43 11 have have VERB iajs-2475 43 12 k k PROPN iajs-2475 43 13 has have VERB iajs-2475 43 14 a a DET iajs-2475 43 15 weakly weakly ADJ iajs-2475 43 16 supplement supplement NOUN iajs-2475 43 17 which which PRON iajs-2475 43 18 is be AUX iajs-2475 43 19 direct direct ADJ iajs-2475 43 20 summand summand NOUN iajs-2475 43 21 . . PUNCT iajs-2475 44 1 conversely conversely ADV iajs-2475 44 2 , , PUNCT iajs-2475 44 3 let let VERB iajs-2475 44 4 n n PRON iajs-2475 44 5 be be AUX iajs-2475 44 6 a a DET iajs-2475 44 7 submodule submodule NOUN iajs-2475 44 8 of of ADP iajs-2475 44 9 a a DET iajs-2475 44 10 module module NOUN iajs-2475 44 11 m. m. NOUN iajs-2475 44 12 by by ADP iajs-2475 44 13 hypothesis hypothesis NOUN iajs-2475 44 14 , , PUNCT iajs-2475 44 15 n n PRON iajs-2475 44 16 is be AUX iajs-2475 44 17 essential essential ADJ iajs-2475 44 18 in in ADP iajs-2475 44 19 submodule submodule PROPN iajs-2475 44 20 k k PROPN iajs-2475 44 21 has have VERB iajs-2475 44 22 a a DET iajs-2475 44 23 weakly weakly ADJ iajs-2475 44 24 supplement supplement NOUN iajs-2475 44 25 l l NOUN iajs-2475 44 26 which which PRON iajs-2475 44 27 is be AUX iajs-2475 44 28 direct direct ADJ iajs-2475 44 29 summand summand NOUN iajs-2475 44 30 ( ( PUNCT iajs-2475 44 31 i.e i.e PROPN iajs-2475 44 32 ) ) PUNCT iajs-2475 44 33 m m PROPN iajs-2475 44 34 = = PROPN iajs-2475 44 35 k+l k+l X iajs-2475 44 36 and and CCONJ iajs-2475 44 37 k∩lβ‰ͺm k∩lβ‰ͺm PROPN iajs-2475 44 38 where where SCONJ iajs-2475 44 39 l l NOUN iajs-2475 44 40 is be AUX iajs-2475 44 41 direct direct ADJ iajs-2475 44 42 summand summand NOUN iajs-2475 44 43 in in ADP iajs-2475 44 44 m. m. NOUN iajs-2475 44 45 since since SCONJ iajs-2475 44 46 k∩lβŠ†lβŠ†m k∩lβŠ†lβŠ†m ADJ iajs-2475 44 47 and and CCONJ iajs-2475 44 48 l l NOUN iajs-2475 44 49 is be AUX iajs-2475 44 50 direct direct ADJ iajs-2475 44 51 summand summand NOUN iajs-2475 44 52 in in ADP iajs-2475 44 53 m m PROPN iajs-2475 44 54 , , PUNCT iajs-2475 44 55 thus thus ADV iajs-2475 44 56 we we PRON iajs-2475 44 57 have have VERB iajs-2475 44 58 k∩lβ‰ͺl k∩lβ‰ͺl PROPN iajs-2475 44 59 . . PUNCT iajs-2475 45 1 then then ADV iajs-2475 45 2 n n PROPN iajs-2475 45 3 is be AUX iajs-2475 45 4 essential essential ADJ iajs-2475 45 5 in in ADP iajs-2475 45 6 submodule submodule PROPN iajs-2475 45 7 k k PROPN iajs-2475 45 8 has have VERB iajs-2475 45 9 a a DET iajs-2475 45 10 supplement supplement NOUN iajs-2475 45 11 l l NOUN iajs-2475 45 12 which which PRON iajs-2475 45 13 is be AUX iajs-2475 45 14 direct direct ADJ iajs-2475 45 15 summand summand NOUN iajs-2475 45 16 . . PUNCT iajs-2475 46 1 hence hence ADV iajs-2475 46 2 m m PROPN iajs-2475 46 3 is be AUX iajs-2475 46 4 βŠ•-s βŠ•-s ADJ iajs-2475 46 5 - - PUNCT iajs-2475 46 6 extending extending NOUN iajs-2475 46 7 . . PUNCT iajs-2475 47 1 proposition proposition NOUN iajs-2475 47 2 ( ( PUNCT iajs-2475 47 3 3 3 X iajs-2475 47 4 ) ) PUNCT iajs-2475 47 5 a a DET iajs-2475 47 6 module module NOUN iajs-2475 47 7 m m VERB iajs-2475 47 8 is be AUX iajs-2475 47 9 βŠ•-s βŠ•-s ADV iajs-2475 47 10 - - PUNCT iajs-2475 47 11 extending extend VERB iajs-2475 47 12 if if SCONJ iajs-2475 48 1 and and CCONJ iajs-2475 48 2 only only ADV iajs-2475 48 3 if if SCONJ iajs-2475 48 4 each each DET iajs-2475 48 5 closed closed ADJ iajs-2475 48 6 submodule submodule NOUN iajs-2475 48 7 in in ADP iajs-2475 48 8 m m PROPN iajs-2475 48 9 has have VERB iajs-2475 48 10 a a DET iajs-2475 48 11 ( ( PUNCT iajs-2475 48 12 weakly weakly ADJ iajs-2475 48 13 ) ) PUNCT iajs-2475 48 14 supplement supplement NOUN iajs-2475 48 15 which which PRON iajs-2475 48 16 is be AUX iajs-2475 48 17 direct direct ADJ iajs-2475 48 18 summand summand NOUN iajs-2475 48 19 . . PUNCT iajs-2475 49 1 82 82 NUM iajs-2475 50 1 ibn ibn PROPN iajs-2475 50 2 al al PROPN iajs-2475 50 3 - - PUNCT iajs-2475 50 4 haitham haitham PROPN iajs-2475 50 5 jour jour X iajs-2475 50 6 . . PROPN iajs-2475 50 7 for for ADP iajs-2475 50 8 pure pure ADJ iajs-2475 50 9 & & CCONJ iajs-2475 50 10 appl appl PROPN iajs-2475 50 11 . . PUNCT iajs-2475 51 1 sci sci PROPN iajs-2475 51 2 . . PROPN iajs-2475 52 1 33 33 NUM iajs-2475 52 2 ( ( PUNCT iajs-2475 52 3 3 3 NUM iajs-2475 52 4 ) ) PUNCT iajs-2475 52 5 2020 2020 NUM iajs-2475 53 1 proof proof NOUN iajs-2475 53 2 ( ( PUNCT iajs-2475 53 3 ⟹ ⟹ X iajs-2475 53 4 ) ) PUNCT iajs-2475 53 5 let let VERB iajs-2475 53 6 b b X iajs-2475 53 7 be be AUX iajs-2475 53 8 a a DET iajs-2475 53 9 closed closed ADJ iajs-2475 53 10 submodule submodule NOUN iajs-2475 53 11 in in ADP iajs-2475 53 12 m. m. NOUN iajs-2475 53 13 since since SCONJ iajs-2475 53 14 m m PROPN iajs-2475 53 15 is be AUX iajs-2475 53 16 βŠ•-s βŠ•-s ADJ iajs-2475 53 17 - - PUNCT iajs-2475 53 18 extending extending NOUN iajs-2475 53 19 , , PUNCT iajs-2475 53 20 so so CCONJ iajs-2475 53 21 b b PROPN iajs-2475 53 22 is be AUX iajs-2475 53 23 essential essential ADJ iajs-2475 53 24 in in ADP iajs-2475 53 25 submodule submodule PROPN iajs-2475 53 26 k k PROPN iajs-2475 53 27 has have VERB iajs-2475 53 28 a a DET iajs-2475 53 29 supplement supplement NOUN iajs-2475 53 30 which which PRON iajs-2475 53 31 is be AUX iajs-2475 53 32 direct direct ADJ iajs-2475 53 33 summand summand NOUN iajs-2475 53 34 . . PUNCT iajs-2475 54 1 but but CCONJ iajs-2475 54 2 b b NOUN iajs-2475 54 3 is be AUX iajs-2475 54 4 closed closed ADJ iajs-2475 54 5 , , PUNCT iajs-2475 54 6 so so ADV iajs-2475 54 7 we we PRON iajs-2475 54 8 have have VERB iajs-2475 54 9 b b NOUN iajs-2475 54 10 has have VERB iajs-2475 54 11 a a DET iajs-2475 54 12 supplement supplement NOUN iajs-2475 54 13 which which PRON iajs-2475 54 14 is be AUX iajs-2475 54 15 direct direct ADJ iajs-2475 54 16 summand summand NOUN iajs-2475 54 17 . . PUNCT iajs-2475 55 1 ( ( PUNCT iajs-2475 55 2 ⟸ ⟸ ADV iajs-2475 55 3 ) ) PUNCT iajs-2475 55 4 let let VERB iajs-2475 55 5 b b NOUN iajs-2475 55 6 be be AUX iajs-2475 55 7 submodule submodule NOUN iajs-2475 55 8 in in ADP iajs-2475 55 9 m. m. NOUN iajs-2475 55 10 then then ADV iajs-2475 55 11 by by ADP iajs-2475 55 12 using use VERB iajs-2475 55 13 zorn zorn PROPN iajs-2475 55 14 's 's PART iajs-2475 55 15 lemma lemma PROPN iajs-2475 55 16 , , PUNCT iajs-2475 55 17 so so SCONJ iajs-2475 55 18 there there PRON iajs-2475 55 19 is be VERB iajs-2475 55 20 a a DET iajs-2475 55 21 closed closed ADJ iajs-2475 55 22 submodule submodule NOUN iajs-2475 55 23 d d NOUN iajs-2475 55 24 in in ADP iajs-2475 55 25 m m PRON iajs-2475 55 26 such such ADJ iajs-2475 55 27 that that SCONJ iajs-2475 55 28 b b NOUN iajs-2475 55 29 is be AUX iajs-2475 55 30 essential essential ADJ iajs-2475 55 31 in in ADP iajs-2475 55 32 d. d. PROPN iajs-2475 55 33 by by ADP iajs-2475 55 34 hypothesis hypothesis NOUN iajs-2475 55 35 , , PUNCT iajs-2475 55 36 d d PROPN iajs-2475 55 37 has have VERB iajs-2475 55 38 a a DET iajs-2475 55 39 supplement supplement NOUN iajs-2475 55 40 which which PRON iajs-2475 55 41 is be AUX iajs-2475 55 42 direct direct ADJ iajs-2475 55 43 summand summand NOUN iajs-2475 55 44 . . PUNCT iajs-2475 56 1 following follow VERB iajs-2475 56 2 [ [ X iajs-2475 56 3 1 1 NUM iajs-2475 56 4 ] ] PUNCT iajs-2475 56 5 . . PUNCT iajs-2475 57 1 a a DET iajs-2475 57 2 module module NOUN iajs-2475 57 3 m m VERB iajs-2475 57 4 is be AUX iajs-2475 57 5 closed close VERB iajs-2475 57 6 ⨁-supplemented ⨁-supplemente VERB iajs-2475 57 7 if if SCONJ iajs-2475 57 8 each each DET iajs-2475 57 9 closed closed ADJ iajs-2475 57 10 submodule submodule NOUN iajs-2475 57 11 in in ADP iajs-2475 57 12 m m PROPN iajs-2475 57 13 has have VERB iajs-2475 57 14 a a DET iajs-2475 57 15 weakly weakly ADJ iajs-2475 57 16 supplement supplement NOUN iajs-2475 57 17 which which PRON iajs-2475 57 18 is be AUX iajs-2475 57 19 direct direct ADJ iajs-2475 57 20 summand summand NOUN iajs-2475 57 21 . . PUNCT iajs-2475 58 1 the the DET iajs-2475 58 2 next next ADJ iajs-2475 58 3 result result NOUN iajs-2475 58 4 is be AUX iajs-2475 58 5 directly directly ADV iajs-2475 58 6 by by ADP iajs-2475 58 7 proposition proposition NOUN iajs-2475 58 8 ( ( PUNCT iajs-2475 58 9 2 2 NUM iajs-2475 58 10 ) ) PUNCT iajs-2475 58 11 . . PUNCT iajs-2475 59 1 corollary corollary ADJ iajs-2475 59 2 ( ( PUNCT iajs-2475 59 3 4 4 NUM iajs-2475 59 4 ) ) PUNCT iajs-2475 59 5 a a DET iajs-2475 59 6 module module NOUN iajs-2475 59 7 m m VERB iajs-2475 59 8 is be AUX iajs-2475 59 9 βŠ•-s βŠ•-s ADV iajs-2475 59 10 - - PUNCT iajs-2475 59 11 extending extend VERB iajs-2475 59 12 if if SCONJ iajs-2475 60 1 and and CCONJ iajs-2475 60 2 only only ADV iajs-2475 60 3 if if SCONJ iajs-2475 60 4 m m NOUN iajs-2475 60 5 is be AUX iajs-2475 60 6 closed close VERB iajs-2475 60 7 ⨁-supplemented ⨁-supplemente VERB iajs-2475 60 8 . . PUNCT iajs-2475 61 1 remarks remark NOUN iajs-2475 61 2 and and CCONJ iajs-2475 61 3 examples example NOUN iajs-2475 61 4 ( ( PUNCT iajs-2475 61 5 5 5 NUM iajs-2475 61 6 ) ) PUNCT iajs-2475 61 7 1 1 NUM iajs-2475 61 8 . . PUNCT iajs-2475 62 1 each each DET iajs-2475 62 2 extending extend VERB iajs-2475 62 3 module module NOUN iajs-2475 62 4 is be AUX iajs-2475 62 5 βŠ•-s βŠ•-s ADJ iajs-2475 62 6 - - PUNCT iajs-2475 62 7 extending extending NOUN iajs-2475 62 8 , , PUNCT iajs-2475 62 9 while while SCONJ iajs-2475 62 10 it it PRON iajs-2475 62 11 is be AUX iajs-2475 62 12 not not PART iajs-2475 62 13 conversely conversely ADV iajs-2475 62 14 . . PUNCT iajs-2475 63 1 for for ADP iajs-2475 63 2 example example NOUN iajs-2475 63 3 , , PUNCT iajs-2475 63 4 m=𝑍8⨁𝑍2 m=𝑍8⨁𝑍2 ADJ iajs-2475 63 5 as as ADP iajs-2475 63 6 z z NOUN iajs-2475 63 7 - - PUNCT iajs-2475 63 8 module module NOUN iajs-2475 63 9 is be AUX iajs-2475 63 10 βŠ•-s βŠ•-s ADV iajs-2475 63 11 - - PUNCT iajs-2475 63 12 extending extending NOUN iajs-2475 63 13 which which PRON iajs-2475 63 14 it it PRON iajs-2475 63 15 is be AUX iajs-2475 63 16 not not PART iajs-2475 63 17 extending extend VERB iajs-2475 63 18 [ [ X iajs-2475 63 19 1 1 NUM iajs-2475 63 20 ] ] PUNCT iajs-2475 63 21 . . PUNCT iajs-2475 64 1 2 2 X iajs-2475 64 2 . . X iajs-2475 64 3 every every DET iajs-2475 64 4 βŠ•-s βŠ•-s ADJ iajs-2475 64 5 - - PUNCT iajs-2475 64 6 extending extend VERB iajs-2475 64 7 module module NOUN iajs-2475 64 8 is be AUX iajs-2475 64 9 weakly weakly ADJ iajs-2475 64 10 supplement supplement NOUN iajs-2475 64 11 extending extending NOUN iajs-2475 64 12 , , PUNCT iajs-2475 64 13 while while SCONJ iajs-2475 64 14 the the DET iajs-2475 64 15 other other ADJ iajs-2475 64 16 direction direction NOUN iajs-2475 64 17 is be AUX iajs-2475 64 18 not not PART iajs-2475 64 19 true true ADJ iajs-2475 64 20 in in ADP iajs-2475 64 21 general general ADJ iajs-2475 64 22 . . PUNCT iajs-2475 65 1 3 3 X iajs-2475 65 2 . . X iajs-2475 65 3 every every DET iajs-2475 65 4 ⨁-supplemented ⨁-supplemente VERB iajs-2475 65 5 module module NOUN iajs-2475 65 6 is be AUX iajs-2475 65 7 βŠ•-s βŠ•-s NOUN iajs-2475 65 8 - - PUNCT iajs-2475 65 9 extending extending NOUN iajs-2475 65 10 , , PUNCT iajs-2475 65 11 while while SCONJ iajs-2475 65 12 it it PRON iajs-2475 65 13 is be AUX iajs-2475 65 14 not not PART iajs-2475 65 15 conversely conversely ADV iajs-2475 65 16 . . PUNCT iajs-2475 66 1 in in ADP iajs-2475 66 2 fact fact NOUN iajs-2475 66 3 , , PUNCT iajs-2475 66 4 z z PROPN iajs-2475 66 5 is be AUX iajs-2475 66 6 βŠ•-s βŠ•-s ADV iajs-2475 66 7 - - PUNCT iajs-2475 66 8 extending extend VERB iajs-2475 66 9 z z NOUN iajs-2475 66 10 - - PUNCT iajs-2475 66 11 module module NOUN iajs-2475 66 12 which which PRON iajs-2475 66 13 it it PRON iajs-2475 66 14 is be AUX iajs-2475 66 15 not not PART iajs-2475 66 16 ⨁-supplemented ⨁-supplemente VERB iajs-2475 66 17 ( ( PUNCT iajs-2475 66 18 because because SCONJ iajs-2475 66 19 a a DET iajs-2475 66 20 submodule submodule NOUN iajs-2475 66 21 2z 2z NUM iajs-2475 66 22 in in ADP iajs-2475 66 23 z z NOUN iajs-2475 66 24 has have VERB iajs-2475 66 25 no no DET iajs-2475 66 26 a a DET iajs-2475 66 27 supplement supplement NOUN iajs-2475 66 28 submodule submodule NOUN iajs-2475 66 29 in in ADP iajs-2475 66 30 z z PROPN iajs-2475 66 31 ) ) PUNCT iajs-2475 66 32 . . PUNCT iajs-2475 67 1 4 4 X iajs-2475 67 2 . . X iajs-2475 68 1 every every DET iajs-2475 68 2 uniform uniform ADJ iajs-2475 68 3 module module NOUN iajs-2475 68 4 is be AUX iajs-2475 68 5 βŠ•-s βŠ•-s ADJ iajs-2475 68 6 - - PUNCT iajs-2475 68 7 extending extending NOUN iajs-2475 68 8 , , PUNCT iajs-2475 68 9 while while SCONJ iajs-2475 68 10 the the DET iajs-2475 68 11 converse converse NOUN iajs-2475 68 12 is be AUX iajs-2475 68 13 not not PART iajs-2475 68 14 true true ADJ iajs-2475 68 15 . . PUNCT iajs-2475 69 1 for for ADP iajs-2475 69 2 example example NOUN iajs-2475 69 3 , , PUNCT iajs-2475 69 4 𝑍10 𝑍10 PROPN iajs-2475 69 5 is be AUX iajs-2475 69 6 βŠ•-s βŠ•-s ADV iajs-2475 69 7 - - PUNCT iajs-2475 69 8 extending extend VERB iajs-2475 69 9 z z NOUN iajs-2475 69 10 - - PUNCT iajs-2475 69 11 module module NOUN iajs-2475 69 12 which which PRON iajs-2475 69 13 it it PRON iajs-2475 69 14 is be AUX iajs-2475 69 15 not not PART iajs-2475 69 16 uniform uniform ADJ iajs-2475 69 17 . . PUNCT iajs-2475 70 1 5 5 X iajs-2475 70 2 . . X iajs-2475 70 3 in in ADP iajs-2475 70 4 [ [ X iajs-2475 70 5 1 1 NUM iajs-2475 70 6 ] ] PUNCT iajs-2475 70 7 . . PUNCT iajs-2475 71 1 every every DET iajs-2475 71 2 lifting lifting NOUN iajs-2475 71 3 module module NOUN iajs-2475 71 4 is be AUX iajs-2475 71 5 weakly weakly ADJ iajs-2475 71 6 supplement supplement ADJ iajs-2475 71 7 extending extending NOUN iajs-2475 71 8 . . PUNCT iajs-2475 72 1 this this DET iajs-2475 72 2 result result NOUN iajs-2475 72 3 can can AUX iajs-2475 72 4 be be AUX iajs-2475 72 5 generalized generalize VERB iajs-2475 72 6 to to ADP iajs-2475 72 7 βŠ•-s βŠ•-s NOUN iajs-2475 72 8 - - PUNCT iajs-2475 72 9 extending extend VERB iajs-2475 72 10 ( ( PUNCT iajs-2475 72 11 i.e i.e NOUN iajs-2475 72 12 ) ) PUNCT iajs-2475 72 13 , , PUNCT iajs-2475 72 14 every every DET iajs-2475 72 15 lifting lifting NOUN iajs-2475 72 16 module module NOUN iajs-2475 72 17 is be AUX iajs-2475 72 18 βŠ•-s βŠ•-s NOUN iajs-2475 72 19 - - PUNCT iajs-2475 72 20 extending extend VERB iajs-2475 72 21 ( ( PUNCT iajs-2475 72 22 since since SCONJ iajs-2475 72 23 every every DET iajs-2475 72 24 lifting lifting NOUN iajs-2475 72 25 module module NOUN iajs-2475 72 26 is be AUX iajs-2475 72 27 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 72 28 ) ) PUNCT iajs-2475 72 29 , , PUNCT iajs-2475 72 30 while while SCONJ iajs-2475 72 31 the the DET iajs-2475 72 32 converse converse NOUN iajs-2475 72 33 is be AUX iajs-2475 72 34 not not PART iajs-2475 72 35 true true ADJ iajs-2475 72 36 . . PUNCT iajs-2475 73 1 in in ADP iajs-2475 73 2 fact fact NOUN iajs-2475 73 3 , , PUNCT iajs-2475 73 4 z z PROPN iajs-2475 73 5 is be AUX iajs-2475 73 6 βŠ•-sextending βŠ•-sextende VERB iajs-2475 73 7 z z NOUN iajs-2475 73 8 - - PUNCT iajs-2475 73 9 module module NOUN iajs-2475 73 10 which which PRON iajs-2475 73 11 is be AUX iajs-2475 73 12 not not PART iajs-2475 73 13 lifting lift VERB iajs-2475 73 14 . . PUNCT iajs-2475 74 1 6 6 X iajs-2475 74 2 . . X iajs-2475 74 3 every every DET iajs-2475 74 4 weakly weakly ADJ iajs-2475 74 5 ( ( PUNCT iajs-2475 74 6 supplemented supplement VERB iajs-2475 74 7 ) ) PUNCT iajs-2475 74 8 module module NOUN iajs-2475 74 9 is be AUX iajs-2475 74 10 weakly weakly ADJ iajs-2475 74 11 supplement supplement NOUN iajs-2475 74 12 extending extend VERB iajs-2475 74 13 [ [ X iajs-2475 74 14 1 1 NUM iajs-2475 74 15 ] ] PUNCT iajs-2475 74 16 . . PUNCT iajs-2475 75 1 this this DET iajs-2475 75 2 result result NOUN iajs-2475 75 3 is be AUX iajs-2475 75 4 not not PART iajs-2475 75 5 still still ADV iajs-2475 75 6 valid valid ADJ iajs-2475 75 7 for for ADP iajs-2475 75 8 βŠ•-s βŠ•-s NOUN iajs-2475 75 9 - - PUNCT iajs-2475 75 10 extending extending NOUN iajs-2475 75 11 . . PUNCT iajs-2475 76 1 not not PART iajs-2475 76 2 every every DET iajs-2475 76 3 weakly weakly ADJ iajs-2475 76 4 supplemented supplement VERB iajs-2475 76 5 module module NOUN iajs-2475 76 6 is be AUX iajs-2475 76 7 βŠ•-sextending βŠ•-sextende VERB iajs-2475 76 8 . . PUNCT iajs-2475 77 1 moreover moreover ADV iajs-2475 77 2 , , PUNCT iajs-2475 77 3 not not PART iajs-2475 77 4 every every DET iajs-2475 77 5 βŠ•-s βŠ•-s ADJ iajs-2475 77 6 - - PUNCT iajs-2475 77 7 extending extend VERB iajs-2475 77 8 module module NOUN iajs-2475 77 9 is be AUX iajs-2475 77 10 weakly weakly ADV iajs-2475 77 11 supplemented supplement VERB iajs-2475 77 12 , , PUNCT iajs-2475 77 13 z z PROPN iajs-2475 77 14 is be AUX iajs-2475 77 15 βŠ•-sextending βŠ•-sextende VERB iajs-2475 77 16 z z NOUN iajs-2475 77 17 - - PUNCT iajs-2475 77 18 module module NOUN iajs-2475 77 19 which which PRON iajs-2475 77 20 it it PRON iajs-2475 77 21 is be AUX iajs-2475 77 22 not not PART iajs-2475 77 23 weakly weakly ADJ iajs-2475 77 24 supplemented supplement VERB iajs-2475 77 25 . . PUNCT iajs-2475 78 1 7 7 X iajs-2475 78 2 . . X iajs-2475 78 3 following follow VERB iajs-2475 78 4 [ [ X iajs-2475 78 5 7 7 NUM iajs-2475 78 6 ] ] PUNCT iajs-2475 78 7 . . PUNCT iajs-2475 79 1 every every DET iajs-2475 79 2 semi semi ADJ iajs-2475 79 3 - - ADJ iajs-2475 79 4 simple simple ADJ iajs-2475 79 5 module module NOUN iajs-2475 79 6 is be AUX iajs-2475 79 7 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 79 8 . . PUNCT iajs-2475 80 1 so so ADV iajs-2475 80 2 every every DET iajs-2475 80 3 semi semi ADJ iajs-2475 80 4 - - ADJ iajs-2475 80 5 simple simple ADJ iajs-2475 80 6 is be AUX iajs-2475 80 7 βŠ•-s βŠ•-s NOUN iajs-2475 80 8 - - PUNCT iajs-2475 80 9 extending extending NOUN iajs-2475 80 10 , , PUNCT iajs-2475 80 11 while while SCONJ iajs-2475 80 12 the the DET iajs-2475 80 13 converse converse NOUN iajs-2475 80 14 is be AUX iajs-2475 80 15 not not PART iajs-2475 80 16 true true ADJ iajs-2475 80 17 . . PUNCT iajs-2475 81 1 q q PUNCT iajs-2475 81 2 is be AUX iajs-2475 81 3 βŠ•-s βŠ•-s ADV iajs-2475 81 4 - - PUNCT iajs-2475 81 5 extending extend VERB iajs-2475 81 6 z z NOUN iajs-2475 81 7 - - PUNCT iajs-2475 81 8 module module NOUN iajs-2475 81 9 which which PRON iajs-2475 81 10 it it PRON iajs-2475 81 11 is be AUX iajs-2475 81 12 not not PART iajs-2475 81 13 semi semi ADJ iajs-2475 81 14 - - ADJ iajs-2475 81 15 simple simple ADJ iajs-2475 81 16 . . PUNCT iajs-2475 82 1 recall recall NOUN iajs-2475 82 2 that that PRON iajs-2475 82 3 , , PUNCT iajs-2475 82 4 a a DET iajs-2475 82 5 module module NOUN iajs-2475 82 6 m m VERB iajs-2475 82 7 is be AUX iajs-2475 82 8 called call VERB iajs-2475 82 9 refinable refinable ADJ iajs-2475 82 10 if if SCONJ iajs-2475 82 11 for for ADP iajs-2475 82 12 every every DET iajs-2475 82 13 submodule submodule PROPN iajs-2475 82 14 w w PROPN iajs-2475 82 15 , , PUNCT iajs-2475 82 16 v v NOUN iajs-2475 82 17 in in ADP iajs-2475 82 18 m m PROPN iajs-2475 82 19 with with ADP iajs-2475 82 20 w+v w+v NOUN iajs-2475 82 21 = = SYM iajs-2475 82 22 m m PROPN iajs-2475 82 23 , , PUNCT iajs-2475 82 24 then then ADV iajs-2475 82 25 there there PRON iajs-2475 82 26 is be VERB iajs-2475 82 27 a a DET iajs-2475 82 28 direct direct ADJ iajs-2475 82 29 summand summand NOUN iajs-2475 82 30 w w NOUN iajs-2475 82 31 ' ' PUNCT iajs-2475 82 32 in in ADP iajs-2475 82 33 m m PROPN iajs-2475 82 34 such such ADJ iajs-2475 82 35 that that SCONJ iajs-2475 82 36 w'βŠ†w w'βŠ†w NOUN iajs-2475 82 37 and and CCONJ iajs-2475 82 38 w'+v w'+v NOUN iajs-2475 82 39 = = NOUN iajs-2475 82 40 m m PROPN iajs-2475 83 1 [ [ X iajs-2475 83 2 2 2 NUM iajs-2475 83 3 ] ] PUNCT iajs-2475 83 4 . . PUNCT iajs-2475 84 1 following follow VERB iajs-2475 84 2 [ [ X iajs-2475 84 3 1 1 NUM iajs-2475 84 4 ] ] PUNCT iajs-2475 84 5 . . PUNCT iajs-2475 85 1 every every DET iajs-2475 85 2 closed close VERB iajs-2475 85 3 ⨁-supplemented ⨁-supplemente VERB iajs-2475 85 4 module module NOUN iajs-2475 85 5 is be AUX iajs-2475 85 6 weakly weakly ADJ iajs-2475 85 7 supplement supplement ADJ iajs-2475 85 8 extending extending NOUN iajs-2475 85 9 . . PUNCT iajs-2475 86 1 also also ADV iajs-2475 86 2 , , PUNCT iajs-2475 86 3 from from ADP iajs-2475 86 4 [ [ X iajs-2475 86 5 1 1 NUM iajs-2475 86 6 ] ] PUNCT iajs-2475 86 7 . . PUNCT iajs-2475 87 1 studied study VERB iajs-2475 87 2 when when SCONJ iajs-2475 87 3 the the DET iajs-2475 87 4 converse converse NOUN iajs-2475 87 5 is be AUX iajs-2475 87 6 true true ADJ iajs-2475 87 7 . . PUNCT iajs-2475 88 1 moreover moreover ADV iajs-2475 88 2 , , PUNCT iajs-2475 88 3 we we PRON iajs-2475 88 4 have have VERB iajs-2475 88 5 this this DET iajs-2475 88 6 corollary corollary ADJ iajs-2475 88 7 : : PUNCT iajs-2475 88 8 corollary corollary ADJ iajs-2475 88 9 ( ( PUNCT iajs-2475 88 10 6 6 NUM iajs-2475 88 11 ) ) PUNCT iajs-2475 88 12 let let VERB iajs-2475 88 13 𝑀 𝑀 PRON iajs-2475 88 14 be be AUX iajs-2475 88 15 a a DET iajs-2475 88 16 refinable refinable ADJ iajs-2475 88 17 module module NOUN iajs-2475 88 18 . . PUNCT iajs-2475 89 1 then then ADV iajs-2475 89 2 the the DET iajs-2475 89 3 following follow VERB iajs-2475 89 4 are be AUX iajs-2475 89 5 equivalent equivalent ADJ iajs-2475 89 6 : : PUNCT iajs-2475 89 7 1 1 X iajs-2475 89 8 . . X iajs-2475 89 9 𝑀 𝑀 PROPN iajs-2475 89 10 is be AUX iajs-2475 89 11 weakly weakly ADV iajs-2475 89 12 supplement supplement ADJ iajs-2475 89 13 extending extend VERB iajs-2475 89 14 module module NOUN iajs-2475 89 15 . . PUNCT iajs-2475 90 1 2 2 X iajs-2475 90 2 . . X iajs-2475 90 3 𝑀 𝑀 PROPN iajs-2475 90 4 is be AUX iajs-2475 90 5 βŠ•-s βŠ•-s ADJ iajs-2475 90 6 - - PUNCT iajs-2475 90 7 extending extend VERB iajs-2475 90 8 module module NOUN iajs-2475 90 9 . . PUNCT iajs-2475 91 1 83 83 NUM iajs-2475 91 2 ibn ibn PROPN iajs-2475 91 3 al al PROPN iajs-2475 91 4 - - PUNCT iajs-2475 91 5 haitham haitham PROPN iajs-2475 91 6 jour jour X iajs-2475 91 7 . . PROPN iajs-2475 91 8 for for ADP iajs-2475 91 9 pure pure ADJ iajs-2475 91 10 & & CCONJ iajs-2475 91 11 appl appl PROPN iajs-2475 91 12 . . PUNCT iajs-2475 92 1 sci sci PROPN iajs-2475 92 2 . . PROPN iajs-2475 93 1 33 33 NUM iajs-2475 93 2 ( ( PUNCT iajs-2475 93 3 3 3 NUM iajs-2475 93 4 ) ) PUNCT iajs-2475 93 5 2020 2020 NUM iajs-2475 93 6 recall recall VERB iajs-2475 93 7 that that PRON iajs-2475 93 8 , , PUNCT iajs-2475 93 9 a a DET iajs-2475 93 10 module module NOUN iajs-2475 93 11 m m VERB iajs-2475 93 12 is be AUX iajs-2475 93 13 said say VERB iajs-2475 93 14 to to PART iajs-2475 93 15 be be AUX iajs-2475 93 16 wd wd NOUN iajs-2475 93 17 - - NOUN iajs-2475 93 18 module module NOUN iajs-2475 93 19 if if SCONJ iajs-2475 93 20 each each DET iajs-2475 93 21 weakly weakly ADJ iajs-2475 93 22 supplement supplement NOUN iajs-2475 93 23 submodule submodule NOUN iajs-2475 93 24 is be AUX iajs-2475 93 25 direct direct ADJ iajs-2475 93 26 summand summand NOUN iajs-2475 94 1 [ [ X iajs-2475 94 2 1 1 NUM iajs-2475 94 3 ] ] PUNCT iajs-2475 94 4 . . PUNCT iajs-2475 95 1 proposition proposition NOUN iajs-2475 95 2 ( ( PUNCT iajs-2475 95 3 7 7 X iajs-2475 95 4 ) ) PUNCT iajs-2475 95 5 let let VERB iajs-2475 95 6 𝑀 𝑀 PRON iajs-2475 95 7 be be AUX iajs-2475 95 8 a a DET iajs-2475 95 9 wd wd NOUN iajs-2475 95 10 - - PUNCT iajs-2475 95 11 module module NOUN iajs-2475 95 12 . . PUNCT iajs-2475 96 1 then then ADV iajs-2475 96 2 the the DET iajs-2475 96 3 following follow VERB iajs-2475 96 4 statement statement NOUN iajs-2475 96 5 are be AUX iajs-2475 96 6 equivalent equivalent ADJ iajs-2475 96 7 : : PUNCT iajs-2475 96 8 1 1 X iajs-2475 96 9 . . X iajs-2475 96 10 m m PROPN iajs-2475 96 11 is be AUX iajs-2475 96 12 βŠ•-s βŠ•-s ADJ iajs-2475 96 13 - - PUNCT iajs-2475 96 14 extending extend VERB iajs-2475 96 15 module module NOUN iajs-2475 96 16 . . PUNCT iajs-2475 97 1 2 2 X iajs-2475 97 2 . . X iajs-2475 97 3 m m PROPN iajs-2475 97 4 is be AUX iajs-2475 97 5 weakly weakly ADV iajs-2475 97 6 supplement supplement ADJ iajs-2475 97 7 extending extend VERB iajs-2475 97 8 module module NOUN iajs-2475 97 9 . . PUNCT iajs-2475 98 1 proof proof NOUN iajs-2475 98 2 ( ( PUNCT iajs-2475 98 3 1⟹2 1⟹2 NUM iajs-2475 98 4 ) ) PUNCT iajs-2475 98 5 directly directly ADV iajs-2475 98 6 by by ADP iajs-2475 98 7 ( ( PUNCT iajs-2475 98 8 remarks remark NOUN iajs-2475 98 9 and and CCONJ iajs-2475 98 10 examples example NOUN iajs-2475 98 11 ( ( PUNCT iajs-2475 98 12 5 5 NUM iajs-2475 98 13 ) ) PUNCT iajs-2475 98 14 ) ) PUNCT iajs-2475 98 15 . . PUNCT iajs-2475 99 1 ( ( PUNCT iajs-2475 99 2 2⟹1 2⟹1 NUM iajs-2475 99 3 ) ) PUNCT iajs-2475 99 4 let let VERB iajs-2475 99 5 w w NOUN iajs-2475 99 6 be be AUX iajs-2475 99 7 a a DET iajs-2475 99 8 closed closed ADJ iajs-2475 99 9 submodule submodule NOUN iajs-2475 99 10 of of ADP iajs-2475 99 11 weakly weakly ADJ iajs-2475 99 12 supplement supplement NOUN iajs-2475 99 13 extending extend VERB iajs-2475 99 14 module module NOUN iajs-2475 99 15 m m NOUN iajs-2475 99 16 , , PUNCT iajs-2475 99 17 so so ADV iajs-2475 99 18 w w PROPN iajs-2475 99 19 is be AUX iajs-2475 99 20 a a DET iajs-2475 99 21 weakly weakly ADJ iajs-2475 99 22 supplement supplement NOUN iajs-2475 99 23 submodule submodule NOUN iajs-2475 99 24 of of ADP iajs-2475 99 25 v v NOUN iajs-2475 99 26 in in ADP iajs-2475 99 27 m. m. NOUN iajs-2475 99 28 also also ADV iajs-2475 99 29 v v VERB iajs-2475 99 30 is be AUX iajs-2475 99 31 weakly weakly ADJ iajs-2475 99 32 supplements supplement NOUN iajs-2475 99 33 of of ADP iajs-2475 99 34 w w NOUN iajs-2475 99 35 in in ADP iajs-2475 99 36 m. m. NOUN iajs-2475 99 37 but but CCONJ iajs-2475 99 38 m m NOUN iajs-2475 99 39 is be AUX iajs-2475 99 40 wd wd NOUN iajs-2475 99 41 - - NOUN iajs-2475 99 42 module module NOUN iajs-2475 99 43 . . PUNCT iajs-2475 100 1 hence hence ADV iajs-2475 100 2 we we PRON iajs-2475 100 3 have have VERB iajs-2475 100 4 v v NOUN iajs-2475 100 5 is be AUX iajs-2475 100 6 direct direct ADJ iajs-2475 100 7 summand summand NOUN iajs-2475 100 8 of of ADP iajs-2475 100 9 m m PROPN iajs-2475 101 1 and and CCONJ iajs-2475 101 2 so so ADV iajs-2475 101 3 m m AUX iajs-2475 101 4 is be AUX iajs-2475 101 5 βŠ•-sextending βŠ•-sextende VERB iajs-2475 101 6 module module NOUN iajs-2475 101 7 . . PUNCT iajs-2475 102 1 proposition proposition NOUN iajs-2475 102 2 ( ( PUNCT iajs-2475 102 3 8) 8) NUM iajs-2475 102 4 let let VERB iajs-2475 102 5 m m PRON iajs-2475 102 6 be be AUX iajs-2475 102 7 a a DET iajs-2475 102 8 wd wd NOUN iajs-2475 102 9 - - PUNCT iajs-2475 102 10 module module NOUN iajs-2475 102 11 . . PUNCT iajs-2475 103 1 then then ADV iajs-2475 103 2 the the DET iajs-2475 103 3 following follow VERB iajs-2475 103 4 statement statement NOUN iajs-2475 103 5 are be AUX iajs-2475 103 6 equivalent equivalent ADJ iajs-2475 103 7 : : PUNCT iajs-2475 103 8 1 1 X iajs-2475 103 9 . . X iajs-2475 103 10 m m PROPN iajs-2475 103 11 is be AUX iajs-2475 103 12 extending extend VERB iajs-2475 103 13 module module NOUN iajs-2475 103 14 . . PUNCT iajs-2475 104 1 2 2 X iajs-2475 104 2 . . X iajs-2475 104 3 m m PROPN iajs-2475 104 4 is be AUX iajs-2475 104 5 βŠ•-s βŠ•-s ADJ iajs-2475 104 6 - - PUNCT iajs-2475 104 7 extending extend VERB iajs-2475 104 8 module module NOUN iajs-2475 104 9 . . PUNCT iajs-2475 105 1 proof proof NOUN iajs-2475 105 2 ( ( PUNCT iajs-2475 105 3 1⟹2 1⟹2 NUM iajs-2475 105 4 ) ) PUNCT iajs-2475 105 5 directly directly ADV iajs-2475 105 6 by by ADP iajs-2475 105 7 ( ( PUNCT iajs-2475 105 8 remarks remark NOUN iajs-2475 105 9 and and CCONJ iajs-2475 105 10 examples example NOUN iajs-2475 105 11 ( ( PUNCT iajs-2475 105 12 5 5 NUM iajs-2475 105 13 ) ) PUNCT iajs-2475 105 14 ) ) PUNCT iajs-2475 105 15 . . PUNCT iajs-2475 106 1 ( ( PUNCT iajs-2475 106 2 2⟹1 2⟹1 NUM iajs-2475 106 3 ) ) PUNCT iajs-2475 106 4 let let VERB iajs-2475 106 5 a a PRON iajs-2475 106 6 be be AUX iajs-2475 106 7 a a DET iajs-2475 106 8 closed closed ADJ iajs-2475 106 9 submodule submodule NOUN iajs-2475 106 10 of of ADP iajs-2475 106 11 βŠ•-s βŠ•-s ADJ iajs-2475 106 12 - - PUNCT iajs-2475 106 13 extending extend VERB iajs-2475 106 14 module module NOUN iajs-2475 106 15 m m NOUN iajs-2475 106 16 , , PUNCT iajs-2475 106 17 so so ADV iajs-2475 106 18 a a PRON iajs-2475 106 19 has have VERB iajs-2475 106 20 a a DET iajs-2475 106 21 weakly weakly ADJ iajs-2475 106 22 supplement supplement NOUN iajs-2475 106 23 submodule submodule NOUN iajs-2475 106 24 k k PROPN iajs-2475 106 25 in in ADP iajs-2475 106 26 m m PROPN iajs-2475 106 27 which which PRON iajs-2475 106 28 is be AUX iajs-2475 106 29 direct direct ADJ iajs-2475 106 30 summand summand NOUN iajs-2475 106 31 . . PUNCT iajs-2475 107 1 also also ADV iajs-2475 107 2 k k PROPN iajs-2475 107 3 is be AUX iajs-2475 107 4 weakly weakly ADJ iajs-2475 107 5 supplement supplement NOUN iajs-2475 107 6 of of ADP iajs-2475 107 7 a a PRON iajs-2475 107 8 in in ADP iajs-2475 107 9 m. m. NOUN iajs-2475 107 10 but but CCONJ iajs-2475 107 11 m m NOUN iajs-2475 107 12 is be AUX iajs-2475 107 13 wd wd NOUN iajs-2475 107 14 - - NOUN iajs-2475 107 15 module module NOUN iajs-2475 107 16 . . PUNCT iajs-2475 108 1 then then ADV iajs-2475 108 2 we we PRON iajs-2475 108 3 have have VERB iajs-2475 108 4 a a PRON iajs-2475 108 5 is be AUX iajs-2475 108 6 direct direct ADJ iajs-2475 108 7 summand summand NOUN iajs-2475 108 8 of of ADP iajs-2475 108 9 m. m. NOUN iajs-2475 108 10 hence hence ADV iajs-2475 108 11 m m VERB iajs-2475 108 12 is be AUX iajs-2475 108 13 βŠ•-s βŠ•-s ADJ iajs-2475 108 14 - - PUNCT iajs-2475 108 15 extending extend VERB iajs-2475 108 16 module module NOUN iajs-2475 108 17 . . PUNCT iajs-2475 109 1 the the DET iajs-2475 109 2 following follow VERB iajs-2475 109 3 lemma lemma PROPN iajs-2475 109 4 helps help VERB iajs-2475 109 5 us we PRON iajs-2475 109 6 in in ADP iajs-2475 109 7 the the DET iajs-2475 109 8 next next ADJ iajs-2475 109 9 results result NOUN iajs-2475 109 10 lemma lemma PROPN iajs-2475 109 11 ( ( PUNCT iajs-2475 109 12 9 9 X iajs-2475 109 13 ) ) PUNCT iajs-2475 109 14 let let VERB iajs-2475 109 15 m m PRON iajs-2475 109 16 be be AUX iajs-2475 109 17 a a DET iajs-2475 109 18 βŠ•-s βŠ•-s ADJ iajs-2475 109 19 - - PUNCT iajs-2475 109 20 extending extend VERB iajs-2475 109 21 module module NOUN iajs-2475 109 22 . . PUNCT iajs-2475 110 1 suppose suppose VERB iajs-2475 110 2 that that SCONJ iajs-2475 110 3 w w PROPN iajs-2475 110 4 be be AUX iajs-2475 110 5 a a DET iajs-2475 110 6 closed closed ADJ iajs-2475 110 7 submodule submodule NOUN iajs-2475 110 8 in in ADP iajs-2475 110 9 m m PROPN iajs-2475 110 10 and and CCONJ iajs-2475 110 11 b b PROPN iajs-2475 110 12 is be AUX iajs-2475 110 13 small small ADJ iajs-2475 110 14 in in ADP iajs-2475 110 15 m. m. NOUN iajs-2475 110 16 t t PROPN iajs-2475 110 17 then then ADV iajs-2475 110 18 there there PRON iajs-2475 110 19 is be VERB iajs-2475 110 20 a a DET iajs-2475 110 21 submodule submodule NOUN iajs-2475 110 22 c c NOUN iajs-2475 110 23 in in ADP iajs-2475 110 24 m m PRON iajs-2475 110 25 such such ADJ iajs-2475 110 26 that that SCONJ iajs-2475 110 27 m m NOUN iajs-2475 110 28 = = ADJ iajs-2475 110 29 w+c w+c ADP iajs-2475 110 30 = = SYM iajs-2475 110 31 w+c+b w+c+b NOUN iajs-2475 110 32 and and CCONJ iajs-2475 110 33 w∩cβ‰ͺm w∩cβ‰ͺm PROPN iajs-2475 110 34 , , PUNCT iajs-2475 110 35 so so ADV iajs-2475 110 36 c∩(w+b c∩(w+b PROPN iajs-2475 110 37 ) ) PUNCT iajs-2475 110 38 β‰ͺm β‰ͺm ADV iajs-2475 110 39 . . PUNCT iajs-2475 111 1 proof proof NOUN iajs-2475 111 2 let let VERB iajs-2475 111 3 w w NOUN iajs-2475 111 4 be be AUX iajs-2475 111 5 a a DET iajs-2475 111 6 closed closed ADJ iajs-2475 111 7 submodule submodule NOUN iajs-2475 111 8 of of ADP iajs-2475 111 9 βŠ•-s βŠ•-s ADJ iajs-2475 111 10 - - PUNCT iajs-2475 111 11 extending extend VERB iajs-2475 111 12 module module NOUN iajs-2475 111 13 m m NOUN iajs-2475 111 14 , , PUNCT iajs-2475 111 15 then then ADV iajs-2475 111 16 w w PROPN iajs-2475 111 17 has have VERB iajs-2475 111 18 a a DET iajs-2475 111 19 weakly weakly ADJ iajs-2475 111 20 supplements supplement NOUN iajs-2475 111 21 c c NOUN iajs-2475 111 22 in in ADP iajs-2475 111 23 m m PROPN iajs-2475 111 24 that that PRON iajs-2475 111 25 is be AUX iajs-2475 111 26 direct direct ADJ iajs-2475 111 27 summand summand NOUN iajs-2475 111 28 ( ( PUNCT iajs-2475 111 29 i.e i.e PROPN iajs-2475 111 30 ) ) PUNCT iajs-2475 111 31 m m PROPN iajs-2475 111 32 = = NOUN iajs-2475 111 33 w+c w+c X iajs-2475 111 34 and and CCONJ iajs-2475 111 35 w∩cβ‰ͺm w∩cβ‰ͺm PROPN iajs-2475 111 36 . . PUNCT iajs-2475 112 1 now now ADV iajs-2475 112 2 let let VERB iajs-2475 112 3 h h NOUN iajs-2475 112 4 : : PUNCT iajs-2475 112 5 mβ†’(mβˆ• mβ†’(mβˆ• PROPN iajs-2475 112 6 π‘Š)⨁(mβˆ•c π‘Š)⨁(mβˆ•c PROPN iajs-2475 112 7 ) ) PUNCT iajs-2475 112 8 is be AUX iajs-2475 112 9 defined define VERB iajs-2475 112 10 by by ADP iajs-2475 112 11 h(d)=(d+w h(d)=(d+w PROPN iajs-2475 112 12 , , PUNCT iajs-2475 112 13 d+c d+c NUM iajs-2475 112 14 ) ) PUNCT iajs-2475 112 15 and and CCONJ iajs-2475 112 16 let let VERB iajs-2475 112 17 j j NOUN iajs-2475 112 18 : : PUNCT iajs-2475 112 19 ( ( PUNCT iajs-2475 112 20 mβˆ• mβˆ• PROPN iajs-2475 112 21 π‘Š)⨁(mβˆ•c π‘Š)⨁(mβˆ•c PROPN iajs-2475 112 22 ) ) PUNCT iajs-2475 113 1 β†’ β†’ PUNCT iajs-2475 113 2 ( ( PUNCT iajs-2475 113 3 mβˆ• mβˆ• PROPN iajs-2475 113 4 π‘Š π‘Š PROPN iajs-2475 113 5 + + NOUN iajs-2475 113 6 𝐡)⨁(mβˆ•c 𝐡)⨁(mβˆ•c PROPN iajs-2475 113 7 ) ) PUNCT iajs-2475 113 8 is be AUX iajs-2475 113 9 defined define VERB iajs-2475 113 10 by by ADP iajs-2475 113 11 j(d+w j(d+w PROPN iajs-2475 113 12 , , PUNCT iajs-2475 113 13 m'+c)=(d+w+b m'+c)=(d+w+b PROPN iajs-2475 113 14 , , PUNCT iajs-2475 113 15 m'+c m'+c NUM iajs-2475 113 16 ) ) PUNCT iajs-2475 113 17 . . PUNCT iajs-2475 114 1 since since SCONJ iajs-2475 114 2 w∩cβ‰ͺm w∩cβ‰ͺm PROPN iajs-2475 114 3 , , PUNCT iajs-2475 114 4 then then ADV iajs-2475 114 5 h h PROPN iajs-2475 114 6 is be AUX iajs-2475 114 7 epimorphism epimorphism NOUN iajs-2475 114 8 and and CCONJ iajs-2475 114 9 kerh= kerh= PROPN iajs-2475 114 10 w∩cβ‰ͺm w∩cβ‰ͺm PROPN iajs-2475 114 11 , , PUNCT iajs-2475 114 12 and and CCONJ iajs-2475 114 13 since since SCONJ iajs-2475 114 14 kerj=((w+bβˆ•w)βŠ•0 kerj=((w+bβˆ•w)βŠ•0 PROPN iajs-2475 114 15 and and CCONJ iajs-2475 114 16 ( ( PUNCT iajs-2475 114 17 w+b)βˆ•w=(b)β‰ͺmβˆ•w w+b)βˆ•w=(b)β‰ͺmβˆ•w PROPN iajs-2475 114 18 when when SCONJ iajs-2475 114 19 the the DET iajs-2475 114 20 canonical canonical ADJ iajs-2475 114 21 epimorphism epimorphism NOUN iajs-2475 114 22 𝜈 𝜈 X iajs-2475 114 23 : : PUNCT iajs-2475 114 24 mβ†’mβˆ•w mβ†’mβˆ•w NOUN iajs-2475 114 25 . . PUNCT iajs-2475 115 1 so so ADV iajs-2475 115 2 we we PRON iajs-2475 115 3 have have VERB iajs-2475 115 4 j j PROPN iajs-2475 115 5 is be AUX iajs-2475 115 6 a a DET iajs-2475 115 7 small small ADJ iajs-2475 115 8 epimorphism epimorphism NOUN iajs-2475 115 9 and and CCONJ iajs-2475 115 10 jh jh PROPN iajs-2475 115 11 is be AUX iajs-2475 115 12 small small ADJ iajs-2475 115 13 epimorphism epimorphism NOUN iajs-2475 115 14 since since SCONJ iajs-2475 115 15 kerjh= kerjh= PROPN iajs-2475 115 16 c∩(b+w)β‰ͺm c∩(b+w)β‰ͺm PROPN iajs-2475 115 17 . . PUNCT iajs-2475 116 1 we we PRON iajs-2475 116 2 noticed notice VERB iajs-2475 116 3 that that SCONJ iajs-2475 116 4 , , PUNCT iajs-2475 116 5 every every DET iajs-2475 116 6 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 116 7 module module NOUN iajs-2475 116 8 is be AUX iajs-2475 116 9 βŠ•-s βŠ•-s ADV iajs-2475 116 10 - - PUNCT iajs-2475 116 11 extending extend VERB iajs-2475 116 12 see see NOUN iajs-2475 116 13 ( ( PUNCT iajs-2475 116 14 remarks remark NOUN iajs-2475 116 15 and and CCONJ iajs-2475 116 16 example example NOUN iajs-2475 116 17 ( ( PUNCT iajs-2475 116 18 5 5 NUM iajs-2475 116 19 ) ) PUNCT iajs-2475 116 20 ) ) PUNCT iajs-2475 116 21 . . PUNCT iajs-2475 117 1 next next ADV iajs-2475 117 2 we we PRON iajs-2475 117 3 explained explain VERB iajs-2475 117 4 when when SCONJ iajs-2475 117 5 the the DET iajs-2475 117 6 convers conver NOUN iajs-2475 117 7 is be AUX iajs-2475 117 8 true true ADJ iajs-2475 117 9 . . PUNCT iajs-2475 118 1 proposition proposition NOUN iajs-2475 118 2 ( ( PUNCT iajs-2475 118 3 10 10 NUM iajs-2475 118 4 ) ) PUNCT iajs-2475 118 5 let let VERB iajs-2475 118 6 m m PRON iajs-2475 118 7 be be AUX iajs-2475 118 8 a a DET iajs-2475 118 9 module module NOUN iajs-2475 118 10 in in ADP iajs-2475 118 11 which which PRON iajs-2475 118 12 each each DET iajs-2475 118 13 submodule submodule NOUN iajs-2475 118 14 l l PROPN iajs-2475 118 15 in in ADP iajs-2475 118 16 m m NOUN iajs-2475 118 17 there there PRON iajs-2475 118 18 exists exist VERB iajs-2475 118 19 a a DET iajs-2475 118 20 closed closed ADJ iajs-2475 118 21 submodule submodule NOUN iajs-2475 118 22 w w NOUN iajs-2475 118 23 ( ( PUNCT iajs-2475 118 24 depending depend VERB iajs-2475 118 25 on on ADP iajs-2475 118 26 l l NOUN iajs-2475 118 27 ) ) PUNCT iajs-2475 118 28 of of ADP iajs-2475 118 29 m m PRON iajs-2475 118 30 such such ADJ iajs-2475 118 31 that that SCONJ iajs-2475 118 32 l l NOUN iajs-2475 118 33 = = NOUN iajs-2475 118 34 w+c w+c NOUN iajs-2475 118 35 or or CCONJ iajs-2475 118 36 w w NOUN iajs-2475 118 37 = = NOUN iajs-2475 118 38 l+c l+c PROPN iajs-2475 118 39 for for ADP iajs-2475 118 40 some some DET iajs-2475 118 41 c c PROPN iajs-2475 118 42 small small ADJ iajs-2475 118 43 in in ADP iajs-2475 118 44 m. m. NOUN iajs-2475 118 45 then then ADV iajs-2475 118 46 m m VERB iajs-2475 118 47 is be AUX iajs-2475 118 48 βŠ•-s βŠ•-s ADJ iajs-2475 118 49 - - PUNCT iajs-2475 118 50 extending extend VERB iajs-2475 118 51 module module NOUN iajs-2475 119 1 if if SCONJ iajs-2475 119 2 and and CCONJ iajs-2475 119 3 only only ADV iajs-2475 119 4 if if SCONJ iajs-2475 119 5 m m VERB iajs-2475 119 6 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 119 7 . . PUNCT iajs-2475 120 1 84 84 NUM iajs-2475 121 1 ibn ibn PROPN iajs-2475 121 2 al al PROPN iajs-2475 121 3 - - PUNCT iajs-2475 121 4 haitham haitham PROPN iajs-2475 121 5 jour jour X iajs-2475 121 6 . . PROPN iajs-2475 121 7 for for ADP iajs-2475 121 8 pure pure ADJ iajs-2475 121 9 & & CCONJ iajs-2475 121 10 appl appl PROPN iajs-2475 121 11 . . PUNCT iajs-2475 122 1 sci sci PROPN iajs-2475 122 2 . . PROPN iajs-2475 123 1 33 33 NUM iajs-2475 123 2 ( ( PUNCT iajs-2475 123 3 3 3 NUM iajs-2475 123 4 ) ) PUNCT iajs-2475 123 5 2020 2020 NUM iajs-2475 123 6 proof proof NOUN iajs-2475 123 7 let let VERB iajs-2475 123 8 l l NOUN iajs-2475 123 9 be be AUX iajs-2475 123 10 a a DET iajs-2475 123 11 submodule submodule NOUN iajs-2475 123 12 in in ADP iajs-2475 123 13 βŠ•-s βŠ•-s ADJ iajs-2475 123 14 - - PUNCT iajs-2475 123 15 extending extend VERB iajs-2475 123 16 module module NOUN iajs-2475 123 17 m m NOUN iajs-2475 123 18 , , PUNCT iajs-2475 123 19 so so SCONJ iajs-2475 123 20 there there PRON iajs-2475 123 21 exists exist VERB iajs-2475 123 22 a a DET iajs-2475 123 23 closed closed ADJ iajs-2475 123 24 submodule submodule NOUN iajs-2475 123 25 w w NOUN iajs-2475 123 26 in in ADP iajs-2475 123 27 m m PRON iajs-2475 123 28 such such ADJ iajs-2475 123 29 that that SCONJ iajs-2475 123 30 l l NOUN iajs-2475 123 31 = = NOUN iajs-2475 123 32 w+c w+c X iajs-2475 123 33 where where SCONJ iajs-2475 123 34 cβ‰ͺm cβ‰ͺm PROPN iajs-2475 123 35 . . PUNCT iajs-2475 124 1 but but CCONJ iajs-2475 124 2 m m PROPN iajs-2475 124 3 is be AUX iajs-2475 124 4 βŠ•-s βŠ•-s ADJ iajs-2475 124 5 - - PUNCT iajs-2475 124 6 extending extending ADJ iajs-2475 124 7 , , PUNCT iajs-2475 124 8 thus thus ADV iajs-2475 124 9 w w PUNCT iajs-2475 124 10 has have VERB iajs-2475 124 11 a a DET iajs-2475 124 12 supplements supplement NOUN iajs-2475 124 13 d d NOUN iajs-2475 124 14 in in ADP iajs-2475 124 15 m m PRON iajs-2475 124 16 that that PRON iajs-2475 124 17 is be AUX iajs-2475 124 18 direct direct ADJ iajs-2475 124 19 summand summand NOUN iajs-2475 124 20 ( ( PUNCT iajs-2475 124 21 i.e i.e PROPN iajs-2475 124 22 ) ) PUNCT iajs-2475 124 23 m m PROPN iajs-2475 124 24 = = NOUN iajs-2475 124 25 w+d w+d NUM iajs-2475 124 26 = = SYM iajs-2475 124 27 w+c+d w+c+d X iajs-2475 124 28 = = X iajs-2475 124 29 l+d l+d X iajs-2475 124 30 where where SCONJ iajs-2475 124 31 cβ‰ͺm cβ‰ͺm PROPN iajs-2475 124 32 and and CCONJ iajs-2475 124 33 w∩dβ‰ͺd w∩dβ‰ͺd PROPN iajs-2475 124 34 , , PUNCT iajs-2475 124 35 then then ADV iajs-2475 124 36 l∩d l∩d VERB iajs-2475 124 37 βŠ† βŠ† NUM iajs-2475 124 38 ( ( PUNCT iajs-2475 124 39 w+c)∩d w+c)∩d X iajs-2475 124 40 . . PUNCT iajs-2475 125 1 since since SCONJ iajs-2475 125 2 w∩dβ‰ͺd w∩dβ‰ͺd PROPN iajs-2475 125 3 , , PUNCT iajs-2475 125 4 then then ADV iajs-2475 125 5 w∩dβ‰ͺm w∩dβ‰ͺm PROPN iajs-2475 125 6 so so ADV iajs-2475 125 7 by by ADP iajs-2475 125 8 lemma lemma PROPN iajs-2475 125 9 ( ( PUNCT iajs-2475 125 10 9 9 NUM iajs-2475 125 11 ) ) PUNCT iajs-2475 125 12 ( ( PUNCT iajs-2475 125 13 w+c)∩dβ‰ͺm w+c)∩dβ‰ͺm X iajs-2475 125 14 . . PUNCT iajs-2475 126 1 thus thus ADV iajs-2475 126 2 l∩dβ‰ͺm l∩dβ‰ͺm PROPN iajs-2475 126 3 . . PUNCT iajs-2475 127 1 now now ADV iajs-2475 127 2 since since SCONJ iajs-2475 127 3 l∩d l∩d ADJ iajs-2475 127 4 βŠ†dβŠ†m βŠ†dβŠ†m NOUN iajs-2475 127 5 and and CCONJ iajs-2475 127 6 d d ADP iajs-2475 127 7 direct direct ADJ iajs-2475 127 8 summand summand NOUN iajs-2475 127 9 , , PUNCT iajs-2475 127 10 so so ADV iajs-2475 127 11 we we PRON iajs-2475 127 12 have have VERB iajs-2475 127 13 l∩dβ‰ͺd l∩dβ‰ͺd NOUN iajs-2475 127 14 and and CCONJ iajs-2475 127 15 hence hence ADV iajs-2475 127 16 m m VERB iajs-2475 127 17 is be AUX iajs-2475 127 18 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 127 19 . . PUNCT iajs-2475 128 1 or or CCONJ iajs-2475 128 2 , , PUNCT iajs-2475 128 3 w w PROPN iajs-2475 128 4 = = NOUN iajs-2475 128 5 l+c l+c X iajs-2475 128 6 where where SCONJ iajs-2475 128 7 cβ‰ͺm cβ‰ͺm PROPN iajs-2475 128 8 . . PUNCT iajs-2475 129 1 since since SCONJ iajs-2475 129 2 m m PROPN iajs-2475 129 3 is be AUX iajs-2475 129 4 βŠ•s βŠ•s NOUN iajs-2475 129 5 - - PUNCT iajs-2475 129 6 extending extending ADJ iajs-2475 129 7 , , PUNCT iajs-2475 129 8 thus thus ADV iajs-2475 129 9 w w PUNCT iajs-2475 129 10 has have VERB iajs-2475 129 11 a a DET iajs-2475 129 12 supplement supplement NOUN iajs-2475 129 13 d d NOUN iajs-2475 129 14 in in ADP iajs-2475 129 15 m m PRON iajs-2475 129 16 that that PRON iajs-2475 129 17 is be AUX iajs-2475 129 18 direct direct ADJ iajs-2475 129 19 summand summand NOUN iajs-2475 129 20 ( ( PUNCT iajs-2475 129 21 i.e i.e PROPN iajs-2475 129 22 ) ) PUNCT iajs-2475 129 23 m m PROPN iajs-2475 129 24 = = ADJ iajs-2475 129 25 w+d w+d NUM iajs-2475 129 26 = = SYM iajs-2475 129 27 l+c+d l+c+d X iajs-2475 129 28 = = SYM iajs-2475 129 29 l+d l+d X iajs-2475 129 30 where where SCONJ iajs-2475 129 31 c c AUX iajs-2475 129 32 β‰ͺm β‰ͺm X iajs-2475 129 33 and and CCONJ iajs-2475 129 34 l∩dβŠ†w∩dβ‰ͺd l∩dβŠ†w∩dβ‰ͺd PROPN iajs-2475 129 35 , , PUNCT iajs-2475 129 36 so so ADV iajs-2475 129 37 l∩dβ‰ͺd l∩dβ‰ͺd PROPN iajs-2475 129 38 . . PUNCT iajs-2475 130 1 thus thus ADV iajs-2475 130 2 m m NOUN iajs-2475 130 3 is be AUX iajs-2475 130 4 βŠ•supplemented βŠ•supplemente VERB iajs-2475 130 5 . . PUNCT iajs-2475 131 1 conversely conversely ADV iajs-2475 131 2 see see VERB iajs-2475 131 3 ( ( PUNCT iajs-2475 131 4 remarks remark NOUN iajs-2475 131 5 and and CCONJ iajs-2475 131 6 examples example NOUN iajs-2475 131 7 ( ( PUNCT iajs-2475 131 8 5 5 NUM iajs-2475 131 9 ) ) PUNCT iajs-2475 131 10 ) ) PUNCT iajs-2475 131 11 . . PUNCT iajs-2475 132 1 recall recall VERB iajs-2475 132 2 that that PRON iajs-2475 132 3 , , PUNCT iajs-2475 132 4 a a DET iajs-2475 132 5 module module NOUN iajs-2475 132 6 m m VERB iajs-2475 132 7 is be AUX iajs-2475 132 8 called call VERB iajs-2475 132 9 injective injective ADJ iajs-2475 132 10 hull hull NOUN iajs-2475 132 11 of of ADP iajs-2475 132 12 a a DET iajs-2475 132 13 module module NOUN iajs-2475 132 14 q q NOUN iajs-2475 133 1 if if SCONJ iajs-2475 133 2 it it PRON iajs-2475 133 3 is be AUX iajs-2475 133 4 both both CCONJ iajs-2475 133 5 an an DET iajs-2475 133 6 essential essential ADJ iajs-2475 133 7 extension extension NOUN iajs-2475 133 8 of of ADP iajs-2475 133 9 q q PROPN iajs-2475 133 10 and and CCONJ iajs-2475 133 11 an an DET iajs-2475 133 12 injective injective ADJ iajs-2475 133 13 module module NOUN iajs-2475 133 14 [ [ X iajs-2475 133 15 3 3 NUM iajs-2475 133 16 ] ] PUNCT iajs-2475 133 17 . . PUNCT iajs-2475 134 1 the the DET iajs-2475 134 2 following follow VERB iajs-2475 134 3 proposition proposition NOUN iajs-2475 134 4 gives give VERB iajs-2475 134 5 another another DET iajs-2475 134 6 characterization characterization NOUN iajs-2475 134 7 of of ADP iajs-2475 134 8 βŠ•-s βŠ•-s NOUN iajs-2475 134 9 - - PUNCT iajs-2475 134 10 extending extend VERB iajs-2475 134 11 module module NOUN iajs-2475 134 12 . . PUNCT iajs-2475 135 1 proposition proposition NOUN iajs-2475 135 2 ( ( PUNCT iajs-2475 135 3 11 11 NUM iajs-2475 135 4 ) ) PUNCT iajs-2475 135 5 for for ADP iajs-2475 135 6 any any DET iajs-2475 135 7 a a DET iajs-2475 135 8 module module NOUN iajs-2475 135 9 m. m. NOUN iajs-2475 135 10 t t PROPN iajs-2475 135 11 the the DET iajs-2475 135 12 following follow VERB iajs-2475 135 13 statement statement NOUN iajs-2475 135 14 are be AUX iajs-2475 135 15 equivalent equivalent ADJ iajs-2475 135 16 : : PUNCT iajs-2475 135 17 1 1 X iajs-2475 135 18 . . X iajs-2475 135 19 m m PROPN iajs-2475 135 20 is be AUX iajs-2475 135 21 βŠ•-s βŠ•-s ADJ iajs-2475 135 22 - - PUNCT iajs-2475 135 23 extending extend VERB iajs-2475 135 24 module module NOUN iajs-2475 135 25 . . PUNCT iajs-2475 136 1 2 2 X iajs-2475 136 2 . . X iajs-2475 136 3 the the DET iajs-2475 136 4 intersection intersection NOUN iajs-2475 136 5 of of ADP iajs-2475 136 6 m m PROPN iajs-2475 136 7 with with ADP iajs-2475 136 8 any any DET iajs-2475 136 9 direct direct ADJ iajs-2475 136 10 summand summand NOUN iajs-2475 136 11 of of ADP iajs-2475 136 12 injective injective ADJ iajs-2475 136 13 hull hull NOUN iajs-2475 136 14 of of ADP iajs-2475 136 15 m m PROPN iajs-2475 136 16 , , PUNCT iajs-2475 136 17 has have VERB iajs-2475 136 18 a a DET iajs-2475 136 19 weakly weakly ADJ iajs-2475 136 20 supplements supplement NOUN iajs-2475 136 21 submodule submodule NOUN iajs-2475 136 22 in in ADP iajs-2475 136 23 m m PROPN iajs-2475 136 24 that that PRON iajs-2475 136 25 is be AUX iajs-2475 136 26 direct direct ADJ iajs-2475 136 27 summand summand NOUN iajs-2475 136 28 . . PUNCT iajs-2475 137 1 proof proof NOUN iajs-2475 137 2 ( ( PUNCT iajs-2475 137 3 1 1 NUM iajs-2475 137 4 ) ) PUNCT iajs-2475 137 5 ⟹ ⟹ NOUN iajs-2475 137 6 ( ( PUNCT iajs-2475 137 7 2 2 X iajs-2475 137 8 ) ) PUNCT iajs-2475 137 9 let let VERB iajs-2475 137 10 w w NOUN iajs-2475 137 11 be be AUX iajs-2475 137 12 a a DET iajs-2475 137 13 direct direct ADJ iajs-2475 137 14 summand summand NOUN iajs-2475 137 15 of of ADP iajs-2475 137 16 injective injective ADJ iajs-2475 137 17 hull hull NOUN iajs-2475 137 18 of of ADP iajs-2475 137 19 m m PROPN iajs-2475 137 20 , , PUNCT iajs-2475 137 21 i.e i.e PROPN iajs-2475 137 22 e(m)=w⨁v e(m)=w⨁v NOUN iajs-2475 137 23 , , PUNCT iajs-2475 137 24 where where SCONJ iajs-2475 137 25 v v NOUN iajs-2475 137 26 is be AUX iajs-2475 137 27 a a DET iajs-2475 137 28 submodule submodule NOUN iajs-2475 137 29 of of ADP iajs-2475 137 30 injective injective ADJ iajs-2475 137 31 hull hull NOUN iajs-2475 137 32 of of ADP iajs-2475 137 33 m. m. NOUN iajs-2475 137 34 it it PRON iajs-2475 137 35 is be AUX iajs-2475 137 36 easy easy ADJ iajs-2475 137 37 to to PART iajs-2475 137 38 show show VERB iajs-2475 137 39 that that SCONJ iajs-2475 137 40 w∩m w∩m PROPN iajs-2475 137 41 is be AUX iajs-2475 137 42 closed close VERB iajs-2475 137 43 of of ADP iajs-2475 137 44 m. m. NOUN iajs-2475 137 45 so so ADV iajs-2475 137 46 by by ADP iajs-2475 137 47 proposition proposition NOUN iajs-2475 137 48 ( ( PUNCT iajs-2475 137 49 3 3 X iajs-2475 137 50 ) ) PUNCT iajs-2475 137 51 w∩m w∩m NOUN iajs-2475 137 52 has have VERB iajs-2475 137 53 a a DET iajs-2475 137 54 weakly weakly ADJ iajs-2475 137 55 supplement supplement NOUN iajs-2475 137 56 submodule submodule NOUN iajs-2475 137 57 of of ADP iajs-2475 137 58 m m PRON iajs-2475 137 59 which which PRON iajs-2475 137 60 is be AUX iajs-2475 137 61 direct direct ADJ iajs-2475 137 62 summand summand NOUN iajs-2475 137 63 . . PUNCT iajs-2475 138 1 ( ( PUNCT iajs-2475 138 2 2 2 NUM iajs-2475 138 3 ) ) PUNCT iajs-2475 138 4 ⟹ ⟹ NOUN iajs-2475 138 5 ( ( PUNCT iajs-2475 138 6 1 1 X iajs-2475 138 7 ) ) PUNCT iajs-2475 138 8 let let VERB iajs-2475 138 9 w w NOUN iajs-2475 138 10 be be AUX iajs-2475 138 11 a a DET iajs-2475 138 12 submodule submodule NOUN iajs-2475 138 13 of of ADP iajs-2475 138 14 m. m. NOUN iajs-2475 138 15 and and CCONJ iajs-2475 138 16 let let VERB iajs-2475 138 17 v v PART iajs-2475 138 18 be be AUX iajs-2475 138 19 a a DET iajs-2475 138 20 relative relative ADJ iajs-2475 138 21 complement complement NOUN iajs-2475 138 22 of of ADP iajs-2475 138 23 w w PROPN iajs-2475 138 24 in in ADP iajs-2475 138 25 m m PROPN iajs-2475 138 26 , , PUNCT iajs-2475 138 27 then then ADV iajs-2475 138 28 w⨁v w⨁v PROPN iajs-2475 138 29 is be AUX iajs-2475 138 30 essential essential ADJ iajs-2475 138 31 in in ADP iajs-2475 138 32 m m PROPN iajs-2475 138 33 , , PUNCT iajs-2475 138 34 but but CCONJ iajs-2475 138 35 m m NOUN iajs-2475 138 36 is be AUX iajs-2475 138 37 essential essential ADJ iajs-2475 138 38 in in ADP iajs-2475 138 39 injective injective ADJ iajs-2475 138 40 hull hull NOUN iajs-2475 138 41 of of ADP iajs-2475 138 42 m m PROPN iajs-2475 138 43 , , PUNCT iajs-2475 138 44 therefore therefore ADV iajs-2475 138 45 w⨁v w⨁v PROPN iajs-2475 138 46 is be AUX iajs-2475 138 47 essential essential ADJ iajs-2475 138 48 in in ADP iajs-2475 138 49 injective injective ADJ iajs-2475 138 50 hull hull NOUN iajs-2475 138 51 of of ADP iajs-2475 138 52 m m PROPN iajs-2475 138 53 , , PUNCT iajs-2475 138 54 then then ADV iajs-2475 138 55 e(m e(m PROPN iajs-2475 138 56 ) ) PUNCT iajs-2475 138 57 = = NOUN iajs-2475 138 58 e(w⨁v e(w⨁v NOUN iajs-2475 138 59 ) ) PUNCT iajs-2475 138 60 = = SYM iajs-2475 138 61 e(w e(w ADJ iajs-2475 138 62 ) ) PUNCT iajs-2475 138 63 ⨁e(v ⨁e(v NOUN iajs-2475 138 64 ) ) PUNCT iajs-2475 138 65 . . PUNCT iajs-2475 139 1 since since SCONJ iajs-2475 139 2 e(w e(w PROPN iajs-2475 139 3 ) ) PUNCT iajs-2475 139 4 is be AUX iajs-2475 139 5 direct direct ADJ iajs-2475 139 6 summand summand NOUN iajs-2475 139 7 of of ADP iajs-2475 139 8 e(m e(m PROPN iajs-2475 139 9 ) ) PUNCT iajs-2475 139 10 then then ADV iajs-2475 139 11 e(w)∩m e(w)∩m NOUN iajs-2475 139 12 has have VERB iajs-2475 139 13 a a DET iajs-2475 139 14 weakly weakly ADJ iajs-2475 139 15 supplement supplement NOUN iajs-2475 139 16 submodule submodule NOUN iajs-2475 139 17 in in ADP iajs-2475 139 18 m m PROPN iajs-2475 139 19 that that PRON iajs-2475 139 20 is be AUX iajs-2475 139 21 direct direct ADJ iajs-2475 139 22 summand summand NOUN iajs-2475 139 23 . . PUNCT iajs-2475 140 1 but but CCONJ iajs-2475 140 2 w w NOUN iajs-2475 140 3 is be AUX iajs-2475 140 4 essential essential ADJ iajs-2475 140 5 in in ADP iajs-2475 140 6 e(w e(w ADJ iajs-2475 140 7 ) ) PUNCT iajs-2475 140 8 and and CCONJ iajs-2475 140 9 m m PROPN iajs-2475 140 10 is be AUX iajs-2475 140 11 essential essential ADJ iajs-2475 140 12 in in ADP iajs-2475 140 13 m m PROPN iajs-2475 140 14 , , PUNCT iajs-2475 140 15 so so ADV iajs-2475 140 16 w w NOUN iajs-2475 140 17 = = NOUN iajs-2475 140 18 w∩m w∩m NOUN iajs-2475 140 19 is be AUX iajs-2475 140 20 essential essential ADJ iajs-2475 140 21 in in ADP iajs-2475 140 22 e(w)∩m e(w)∩m NOUN iajs-2475 140 23 which which PRON iajs-2475 140 24 has have VERB iajs-2475 140 25 a a DET iajs-2475 140 26 weakly weakly ADJ iajs-2475 140 27 supplement supplement NOUN iajs-2475 140 28 in in ADP iajs-2475 140 29 m m PROPN iajs-2475 140 30 that that PRON iajs-2475 140 31 is be AUX iajs-2475 140 32 direct direct ADJ iajs-2475 140 33 summand summand NOUN iajs-2475 140 34 . . PUNCT iajs-2475 141 1 hence hence ADV iajs-2475 141 2 m m PROPN iajs-2475 141 3 is be AUX iajs-2475 141 4 βŠ•-s βŠ•-s ADJ iajs-2475 141 5 - - PUNCT iajs-2475 141 6 extending extend VERB iajs-2475 141 7 module module NOUN iajs-2475 141 8 . . PUNCT iajs-2475 142 1 recall recall VERB iajs-2475 142 2 that that SCONJ iajs-2475 142 3 , , PUNCT iajs-2475 142 4 if if SCONJ iajs-2475 142 5 each each DET iajs-2475 142 6 submodule submodule NOUN iajs-2475 142 7 a a PRON iajs-2475 142 8 of of ADP iajs-2475 142 9 m m PROPN iajs-2475 142 10 , , PUNCT iajs-2475 142 11 there there PRON iajs-2475 142 12 is be VERB iajs-2475 142 13 an an DET iajs-2475 142 14 ideal ideal ADJ iajs-2475 142 15 j j NOUN iajs-2475 142 16 of of ADP iajs-2475 142 17 r r NOUN iajs-2475 142 18 such such ADJ iajs-2475 142 19 that that SCONJ iajs-2475 142 20 a a DET iajs-2475 142 21 = = NOUN iajs-2475 142 22 jm jm NOUN iajs-2475 142 23 , , PUNCT iajs-2475 142 24 then then ADV iajs-2475 142 25 a a DET iajs-2475 142 26 module module NOUN iajs-2475 142 27 m m VERB iajs-2475 142 28 is be AUX iajs-2475 142 29 said say VERB iajs-2475 142 30 to to PART iajs-2475 142 31 be be AUX iajs-2475 142 32 multiplication multiplication NOUN iajs-2475 143 1 [ [ X iajs-2475 143 2 8 8 NUM iajs-2475 143 3 ] ] PUNCT iajs-2475 143 4 . . PUNCT iajs-2475 144 1 following follow VERB iajs-2475 144 2 [ [ X iajs-2475 144 3 4 4 NUM iajs-2475 144 4 ] ] PUNCT iajs-2475 144 5 . . PUNCT iajs-2475 145 1 let let VERB iajs-2475 145 2 k k PRON iajs-2475 145 3 be be AUX iajs-2475 145 4 a a DET iajs-2475 145 5 submodule submodule NOUN iajs-2475 145 6 of of ADP iajs-2475 145 7 a a DET iajs-2475 145 8 module module NOUN iajs-2475 145 9 m m NOUN iajs-2475 145 10 and and CCONJ iajs-2475 145 11 r r NOUN iajs-2475 145 12 be be AUX iajs-2475 145 13 a a DET iajs-2475 145 14 ring ring NOUN iajs-2475 145 15 . . PUNCT iajs-2475 146 1 the the DET iajs-2475 146 2 ideal ideal NOUN iajs-2475 146 3 { { PUNCT iajs-2475 146 4 x∈r| x∈r| PROPN iajs-2475 146 5 xmβŠ†k}will xmβŠ†k}will X iajs-2475 146 6 be be AUX iajs-2475 146 7 denoted denote VERB iajs-2475 146 8 by by ADP iajs-2475 146 9 [ [ X iajs-2475 146 10 k k X iajs-2475 146 11 : : PUNCT iajs-2475 146 12 m m X iajs-2475 146 13 ] ] X iajs-2475 146 14 and and CCONJ iajs-2475 146 15 the the DET iajs-2475 146 16 annihilater annihilater NOUN iajs-2475 146 17 of of ADP iajs-2475 146 18 m m AUX iajs-2475 146 19 denoted denote VERB iajs-2475 146 20 by by ADP iajs-2475 146 21 annr annr NOUN iajs-2475 146 22 ( ( PUNCT iajs-2475 146 23 m m NOUN iajs-2475 146 24 ) ) PUNCT iajs-2475 146 25 is be AUX iajs-2475 146 26 annr(m)=[0 annr(m)=[0 PROPN iajs-2475 146 27 : : PUNCT iajs-2475 146 28 m m VERB iajs-2475 146 29 ] ] X iajs-2475 146 30 . . PUNCT iajs-2475 147 1 also also ADV iajs-2475 147 2 , , PUNCT iajs-2475 147 3 a a DET iajs-2475 147 4 module module NOUN iajs-2475 147 5 m m VERB iajs-2475 147 6 is be AUX iajs-2475 147 7 called call VERB iajs-2475 147 8 faithful faithful ADJ iajs-2475 147 9 if if SCONJ iajs-2475 147 10 annr(m)=0 annr(m)=0 ADJ iajs-2475 147 11 proposition proposition NOUN iajs-2475 147 12 ( ( PUNCT iajs-2475 147 13 12 12 NUM iajs-2475 147 14 ) ) PUNCT iajs-2475 147 15 let let VERB iajs-2475 147 16 r r PRON iajs-2475 147 17 be be AUX iajs-2475 147 18 a a DET iajs-2475 147 19 commutative commutative ADJ iajs-2475 147 20 ring ring NOUN iajs-2475 147 21 and and CCONJ iajs-2475 147 22 m m DET iajs-2475 147 23 a a DET iajs-2475 147 24 finitely finitely ADV iajs-2475 147 25 generated generate VERB iajs-2475 147 26 faithful faithful ADJ iajs-2475 147 27 multiplication multiplication NOUN iajs-2475 147 28 module module NOUN iajs-2475 147 29 . . PUNCT iajs-2475 148 1 then then ADV iajs-2475 148 2 r r NOUN iajs-2475 148 3 is be AUX iajs-2475 148 4 βŠ•-s βŠ•-s ADV iajs-2475 148 5 - - PUNCT iajs-2475 148 6 extending extend VERB iajs-2475 148 7 if if SCONJ iajs-2475 149 1 and and CCONJ iajs-2475 149 2 only only ADV iajs-2475 149 3 if if SCONJ iajs-2475 149 4 m m NOUN iajs-2475 149 5 is be AUX iajs-2475 149 6 βŠ•-s βŠ•-s ADJ iajs-2475 149 7 - - PUNCT iajs-2475 149 8 extending extending NOUN iajs-2475 149 9 . . PUNCT iajs-2475 150 1 proof proof NOUN iajs-2475 150 2 let let VERB iajs-2475 150 3 n n PRON iajs-2475 150 4 be be AUX iajs-2475 150 5 a a DET iajs-2475 150 6 closed closed ADJ iajs-2475 150 7 submodule submodule NOUN iajs-2475 150 8 in in ADP iajs-2475 150 9 m m PROPN iajs-2475 150 10 , , PUNCT iajs-2475 150 11 then then ADV iajs-2475 150 12 by by ADP iajs-2475 150 13 [ [ PUNCT iajs-2475 150 14 9 9 NUM iajs-2475 150 15 ] ] PUNCT iajs-2475 150 16 . . PUNCT iajs-2475 151 1 there there PRON iajs-2475 151 2 exists exist VERB iajs-2475 151 3 a a DET iajs-2475 151 4 closed closed ADJ iajs-2475 151 5 ideal ideal NOUN iajs-2475 151 6 i i PRON iajs-2475 151 7 in in ADP iajs-2475 151 8 r r NOUN iajs-2475 151 9 such such ADJ iajs-2475 151 10 that that SCONJ iajs-2475 151 11 im im NOUN iajs-2475 152 1 = = NOUN iajs-2475 152 2 n. n. NOUN iajs-2475 152 3 since since SCONJ iajs-2475 152 4 r r NOUN iajs-2475 152 5 is be AUX iajs-2475 152 6 βŠ•-s βŠ•-s ADJ iajs-2475 152 7 - - PUNCT iajs-2475 152 8 extending extending NOUN iajs-2475 152 9 , , PUNCT iajs-2475 152 10 so so CCONJ iajs-2475 152 11 i i PRON iajs-2475 152 12 has have VERB iajs-2475 152 13 a a DET iajs-2475 152 14 weakly weakly ADJ iajs-2475 152 15 supplement supplement NOUN iajs-2475 152 16 j j NOUN iajs-2475 152 17 which which PRON iajs-2475 152 18 is be AUX iajs-2475 152 19 direct direct ADJ iajs-2475 152 20 85 85 NUM iajs-2475 152 21 ibn ibn PROPN iajs-2475 152 22 al al PROPN iajs-2475 152 23 - - PUNCT iajs-2475 152 24 haitham haitham PROPN iajs-2475 152 25 jour jour X iajs-2475 152 26 . . PROPN iajs-2475 153 1 for for ADP iajs-2475 153 2 pure pure ADJ iajs-2475 153 3 & & CCONJ iajs-2475 153 4 appl appl PROPN iajs-2475 153 5 . . PUNCT iajs-2475 154 1 sci sci PROPN iajs-2475 154 2 . . PROPN iajs-2475 155 1 33 33 NUM iajs-2475 155 2 ( ( PUNCT iajs-2475 155 3 3 3 NUM iajs-2475 155 4 ) ) PUNCT iajs-2475 155 5 2020 2020 NUM iajs-2475 155 6 summand summand NOUN iajs-2475 155 7 in in ADP iajs-2475 155 8 r r NOUN iajs-2475 155 9 ( ( PUNCT iajs-2475 155 10 i.e i.e NOUN iajs-2475 155 11 ) ) PUNCT iajs-2475 155 12 r= r= ADJ iajs-2475 155 13 i+j i+j NUM iajs-2475 155 14 and and CCONJ iajs-2475 155 15 i∩jβ‰ͺr i∩jβ‰ͺr NOUN iajs-2475 155 16 . . PUNCT iajs-2475 156 1 let let VERB iajs-2475 156 2 k k NOUN iajs-2475 156 3 = = NOUN iajs-2475 156 4 mj mj X iajs-2475 156 5 where where SCONJ iajs-2475 156 6 k k PROPN iajs-2475 156 7 is be AUX iajs-2475 156 8 submodule submodule NOUN iajs-2475 156 9 in in ADP iajs-2475 156 10 m. m. NOUN iajs-2475 156 11 now now ADV iajs-2475 156 12 since since SCONJ iajs-2475 156 13 m m PROPN iajs-2475 156 14 is be AUX iajs-2475 156 15 multiplication multiplication NOUN iajs-2475 156 16 then then ADV iajs-2475 156 17 m m VERB iajs-2475 156 18 = = NOUN iajs-2475 156 19 rm= rm= ADJ iajs-2475 156 20 ( ( PUNCT iajs-2475 156 21 i+j)m i+j)m NOUN iajs-2475 156 22 = = NOUN iajs-2475 156 23 im+jm im+jm NOUN iajs-2475 156 24 = = NOUN iajs-2475 156 25 n+k n+k NUM iajs-2475 156 26 and and CCONJ iajs-2475 156 27 by by ADP iajs-2475 156 28 [ [ X iajs-2475 156 29 10 10 NUM iajs-2475 156 30 , , PUNCT iajs-2475 156 31 lemma lemma PROPN iajs-2475 156 32 ( ( PUNCT iajs-2475 156 33 4.11 4.11 NUM iajs-2475 156 34 ) ) PUNCT iajs-2475 156 35 ] ] PUNCT iajs-2475 156 36 . . PUNCT iajs-2475 157 1 n∩kβ‰ͺm n∩kβ‰ͺm NOUN iajs-2475 157 2 . . PUNCT iajs-2475 157 3 also also ADV iajs-2475 157 4 , , PUNCT iajs-2475 157 5 since since SCONJ iajs-2475 157 6 j j PROPN iajs-2475 157 7 is be AUX iajs-2475 157 8 direct direct ADJ iajs-2475 157 9 summand summand NOUN iajs-2475 157 10 , , PUNCT iajs-2475 157 11 so so ADV iajs-2475 157 12 we we PRON iajs-2475 157 13 have have VERB iajs-2475 157 14 j+s j+s NOUN iajs-2475 157 15 = = NOUN iajs-2475 157 16 r r NOUN iajs-2475 157 17 and and CCONJ iajs-2475 157 18 j∩s=0 j∩s=0 INTJ iajs-2475 157 19 where where SCONJ iajs-2475 157 20 s s VERB iajs-2475 157 21 is be AUX iajs-2475 157 22 ideal ideal ADJ iajs-2475 157 23 in in ADP iajs-2475 157 24 r r NOUN iajs-2475 157 25 and and CCONJ iajs-2475 157 26 such such ADJ iajs-2475 157 27 that that SCONJ iajs-2475 157 28 sm sm PROPN iajs-2475 157 29 = = PROPN iajs-2475 157 30 f. f. PROPN iajs-2475 157 31 now now ADV iajs-2475 157 32 since since SCONJ iajs-2475 157 33 m m PROPN iajs-2475 157 34 is be AUX iajs-2475 157 35 multiplication multiplication NOUN iajs-2475 157 36 , , PUNCT iajs-2475 157 37 then then ADV iajs-2475 157 38 m m VERB iajs-2475 157 39 = = NOUN iajs-2475 157 40 rm=(j+s)m rm=(j+s)m NOUN iajs-2475 157 41 = = SYM iajs-2475 157 42 jm+sm jm+sm NOUN iajs-2475 157 43 = = ADJ iajs-2475 157 44 k+f k+f NUM iajs-2475 157 45 and and CCONJ iajs-2475 157 46 k∩f k∩f PROPN iajs-2475 157 47 = = SYM iajs-2475 157 48 jm∩sm=(j∩s)m=0 jm∩sm=(j∩s)m=0 PROPN iajs-2475 157 49 . . PUNCT iajs-2475 158 1 thus thus ADV iajs-2475 158 2 we we PRON iajs-2475 158 3 have have VERB iajs-2475 158 4 k k PROPN iajs-2475 158 5 is be AUX iajs-2475 158 6 direct direct ADJ iajs-2475 158 7 summand summand NOUN iajs-2475 158 8 in in ADP iajs-2475 158 9 m. m. NOUN iajs-2475 158 10 hence hence ADV iajs-2475 158 11 m m VERB iajs-2475 158 12 is be AUX iajs-2475 158 13 βŠ•-s βŠ•-s ADJ iajs-2475 158 14 - - PUNCT iajs-2475 158 15 extending extending NOUN iajs-2475 158 16 . . PUNCT iajs-2475 159 1 conversely conversely ADV iajs-2475 159 2 , , PUNCT iajs-2475 159 3 let let VERB iajs-2475 159 4 i i PRON iajs-2475 159 5 be be AUX iajs-2475 159 6 a a DET iajs-2475 159 7 closed closed ADJ iajs-2475 159 8 ideal ideal NOUN iajs-2475 159 9 in in ADP iajs-2475 159 10 r r NOUN iajs-2475 159 11 , , PUNCT iajs-2475 159 12 then then ADV iajs-2475 159 13 by by ADP iajs-2475 159 14 [ [ X iajs-2475 159 15 10 10 NUM iajs-2475 159 16 , , PUNCT iajs-2475 159 17 lemma lemma PROPN iajs-2475 159 18 ( ( PUNCT iajs-2475 159 19 4.10 4.10 NUM iajs-2475 159 20 ) ) PUNCT iajs-2475 159 21 ] ] PUNCT iajs-2475 159 22 . . PUNCT iajs-2475 160 1 there there PRON iajs-2475 160 2 is be VERB iajs-2475 160 3 a a DET iajs-2475 160 4 closed closed ADJ iajs-2475 160 5 submodule submodule NOUN iajs-2475 160 6 n n PRON iajs-2475 160 7 be be VERB iajs-2475 160 8 in in ADP iajs-2475 160 9 m m PRON iajs-2475 160 10 such such ADJ iajs-2475 160 11 that that SCONJ iajs-2475 160 12 n n NOUN iajs-2475 160 13 = = NOUN iajs-2475 160 14 im im NOUN iajs-2475 160 15 . . PUNCT iajs-2475 161 1 since since SCONJ iajs-2475 161 2 m m PROPN iajs-2475 161 3 is be AUX iajs-2475 161 4 βŠ•-s βŠ•-s ADJ iajs-2475 161 5 - - PUNCT iajs-2475 161 6 extending extending ADJ iajs-2475 161 7 , , PUNCT iajs-2475 161 8 so so CCONJ iajs-2475 161 9 n n PROPN iajs-2475 161 10 has have VERB iajs-2475 161 11 a a DET iajs-2475 161 12 weakly weakly ADJ iajs-2475 161 13 supplement supplement NOUN iajs-2475 161 14 k k PROPN iajs-2475 161 15 in in ADP iajs-2475 161 16 m m PROPN iajs-2475 161 17 which which PRON iajs-2475 161 18 is be AUX iajs-2475 161 19 direct direct ADJ iajs-2475 161 20 summand summand NOUN iajs-2475 161 21 ( ( PUNCT iajs-2475 161 22 i.e i.e PROPN iajs-2475 161 23 ) ) PUNCT iajs-2475 161 24 m m PROPN iajs-2475 161 25 = = PROPN iajs-2475 161 26 k+n k+n X iajs-2475 161 27 and and CCONJ iajs-2475 161 28 k∩nβ‰ͺm k∩nβ‰ͺm PROPN iajs-2475 161 29 . . PUNCT iajs-2475 162 1 let let AUX iajs-2475 162 2 k k PROPN iajs-2475 162 3 = = NOUN iajs-2475 162 4 jm jm NOUN iajs-2475 162 5 where where SCONJ iajs-2475 162 6 j j PROPN iajs-2475 162 7 is be AUX iajs-2475 162 8 ideal ideal ADJ iajs-2475 162 9 in in ADP iajs-2475 162 10 r. r. PROPN iajs-2475 162 11 since since SCONJ iajs-2475 162 12 m m PROPN iajs-2475 162 13 is be AUX iajs-2475 162 14 multiplication multiplication NOUN iajs-2475 162 15 then then ADV iajs-2475 162 16 m m VERB iajs-2475 162 17 = = ADJ iajs-2475 162 18 n+k n+k NOUN iajs-2475 162 19 = = SYM iajs-2475 162 20 im+jm=(i+j)m im+jm=(i+j)m PROPN iajs-2475 162 21 = = SYM iajs-2475 162 22 rm rm PROPN iajs-2475 162 23 thus thus ADV iajs-2475 162 24 we we PRON iajs-2475 162 25 have have VERB iajs-2475 162 26 r r NOUN iajs-2475 162 27 = = NOUN iajs-2475 162 28 i+j i+j NUM iajs-2475 162 29 and and CCONJ iajs-2475 162 30 by by ADP iajs-2475 162 31 [ [ X iajs-2475 162 32 10 10 NUM iajs-2475 162 33 , , PUNCT iajs-2475 162 34 lemma lemma PROPN iajs-2475 162 35 ( ( PUNCT iajs-2475 162 36 4.11 4.11 NUM iajs-2475 162 37 ) ) PUNCT iajs-2475 162 38 ] ] PUNCT iajs-2475 162 39 . . PUNCT iajs-2475 163 1 i∩jβ‰ͺr i∩jβ‰ͺr NOUN iajs-2475 163 2 and and CCONJ iajs-2475 163 3 j j PROPN iajs-2475 163 4 is be AUX iajs-2475 163 5 direct direct ADJ iajs-2475 163 6 summand summand NOUN iajs-2475 163 7 in in ADP iajs-2475 163 8 r. r. PROPN iajs-2475 163 9 then then ADV iajs-2475 163 10 r r NOUN iajs-2475 163 11 is be AUX iajs-2475 163 12 βŠ•-s βŠ•-s NOUN iajs-2475 163 13 - - PUNCT iajs-2475 163 14 extending extending NOUN iajs-2475 163 15 . . PUNCT iajs-2475 164 1 recall recall NOUN iajs-2475 164 2 that that PRON iajs-2475 164 3 , , PUNCT iajs-2475 164 4 let let VERB iajs-2475 164 5 f f PRON iajs-2475 164 6 : : PUNCT iajs-2475 164 7 r⟢t r⟢t NOUN iajs-2475 164 8 be be VERB iajs-2475 164 9 a a DET iajs-2475 164 10 ring ring NOUN iajs-2475 164 11 hommorphism hommorphism NOUN iajs-2475 164 12 and and CCONJ iajs-2475 164 13 m m VERB iajs-2475 164 14 a a DET iajs-2475 164 15 right right ADJ iajs-2475 164 16 t t NOUN iajs-2475 164 17 - - PUNCT iajs-2475 164 18 module module NOUN iajs-2475 164 19 . . PUNCT iajs-2475 165 1 on on ADP iajs-2475 165 2 can can AUX iajs-2475 165 3 be be AUX iajs-2475 165 4 defined define VERB iajs-2475 165 5 to to ADP iajs-2475 165 6 as as ADP iajs-2475 165 7 a a DET iajs-2475 165 8 right right ADJ iajs-2475 165 9 r r NOUN iajs-2475 165 10 - - PUNCT iajs-2475 165 11 module module NOUN iajs-2475 165 12 by by ADP iajs-2475 165 13 mr mr PROPN iajs-2475 165 14 = = PROPN iajs-2475 165 15 m m PROPN iajs-2475 165 16 f f X iajs-2475 165 17 ( ( PUNCT iajs-2475 165 18 r r NOUN iajs-2475 165 19 ) ) PUNCT iajs-2475 165 20 for for ADP iajs-2475 165 21 all all DET iajs-2475 165 22 m∈m m∈m NOUN iajs-2475 165 23 and and CCONJ iajs-2475 165 24 r∈r r∈r NOUN iajs-2475 165 25 . . PUNCT iajs-2475 166 1 moreover moreover ADV iajs-2475 166 2 , , PUNCT iajs-2475 166 3 if if SCONJ iajs-2475 166 4 f f PROPN iajs-2475 166 5 is be AUX iajs-2475 166 6 an an DET iajs-2475 166 7 epimorphism epimorphism NOUN iajs-2475 166 8 and and CCONJ iajs-2475 166 9 m m NOUN iajs-2475 166 10 is be AUX iajs-2475 166 11 a a DET iajs-2475 166 12 right right ADJ iajs-2475 166 13 r r NOUN iajs-2475 166 14 - - NOUN iajs-2475 166 15 module module NOUN iajs-2475 166 16 such such ADJ iajs-2475 166 17 that that DET iajs-2475 166 18 kerf kerf NOUN iajs-2475 166 19 βŠ† βŠ† NUM iajs-2475 166 20 r(m r(m NOUN iajs-2475 166 21 ) ) PUNCT iajs-2475 166 22 , , PUNCT iajs-2475 166 23 so so ADV iajs-2475 166 24 also also ADV iajs-2475 166 25 can can AUX iajs-2475 166 26 also also ADV iajs-2475 166 27 be be AUX iajs-2475 166 28 define define VERB iajs-2475 166 29 to to PART iajs-2475 166 30 be be AUX iajs-2475 166 31 a a DET iajs-2475 166 32 right right ADJ iajs-2475 166 33 t t NOUN iajs-2475 166 34 - - PUNCT iajs-2475 166 35 module module NOUN iajs-2475 166 36 by by ADP iajs-2475 166 37 mt mt PROPN iajs-2475 166 38 = = PROPN iajs-2475 166 39 mr mr PROPN iajs-2475 166 40 , , PUNCT iajs-2475 167 1 where where SCONJ iajs-2475 167 2 f f PROPN iajs-2475 167 3 ( ( PUNCT iajs-2475 167 4 r)=t r)=t PROPN iajs-2475 167 5 . . PUNCT iajs-2475 167 6 we we PRON iajs-2475 167 7 denote denote VERB iajs-2475 167 8 by by ADP iajs-2475 167 9 𝑀𝑇 𝑀𝑇 PROPN iajs-2475 167 10 , , PUNCT iajs-2475 167 11 𝑀𝑅 𝑀𝑅 PROPN iajs-2475 167 12 then then ADV iajs-2475 167 13 m m VERB iajs-2475 167 14 is be AUX iajs-2475 167 15 a a DET iajs-2475 167 16 right right ADJ iajs-2475 167 17 t t NOUN iajs-2475 167 18 - - PUNCT iajs-2475 167 19 module module NOUN iajs-2475 167 20 , , PUNCT iajs-2475 167 21 right right ADJ iajs-2475 167 22 r r NOUN iajs-2475 167 23 - - PUNCT iajs-2475 167 24 module module NOUN iajs-2475 167 25 [ [ X iajs-2475 167 26 10 10 NUM iajs-2475 167 27 ] ] PUNCT iajs-2475 167 28 . . PUNCT iajs-2475 168 1 proposition proposition NOUN iajs-2475 168 2 ( ( PUNCT iajs-2475 168 3 13 13 NUM iajs-2475 168 4 ) ) PUNCT iajs-2475 168 5 let let VERB iajs-2475 168 6 f f PRON iajs-2475 168 7 : : PUNCT iajs-2475 168 8 r⟢t r⟢t NOUN iajs-2475 168 9 be be VERB iajs-2475 168 10 a a DET iajs-2475 168 11 ring ring NOUN iajs-2475 168 12 epimorphism epimorphism NOUN iajs-2475 168 13 and and CCONJ iajs-2475 168 14 m m PROPN iajs-2475 168 15 a a DET iajs-2475 168 16 right right ADJ iajs-2475 168 17 r r NOUN iajs-2475 168 18 - - PUNCT iajs-2475 168 19 module module NOUN iajs-2475 168 20 with with ADP iajs-2475 168 21 kerf kerf NOUN iajs-2475 168 22 βŠ† βŠ† NUM iajs-2475 168 23 r(m r(m NOUN iajs-2475 168 24 ) ) PUNCT iajs-2475 168 25 . . PUNCT iajs-2475 169 1 then then ADV iajs-2475 169 2 𝑀𝑅 𝑀𝑅 PROPN iajs-2475 169 3 is be AUX iajs-2475 169 4 βŠ•-s βŠ•-s ADV iajs-2475 169 5 - - PUNCT iajs-2475 169 6 extending extend VERB iajs-2475 169 7 if if SCONJ iajs-2475 170 1 and and CCONJ iajs-2475 170 2 only only ADV iajs-2475 170 3 if if SCONJ iajs-2475 170 4 𝑀𝑇 𝑀𝑇 PROPN iajs-2475 170 5 is be AUX iajs-2475 170 6 βŠ•-s βŠ•-s NOUN iajs-2475 170 7 - - PUNCT iajs-2475 170 8 extending extending NOUN iajs-2475 170 9 . . PUNCT iajs-2475 171 1 proof proof NOUN iajs-2475 171 2 let let VERB iajs-2475 171 3 𝐴𝑇 𝐴𝑇 PROPN iajs-2475 171 4 be be AUX iajs-2475 171 5 a a DET iajs-2475 171 6 closed closed ADJ iajs-2475 171 7 submodule submodule NOUN iajs-2475 171 8 of of ADP iajs-2475 171 9 𝑀𝑇 𝑀𝑇 PROPN iajs-2475 171 10 , , PUNCT iajs-2475 171 11 then then ADV iajs-2475 171 12 𝐴𝑅 𝐴𝑅 PROPN iajs-2475 171 13 is be AUX iajs-2475 171 14 closed close VERB iajs-2475 171 15 submodule submodule NOUN iajs-2475 171 16 in in ADP iajs-2475 171 17 𝑀𝑅 𝑀𝑅 PROPN iajs-2475 171 18 , , PUNCT iajs-2475 171 19 since since SCONJ iajs-2475 171 20 𝑀𝑅 𝑀𝑅 PROPN iajs-2475 171 21 is be AUX iajs-2475 171 22 βŠ•-s βŠ•-s ADJ iajs-2475 171 23 - - PUNCT iajs-2475 171 24 extending extending NOUN iajs-2475 171 25 , , PUNCT iajs-2475 171 26 so so ADV iajs-2475 171 27 𝐴𝑅 𝐴𝑅 PROPN iajs-2475 171 28 has have VERB iajs-2475 171 29 a a DET iajs-2475 171 30 weakly weakly ADJ iajs-2475 171 31 supplements supplement NOUN iajs-2475 171 32 𝐡𝑅 𝐡𝑅 PROPN iajs-2475 171 33 that that PRON iajs-2475 171 34 is be AUX iajs-2475 171 35 direct direct ADJ iajs-2475 171 36 summand summand NOUN iajs-2475 171 37 . . PUNCT iajs-2475 172 1 so so ADV iajs-2475 172 2 𝐡𝑅 𝐡𝑅 PROPN iajs-2475 172 3 we we PRON iajs-2475 172 4 can can AUX iajs-2475 172 5 define define VERB iajs-2475 172 6 to to PART iajs-2475 172 7 be be AUX iajs-2475 172 8 t t NOUN iajs-2475 172 9 - - PUNCT iajs-2475 172 10 module module NOUN iajs-2475 172 11 by by ADP iajs-2475 172 12 mt mt PROPN iajs-2475 172 13 = = PROPN iajs-2475 172 14 mr mr PROPN iajs-2475 172 15 , , PUNCT iajs-2475 172 16 where where SCONJ iajs-2475 172 17 f f PROPN iajs-2475 172 18 ( ( PUNCT iajs-2475 172 19 r)=t r)=t PROPN iajs-2475 172 20 . . PUNCT iajs-2475 173 1 thus thus ADV iajs-2475 173 2 𝐴𝑇 𝐴𝑇 PROPN iajs-2475 173 3 has have VERB iajs-2475 173 4 a a DET iajs-2475 173 5 weakly weakly ADJ iajs-2475 173 6 supplements supplement NOUN iajs-2475 173 7 𝐡𝑇which 𝐡𝑇which NOUN iajs-2475 173 8 is be AUX iajs-2475 173 9 direct direct ADJ iajs-2475 173 10 summand summand NOUN iajs-2475 173 11 in in ADP iajs-2475 173 12 𝑀𝑇. 𝑀𝑇. NOUN iajs-2475 173 13 then then ADV iajs-2475 173 14 𝑀𝑇 𝑀𝑇 PROPN iajs-2475 173 15 is be AUX iajs-2475 173 16 βŠ•-s βŠ•-s ADJ iajs-2475 173 17 - - PUNCT iajs-2475 173 18 extending extending NOUN iajs-2475 173 19 . . PUNCT iajs-2475 174 1 conversely conversely ADV iajs-2475 174 2 , , PUNCT iajs-2475 174 3 let let VERB iajs-2475 174 4 𝐴𝑅 𝐴𝑅 PROPN iajs-2475 174 5 be be AUX iajs-2475 174 6 a a DET iajs-2475 174 7 closed closed ADJ iajs-2475 174 8 submodule submodule NOUN iajs-2475 174 9 of of ADP iajs-2475 174 10 𝑀𝑅 𝑀𝑅 PROPN iajs-2475 174 11 , , PUNCT iajs-2475 174 12 then then ADV iajs-2475 174 13 𝐴𝑇 𝐴𝑇 PROPN iajs-2475 174 14 is be AUX iajs-2475 174 15 closed close VERB iajs-2475 174 16 submodule submodule NOUN iajs-2475 174 17 in in ADP iajs-2475 174 18 𝑀𝑇 𝑀𝑇 PROPN iajs-2475 174 19 , , PUNCT iajs-2475 174 20 since since SCONJ iajs-2475 174 21 𝑀𝑇 𝑀𝑇 PROPN iajs-2475 174 22 is be AUX iajs-2475 174 23 βŠ•-s βŠ•-s ADJ iajs-2475 174 24 - - PUNCT iajs-2475 174 25 extending extending NOUN iajs-2475 174 26 , , PUNCT iajs-2475 174 27 so so ADV iajs-2475 174 28 𝐴𝑇 𝐴𝑇 PROPN iajs-2475 174 29 has have VERB iajs-2475 174 30 a a DET iajs-2475 174 31 weakly weakly ADJ iajs-2475 174 32 suplement suplement NOUN iajs-2475 174 33 𝐡𝑇 𝐡𝑇 PROPN iajs-2475 174 34 that that PRON iajs-2475 174 35 is be AUX iajs-2475 174 36 direct direct ADJ iajs-2475 174 37 summand summand NOUN iajs-2475 174 38 . . PUNCT iajs-2475 175 1 so so ADV iajs-2475 175 2 𝐡𝑇 𝐡𝑇 PROPN iajs-2475 175 3 we we PRON iajs-2475 175 4 can can AUX iajs-2475 175 5 define define VERB iajs-2475 175 6 to to PART iajs-2475 175 7 be be AUX iajs-2475 175 8 rmodule rmodule NOUN iajs-2475 175 9 by by ADP iajs-2475 175 10 mr mr PROPN iajs-2475 175 11 = = PROPN iajs-2475 175 12 mf mf X iajs-2475 175 13 ( ( PUNCT iajs-2475 175 14 r r NOUN iajs-2475 175 15 ) ) PUNCT iajs-2475 175 16 for for ADP iajs-2475 175 17 each each DET iajs-2475 175 18 r∈r r∈r NOUN iajs-2475 175 19 and and CCONJ iajs-2475 175 20 m∈m m∈m NOUN iajs-2475 175 21 . . PUNCT iajs-2475 176 1 thus thus ADV iajs-2475 176 2 𝐴𝑅 𝐴𝑅 PROPN iajs-2475 176 3 has have VERB iajs-2475 176 4 a a DET iajs-2475 176 5 weakly weakly ADJ iajs-2475 176 6 supplement supplement NOUN iajs-2475 176 7 𝐡𝑅which 𝐡𝑅which PROPN iajs-2475 176 8 is be AUX iajs-2475 176 9 direct direct ADJ iajs-2475 176 10 summand summand NOUN iajs-2475 176 11 in in ADP iajs-2475 176 12 𝑀𝑅. 𝑀𝑅. NOUN iajs-2475 176 13 hence hence ADV iajs-2475 176 14 𝑀𝑅 𝑀𝑅 PROPN iajs-2475 176 15 is be AUX iajs-2475 176 16 βŠ•-s βŠ•-s NOUN iajs-2475 176 17 - - PUNCT iajs-2475 176 18 extending extending NOUN iajs-2475 176 19 . . PUNCT iajs-2475 177 1 it it PRON iajs-2475 177 2 is be AUX iajs-2475 177 3 known know VERB iajs-2475 177 4 that that SCONJ iajs-2475 177 5 , , PUNCT iajs-2475 177 6 a a DET iajs-2475 177 7 factor factor NOUN iajs-2475 177 8 module module NOUN iajs-2475 177 9 of of ADP iajs-2475 177 10 βŠ•-supplemented βŠ•-supplemente VERB iajs-2475 177 11 need need NOUN iajs-2475 177 12 not not PART iajs-2475 177 13 necessary necessary ADJ iajs-2475 177 14 βŠ•supplemented βŠ•supplemente VERB iajs-2475 177 15 [ [ X iajs-2475 177 16 7 7 NUM iajs-2475 177 17 ] ] PUNCT iajs-2475 177 18 . . PUNCT iajs-2475 178 1 also also ADV iajs-2475 178 2 , , PUNCT iajs-2475 178 3 in in ADP iajs-2475 178 4 βŠ•-s βŠ•-s ADJ iajs-2475 178 5 - - PUNCT iajs-2475 178 6 extending extend VERB iajs-2475 178 7 module module NOUN iajs-2475 178 8 is be AUX iajs-2475 178 9 not not PART iajs-2475 178 10 verified verify VERB iajs-2475 178 11 . . PUNCT iajs-2475 179 1 thus thus ADV iajs-2475 179 2 in in ADP iajs-2475 179 3 the the DET iajs-2475 179 4 following following ADJ iajs-2475 179 5 result result NOUN iajs-2475 179 6 we we PRON iajs-2475 179 7 obtain obtain VERB iajs-2475 179 8 a a DET iajs-2475 179 9 condition condition NOUN iajs-2475 179 10 under under ADP iajs-2475 179 11 it it PRON iajs-2475 179 12 a a DET iajs-2475 179 13 factor factor NOUN iajs-2475 179 14 module module NOUN iajs-2475 179 15 of of ADP iajs-2475 179 16 βŠ•-s βŠ•-s ADJ iajs-2475 179 17 - - PUNCT iajs-2475 179 18 extending extend VERB iajs-2475 179 19 module module NOUN iajs-2475 179 20 is be AUX iajs-2475 179 21 βŠ•-sextending βŠ•-sextende VERB iajs-2475 179 22 . . PUNCT iajs-2475 180 1 proposition proposition NOUN iajs-2475 180 2 ( ( PUNCT iajs-2475 180 3 14 14 NUM iajs-2475 180 4 ) ) PUNCT iajs-2475 180 5 let let VERB iajs-2475 180 6 m m PRON iajs-2475 180 7 be be AUX iajs-2475 180 8 a a DET iajs-2475 180 9 βŠ•-s βŠ•-s ADJ iajs-2475 180 10 - - PUNCT iajs-2475 180 11 extending extend VERB iajs-2475 180 12 module module NOUN iajs-2475 180 13 . . PUNCT iajs-2475 181 1 then then ADV iajs-2475 181 2 any any DET iajs-2475 181 3 nonsingular nonsingular ADJ iajs-2475 181 4 ( ( PUNCT iajs-2475 181 5 epimorphic epimorphic ADJ iajs-2475 181 6 ) ) PUNCT iajs-2475 181 7 image image NOUN iajs-2475 181 8 of of ADP iajs-2475 181 9 m m PROPN iajs-2475 181 10 is be AUX iajs-2475 181 11 βŠ•-s βŠ•-s ADJ iajs-2475 181 12 - - PUNCT iajs-2475 181 13 extending extend VERB iajs-2475 181 14 module module NOUN iajs-2475 181 15 . . PUNCT iajs-2475 182 1 proof proof NOUN iajs-2475 182 2 let let VERB iajs-2475 182 3 f f PRON iajs-2475 182 4 : : PUNCT iajs-2475 182 5 mβ†’n mβ†’n NOUN iajs-2475 182 6 be be AUX iajs-2475 182 7 an an DET iajs-2475 182 8 epimorphism epimorphism NOUN iajs-2475 182 9 mapping mapping NOUN iajs-2475 182 10 and and CCONJ iajs-2475 182 11 let let VERB iajs-2475 182 12 k k PRON iajs-2475 182 13 be be AUX iajs-2475 182 14 a a DET iajs-2475 182 15 closed closed ADJ iajs-2475 182 16 submodule submodule NOUN iajs-2475 182 17 of of ADP iajs-2475 182 18 n n CCONJ iajs-2475 182 19 , , PUNCT iajs-2475 182 20 so so ADV iajs-2475 182 21 h=π‘“βˆ’1(k h=π‘“βˆ’1(k NOUN iajs-2475 182 22 ) ) PUNCT iajs-2475 182 23 is be AUX iajs-2475 182 24 a a DET iajs-2475 182 25 closed closed ADJ iajs-2475 182 26 submodule submodule NOUN iajs-2475 182 27 of of ADP iajs-2475 182 28 m. m. NOUN iajs-2475 182 29 since since SCONJ iajs-2475 182 30 m m PROPN iajs-2475 182 31 is be AUX iajs-2475 182 32 βŠ•-s βŠ•-s ADJ iajs-2475 182 33 - - PUNCT iajs-2475 182 34 extending extend VERB iajs-2475 182 35 module module NOUN iajs-2475 182 36 , , PUNCT iajs-2475 182 37 so so SCONJ iajs-2475 182 38 h h NOUN iajs-2475 182 39 has have VERB iajs-2475 182 40 a a DET iajs-2475 182 41 weakly weakly ADJ iajs-2475 182 42 supplement supplement NOUN iajs-2475 182 43 submodule submodule NOUN iajs-2475 182 44 w w PROPN iajs-2475 182 45 of of ADP iajs-2475 182 46 m m PRON iajs-2475 182 47 which which PRON iajs-2475 182 48 is be AUX iajs-2475 182 49 direct direct ADJ iajs-2475 182 50 summand summand NOUN iajs-2475 182 51 . . PUNCT iajs-2475 183 1 then then ADV iajs-2475 183 2 m m VERB iajs-2475 183 3 = = X iajs-2475 183 4 h+w h+w X iajs-2475 183 5 and and CCONJ iajs-2475 183 6 h∩wβ‰ͺm h∩wβ‰ͺm PROPN iajs-2475 183 7 . . PUNCT iajs-2475 184 1 now now ADV iajs-2475 184 2 n= n= ADJ iajs-2475 184 3 f f PROPN iajs-2475 184 4 ( ( PUNCT iajs-2475 184 5 m)=f m)=f X iajs-2475 184 6 ( ( PUNCT iajs-2475 184 7 h+w)=f h+w)=f X iajs-2475 184 8 ( ( PUNCT iajs-2475 184 9 h)+f h)+f PROPN iajs-2475 184 10 ( ( PUNCT iajs-2475 184 11 w)=k+ w)=k+ PROPN iajs-2475 184 12 f f PROPN iajs-2475 184 13 ( ( PUNCT iajs-2475 184 14 w w NOUN iajs-2475 184 15 ) ) PUNCT iajs-2475 184 16 and and CCONJ iajs-2475 184 17 ( ( PUNCT iajs-2475 184 18 since since SCONJ iajs-2475 184 19 f f PROPN iajs-2475 184 20 is be AUX iajs-2475 184 21 epimorphism epimorphism NOUN iajs-2475 184 22 and and CCONJ iajs-2475 185 1 ker ker PROPN iajs-2475 185 2 f f PROPN iajs-2475 185 3 βŠ†h βŠ†h PROPN iajs-2475 185 4 ) ) PUNCT iajs-2475 185 5 f f PROPN iajs-2475 185 6 ( ( PUNCT iajs-2475 185 7 h∩w)= h∩w)= PROPN iajs-2475 185 8 f f PROPN iajs-2475 185 9 ( ( PUNCT iajs-2475 185 10 h h NOUN iajs-2475 185 11 ) ) PUNCT iajs-2475 185 12 ∩f ∩f NOUN iajs-2475 185 13 ( ( PUNCT iajs-2475 185 14 w w NOUN iajs-2475 185 15 ) ) PUNCT iajs-2475 185 16 β‰ͺf β‰ͺf NOUN iajs-2475 185 17 ( ( PUNCT iajs-2475 185 18 m m NOUN iajs-2475 185 19 ) ) PUNCT iajs-2475 185 20 , , PUNCT iajs-2475 185 21 so so ADV iajs-2475 185 22 we we PRON iajs-2475 185 23 have have VERB iajs-2475 185 24 k∩ k∩ PROPN iajs-2475 185 25 f f PROPN iajs-2475 185 26 ( ( PUNCT iajs-2475 185 27 w w PROPN iajs-2475 185 28 ) ) PUNCT iajs-2475 185 29 β‰ͺ β‰ͺ PUNCT iajs-2475 186 1 𝑓 𝑓 PROPN iajs-2475 186 2 ( ( PUNCT iajs-2475 186 3 𝑀 𝑀 PROPN iajs-2475 186 4 ) ) PUNCT iajs-2475 187 1 = = X iajs-2475 187 2 n. n. NOUN iajs-2475 187 3 then then ADV iajs-2475 187 4 f f PROPN iajs-2475 187 5 ( ( PUNCT iajs-2475 187 6 w w NOUN iajs-2475 187 7 ) ) PUNCT iajs-2475 187 8 is be AUX iajs-2475 187 9 weakly weakly ADJ iajs-2475 187 10 supplement supplement NOUN iajs-2475 187 11 of of ADP iajs-2475 187 12 k k PROPN iajs-2475 187 13 in in ADP iajs-2475 187 14 n. n. PROPN iajs-2475 187 15 now now ADV iajs-2475 187 16 to to PART iajs-2475 187 17 show show VERB iajs-2475 187 18 that that SCONJ iajs-2475 187 19 f f PROPN iajs-2475 187 20 ( ( PUNCT iajs-2475 187 21 w w NOUN iajs-2475 187 22 ) ) PUNCT iajs-2475 187 23 is be AUX iajs-2475 187 24 direct direct ADJ iajs-2475 187 25 summand summand NOUN iajs-2475 187 26 of of ADP iajs-2475 187 27 n. n. PROPN iajs-2475 187 28 let let VERB iajs-2475 187 29 m m VERB iajs-2475 187 30 = = NOUN iajs-2475 187 31 w+f w+f NUM iajs-2475 187 32 and and CCONJ iajs-2475 187 33 w∩f=0 w∩f=0 VERB iajs-2475 187 34 where where SCONJ iajs-2475 187 35 f f PROPN iajs-2475 187 36 is be AUX iajs-2475 187 37 submodule submodule NOUN iajs-2475 187 38 of of ADP iajs-2475 187 39 m m PROPN iajs-2475 187 40 , , PUNCT iajs-2475 187 41 so so ADV iajs-2475 187 42 n= n= ADJ iajs-2475 187 43 f f PROPN iajs-2475 187 44 ( ( PUNCT iajs-2475 187 45 m)= m)= NOUN iajs-2475 187 46 f f PROPN iajs-2475 187 47 ( ( PUNCT iajs-2475 187 48 f+w)=f f+w)=f ADJ iajs-2475 187 49 ( ( PUNCT iajs-2475 187 50 f)+f f)+f NOUN iajs-2475 187 51 ( ( PUNCT iajs-2475 187 52 w w NOUN iajs-2475 187 53 ) ) PUNCT iajs-2475 187 54 and and CCONJ iajs-2475 187 55 86 86 NUM iajs-2475 187 56 ibn ibn PROPN iajs-2475 187 57 al al PROPN iajs-2475 187 58 - - PUNCT iajs-2475 187 59 haitham haitham PROPN iajs-2475 187 60 jour jour X iajs-2475 187 61 . . PROPN iajs-2475 188 1 for for ADP iajs-2475 188 2 pure pure ADJ iajs-2475 188 3 & & CCONJ iajs-2475 188 4 appl appl PROPN iajs-2475 188 5 . . PUNCT iajs-2475 189 1 sci sci PROPN iajs-2475 189 2 . . PROPN iajs-2475 190 1 33 33 NUM iajs-2475 190 2 ( ( PUNCT iajs-2475 190 3 3 3 NUM iajs-2475 190 4 ) ) PUNCT iajs-2475 190 5 2020 2020 NUM iajs-2475 190 6 f f X iajs-2475 190 7 ( ( PUNCT iajs-2475 190 8 f∩w)= f∩w)= PROPN iajs-2475 190 9 f f PROPN iajs-2475 190 10 ( ( PUNCT iajs-2475 190 11 f f X iajs-2475 190 12 ) ) PUNCT iajs-2475 190 13 ∩f ∩f NOUN iajs-2475 190 14 ( ( PUNCT iajs-2475 190 15 w)=0 w)=0 PROPN iajs-2475 190 16 . . PUNCT iajs-2475 191 1 then then ADV iajs-2475 191 2 f f PROPN iajs-2475 191 3 ( ( PUNCT iajs-2475 191 4 w w NOUN iajs-2475 191 5 ) ) PUNCT iajs-2475 191 6 is be AUX iajs-2475 191 7 direct direct ADJ iajs-2475 191 8 summand summand NOUN iajs-2475 191 9 of of ADP iajs-2475 191 10 n n PROPN iajs-2475 191 11 and and CCONJ iajs-2475 191 12 thus thus ADV iajs-2475 191 13 n n PRON iajs-2475 191 14 is be AUX iajs-2475 191 15 βŠ•-s βŠ•-s ADJ iajs-2475 191 16 - - PUNCT iajs-2475 191 17 extending extend VERB iajs-2475 191 18 module module NOUN iajs-2475 191 19 . . PUNCT iajs-2475 192 1 proposition proposition NOUN iajs-2475 192 2 ( ( PUNCT iajs-2475 192 3 15 15 NUM iajs-2475 192 4 ) ) PUNCT iajs-2475 192 5 if if SCONJ iajs-2475 192 6 a a DET iajs-2475 192 7 module module NOUN iajs-2475 192 8 𝑀1 𝑀1 ADV iajs-2475 192 9 is be AUX iajs-2475 192 10 βŠ•-s βŠ•-s ADJ iajs-2475 192 11 - - PUNCT iajs-2475 192 12 extending extend VERB iajs-2475 192 13 module module NOUN iajs-2475 192 14 and and CCONJ iajs-2475 192 15 𝑀1 𝑀1 PROPN iajs-2475 192 16 β‰… β‰… PROPN iajs-2475 192 17 𝑀2 𝑀2 PROPN iajs-2475 192 18 , , PUNCT iajs-2475 192 19 then then ADV iajs-2475 192 20 𝑀2 𝑀2 PROPN iajs-2475 192 21 is be AUX iajs-2475 192 22 βŠ•-s βŠ•-s ADJ iajs-2475 192 23 - - PUNCT iajs-2475 192 24 extending extend VERB iajs-2475 192 25 module module NOUN iajs-2475 192 26 . . PUNCT iajs-2475 193 1 proof proof NOUN iajs-2475 193 2 let let VERB iajs-2475 193 3 f f X iajs-2475 193 4 : : PUNCT iajs-2475 193 5 𝑀1→𝑀2 𝑀1→𝑀2 ADJ iajs-2475 193 6 be be VERB iajs-2475 193 7 an an DET iajs-2475 193 8 isomorphic isomorphic ADJ iajs-2475 193 9 image image NOUN iajs-2475 193 10 and and CCONJ iajs-2475 193 11 𝑀1is 𝑀1i NOUN iajs-2475 193 12 βŠ•-s βŠ•-s ADJ iajs-2475 193 13 - - PUNCT iajs-2475 193 14 extending extend VERB iajs-2475 193 15 module module NOUN iajs-2475 193 16 . . PUNCT iajs-2475 194 1 let let VERB iajs-2475 194 2 w w NOUN iajs-2475 194 3 is be AUX iajs-2475 194 4 a a DET iajs-2475 194 5 submodule submodule NOUN iajs-2475 194 6 of of ADP iajs-2475 194 7 𝑀2 𝑀2 PROPN iajs-2475 194 8 then then ADV iajs-2475 194 9 we we PRON iajs-2475 194 10 have have VERB iajs-2475 194 11 π‘“βˆ’1(w π‘“βˆ’1(w NOUN iajs-2475 194 12 ) ) PUNCT iajs-2475 194 13 is be AUX iajs-2475 194 14 a a DET iajs-2475 194 15 submodule submodule NOUN iajs-2475 194 16 of of ADP iajs-2475 194 17 𝑀1 𝑀1 PROPN iajs-2475 194 18 . . PUNCT iajs-2475 195 1 since since SCONJ iajs-2475 195 2 𝑀1 𝑀1 PROPN iajs-2475 195 3 is be AUX iajs-2475 195 4 βŠ•-s βŠ•-s ADJ iajs-2475 195 5 - - PUNCT iajs-2475 195 6 extending extend VERB iajs-2475 195 7 module module NOUN iajs-2475 195 8 , , PUNCT iajs-2475 195 9 so so ADV iajs-2475 195 10 π‘“βˆ’1 π‘“βˆ’1 NUM iajs-2475 195 11 ( ( PUNCT iajs-2475 195 12 w w NOUN iajs-2475 195 13 ) ) PUNCT iajs-2475 195 14 is be AUX iajs-2475 195 15 essential essential ADJ iajs-2475 195 16 in in ADP iajs-2475 195 17 v v NUM iajs-2475 195 18 where where SCONJ iajs-2475 195 19 v v NOUN iajs-2475 195 20 has have VERB iajs-2475 195 21 a a DET iajs-2475 195 22 weakly weakly ADJ iajs-2475 195 23 supplement supplement NOUN iajs-2475 195 24 submodule submodule NOUN iajs-2475 195 25 of of ADP iajs-2475 195 26 𝑀1 𝑀1 PROPN iajs-2475 195 27 which which PRON iajs-2475 195 28 is be AUX iajs-2475 195 29 direct direct ADJ iajs-2475 195 30 summand summand NOUN iajs-2475 195 31 , , PUNCT iajs-2475 195 32 so so ADV iajs-2475 195 33 w= w= ADV iajs-2475 195 34 f f X iajs-2475 195 35 ( ( PUNCT iajs-2475 195 36 π‘“βˆ’1 π‘“βˆ’1 PROPN iajs-2475 195 37 ( ( PUNCT iajs-2475 195 38 w w NOUN iajs-2475 195 39 ) ) PUNCT iajs-2475 195 40 ) ) PUNCT iajs-2475 195 41 is be AUX iajs-2475 195 42 essential essential ADJ iajs-2475 195 43 in in ADP iajs-2475 195 44 f f PROPN iajs-2475 195 45 ( ( PUNCT iajs-2475 195 46 v v NOUN iajs-2475 195 47 ) ) PUNCT iajs-2475 195 48 . . PUNCT iajs-2475 196 1 since since SCONJ iajs-2475 196 2 v v NOUN iajs-2475 196 3 has have VERB iajs-2475 196 4 a a DET iajs-2475 196 5 weakly weakly ADJ iajs-2475 196 6 supplement supplement NOUN iajs-2475 196 7 submodule submodule NOUN iajs-2475 196 8 in in ADP iajs-2475 196 9 𝑀1 𝑀1 PROPN iajs-2475 196 10 then then ADV iajs-2475 196 11 there there PRON iajs-2475 196 12 exists exist VERB iajs-2475 196 13 h h NOUN iajs-2475 196 14 is be AUX iajs-2475 196 15 a a DET iajs-2475 196 16 submodule submodule NOUN iajs-2475 196 17 of of ADP iajs-2475 196 18 𝑀1 𝑀1 PROPN iajs-2475 196 19 such such ADJ iajs-2475 196 20 that that SCONJ iajs-2475 196 21 𝑀1 𝑀1 PROPN iajs-2475 196 22 = = X iajs-2475 196 23 v+h v+h X iajs-2475 196 24 and and CCONJ iajs-2475 196 25 v∩h v∩h ADV iajs-2475 196 26 β‰ͺ β‰ͺ ADJ iajs-2475 196 27 𝑀1 𝑀1 PROPN iajs-2475 196 28 . . PUNCT iajs-2475 197 1 then then ADV iajs-2475 197 2 f f PROPN iajs-2475 197 3 ( ( PUNCT iajs-2475 197 4 𝑀1)= 𝑀1)= NOUN iajs-2475 197 5 f f PROPN iajs-2475 197 6 ( ( PUNCT iajs-2475 197 7 v)+ v)+ NOUN iajs-2475 197 8 f f X iajs-2475 197 9 ( ( PUNCT iajs-2475 197 10 h h NOUN iajs-2475 197 11 ) ) PUNCT iajs-2475 197 12 , , PUNCT iajs-2475 197 13 so so ADV iajs-2475 197 14 𝑀2= 𝑀2= VERB iajs-2475 197 15 f f X iajs-2475 197 16 ( ( PUNCT iajs-2475 197 17 v)+ v)+ NOUN iajs-2475 197 18 f f X iajs-2475 197 19 ( ( PUNCT iajs-2475 197 20 h h NOUN iajs-2475 197 21 ) ) PUNCT iajs-2475 197 22 and and CCONJ iajs-2475 197 23 ( ( PUNCT iajs-2475 197 24 since since SCONJ iajs-2475 197 25 f f PROPN iajs-2475 197 26 is be AUX iajs-2475 197 27 monomorphism monomorphism NOUN iajs-2475 197 28 then then ADV iajs-2475 197 29 kerf=0 kerf=0 PROPN iajs-2475 197 30 ) ) PUNCT iajs-2475 198 1 so so ADV iajs-2475 198 2 we we PRON iajs-2475 198 3 have have VERB iajs-2475 198 4 f f X iajs-2475 198 5 ( ( PUNCT iajs-2475 198 6 v∩h)= v∩h)= PROPN iajs-2475 198 7 f f X iajs-2475 198 8 ( ( PUNCT iajs-2475 198 9 v)∩ v)∩ NOUN iajs-2475 198 10 f f PROPN iajs-2475 198 11 ( ( PUNCT iajs-2475 198 12 h h NOUN iajs-2475 198 13 ) ) PUNCT iajs-2475 198 14 β‰ͺ β‰ͺ PUNCT iajs-2475 198 15 𝑀2 𝑀2 PROPN iajs-2475 198 16 . . PUNCT iajs-2475 199 1 then then ADV iajs-2475 199 2 f f PROPN iajs-2475 199 3 ( ( PUNCT iajs-2475 199 4 v v NOUN iajs-2475 199 5 ) ) PUNCT iajs-2475 199 6 has have VERB iajs-2475 199 7 a a DET iajs-2475 199 8 weakly weakly ADJ iajs-2475 199 9 supplement supplement NOUN iajs-2475 199 10 submodule submodule NOUN iajs-2475 199 11 in in ADP iajs-2475 199 12 𝑀2 𝑀2 PROPN iajs-2475 199 13 , , PUNCT iajs-2475 199 14 since since SCONJ iajs-2475 199 15 h h NOUN iajs-2475 199 16 is be AUX iajs-2475 199 17 direct direct ADJ iajs-2475 199 18 summand summand NOUN iajs-2475 199 19 so so ADV iajs-2475 199 20 , , PUNCT iajs-2475 199 21 we we PRON iajs-2475 199 22 have have VERB iajs-2475 199 23 𝑀1 𝑀1 NOUN iajs-2475 199 24 = = NOUN iajs-2475 199 25 h+l h+l X iajs-2475 199 26 and and CCONJ iajs-2475 199 27 h∩l=0 h∩l=0 NOUN iajs-2475 199 28 where where SCONJ iajs-2475 199 29 l l NOUN iajs-2475 199 30 is be AUX iajs-2475 199 31 submodule submodule NOUN iajs-2475 199 32 of of ADP iajs-2475 199 33 𝑀1 𝑀1 PROPN iajs-2475 199 34 then then ADV iajs-2475 199 35 f f PROPN iajs-2475 199 36 ( ( PUNCT iajs-2475 199 37 𝑀1)= 𝑀1)= NOUN iajs-2475 199 38 f f PROPN iajs-2475 199 39 ( ( PUNCT iajs-2475 199 40 h)+ h)+ PROPN iajs-2475 199 41 f f PROPN iajs-2475 199 42 ( ( PUNCT iajs-2475 199 43 l l NOUN iajs-2475 199 44 ) ) PUNCT iajs-2475 199 45 and and CCONJ iajs-2475 199 46 f f PROPN iajs-2475 199 47 ( ( PUNCT iajs-2475 199 48 h h NOUN iajs-2475 199 49 ) ) PUNCT iajs-2475 199 50 ∩ ∩ ADJ iajs-2475 199 51 f f X iajs-2475 199 52 ( ( PUNCT iajs-2475 199 53 l)=0 l)=0 PROPN iajs-2475 199 54 . . PUNCT iajs-2475 200 1 then then ADV iajs-2475 200 2 𝑀2 𝑀2 PROPN iajs-2475 200 3 βŠ•-s βŠ•-s NUM iajs-2475 200 4 - - PUNCT iajs-2475 200 5 extending extend VERB iajs-2475 200 6 module module NOUN iajs-2475 200 7 . . PUNCT iajs-2475 201 1 the the DET iajs-2475 201 2 next next ADJ iajs-2475 201 3 result result NOUN iajs-2475 201 4 gives give VERB iajs-2475 201 5 a a DET iajs-2475 201 6 condition condition NOUN iajs-2475 201 7 under under ADP iajs-2475 201 8 which which PRON iajs-2475 201 9 a a DET iajs-2475 201 10 direct direct ADJ iajs-2475 201 11 summand summand NOUN iajs-2475 201 12 of of ADP iajs-2475 201 13 βŠ•-s βŠ•-s NOUN iajs-2475 201 14 - - PUNCT iajs-2475 201 15 extending extend VERB iajs-2475 201 16 module module NOUN iajs-2475 201 17 is be AUX iajs-2475 201 18 βŠ•-s βŠ•-s NOUN iajs-2475 201 19 - - PUNCT iajs-2475 201 20 extending extending NOUN iajs-2475 201 21 . . PUNCT iajs-2475 202 1 proposition proposition NOUN iajs-2475 202 2 ( ( PUNCT iajs-2475 202 3 16 16 NUM iajs-2475 202 4 ) ) PUNCT iajs-2475 202 5 every every DET iajs-2475 202 6 direct direct ADJ iajs-2475 202 7 summand summand NOUN iajs-2475 202 8 a a PRON iajs-2475 202 9 of of ADP iajs-2475 202 10 βŠ•-s βŠ•-s NOUN iajs-2475 202 11 - - PUNCT iajs-2475 202 12 extending extend VERB iajs-2475 202 13 module module NOUN iajs-2475 202 14 such such ADJ iajs-2475 202 15 that that SCONJ iajs-2475 202 16 the the DET iajs-2475 202 17 intersection intersection NOUN iajs-2475 202 18 of of ADP iajs-2475 202 19 two two NUM iajs-2475 202 20 direct direct ADJ iajs-2475 202 21 summand summand NOUN iajs-2475 202 22 in in ADP iajs-2475 202 23 m m PROPN iajs-2475 202 24 is be AUX iajs-2475 202 25 direct direct ADJ iajs-2475 202 26 summand summand NOUN iajs-2475 202 27 in in ADP iajs-2475 202 28 n n PROPN iajs-2475 202 29 is be AUX iajs-2475 202 30 βŠ•-s βŠ•-s ADJ iajs-2475 202 31 - - PUNCT iajs-2475 202 32 extending extend VERB iajs-2475 202 33 module module NOUN iajs-2475 202 34 . . PUNCT iajs-2475 203 1 proof proof NOUN iajs-2475 203 2 let let VERB iajs-2475 203 3 a a PRON iajs-2475 203 4 be be AUX iajs-2475 203 5 a a DET iajs-2475 203 6 direct direct ADJ iajs-2475 203 7 summand summand NOUN iajs-2475 203 8 of of ADP iajs-2475 203 9 βŠ•-s βŠ•-s NOUN iajs-2475 203 10 - - PUNCT iajs-2475 203 11 extending extend VERB iajs-2475 203 12 module module NOUN iajs-2475 203 13 m m NOUN iajs-2475 203 14 and and CCONJ iajs-2475 203 15 let let VERB iajs-2475 203 16 w w NOUN iajs-2475 203 17 be be AUX iajs-2475 203 18 a a DET iajs-2475 203 19 closed closed ADJ iajs-2475 203 20 submodule submodule NOUN iajs-2475 203 21 in in ADP iajs-2475 203 22 a a PRON iajs-2475 203 23 , , PUNCT iajs-2475 203 24 so so CCONJ iajs-2475 203 25 w w PROPN iajs-2475 203 26 is be AUX iajs-2475 203 27 a a DET iajs-2475 203 28 submodule submodule NOUN iajs-2475 203 29 of of ADP iajs-2475 203 30 m. m. NOUN iajs-2475 203 31 since since SCONJ iajs-2475 203 32 a a PRON iajs-2475 203 33 is be AUX iajs-2475 203 34 a a DET iajs-2475 203 35 direct direct ADJ iajs-2475 203 36 summand summand NOUN iajs-2475 203 37 of of ADP iajs-2475 203 38 m m PROPN iajs-2475 203 39 , , PUNCT iajs-2475 203 40 so so ADV iajs-2475 203 41 a a PRON iajs-2475 203 42 is be AUX iajs-2475 203 43 closed close VERB iajs-2475 203 44 submodule submodule NOUN iajs-2475 203 45 of of ADP iajs-2475 203 46 m m PRON iajs-2475 203 47 and and CCONJ iajs-2475 203 48 we we PRON iajs-2475 203 49 have have VERB iajs-2475 203 50 w w NOUN iajs-2475 203 51 is be AUX iajs-2475 203 52 closed close VERB iajs-2475 203 53 submodule submodule NOUN iajs-2475 203 54 of of ADP iajs-2475 203 55 m m PROPN iajs-2475 203 56 , , PUNCT iajs-2475 203 57 but but CCONJ iajs-2475 203 58 m m PROPN iajs-2475 203 59 is be AUX iajs-2475 203 60 βŠ•-sextending βŠ•-sextende VERB iajs-2475 203 61 module module NOUN iajs-2475 203 62 . . PUNCT iajs-2475 204 1 thus thus ADV iajs-2475 204 2 by by ADP iajs-2475 204 3 using use VERB iajs-2475 204 4 proposition proposition NOUN iajs-2475 204 5 ( ( PUNCT iajs-2475 204 6 2.3 2.3 NUM iajs-2475 204 7 ) ) PUNCT iajs-2475 204 8 , , PUNCT iajs-2475 204 9 w w PROPN iajs-2475 204 10 has have VERB iajs-2475 204 11 a a DET iajs-2475 204 12 weakly weakly ADJ iajs-2475 204 13 supplement supplement NOUN iajs-2475 204 14 submodule submodule NOUN iajs-2475 204 15 of of ADP iajs-2475 204 16 m m PRON iajs-2475 204 17 which which PRON iajs-2475 204 18 is be AUX iajs-2475 204 19 direct direct ADJ iajs-2475 204 20 summand summand NOUN iajs-2475 204 21 then then ADV iajs-2475 204 22 m m PROPN iajs-2475 204 23 = = PROPN iajs-2475 204 24 w+h w+h PROPN iajs-2475 204 25 and and CCONJ iajs-2475 204 26 w∩h w∩h VERB iajs-2475 204 27 < < X iajs-2475 204 28 < < X iajs-2475 204 29 m m PROPN iajs-2475 204 30 , , PUNCT iajs-2475 204 31 where where SCONJ iajs-2475 204 32 h h NOUN iajs-2475 204 33 is be AUX iajs-2475 204 34 a a DET iajs-2475 204 35 direct direct ADJ iajs-2475 204 36 summand summand NOUN iajs-2475 204 37 of of ADP iajs-2475 204 38 m. m. NOUN iajs-2475 204 39 so so ADV iajs-2475 204 40 a∩m a∩m ADJ iajs-2475 204 41 = = SYM iajs-2475 204 42 a∩(w+h a∩(w+h ADJ iajs-2475 204 43 ) ) PUNCT iajs-2475 204 44 then then ADV iajs-2475 204 45 ( ( PUNCT iajs-2475 204 46 by by ADP iajs-2475 204 47 modular modular ADJ iajs-2475 204 48 law law NOUN iajs-2475 204 49 ) ) PUNCT iajs-2475 204 50 a a DET iajs-2475 204 51 = = NOUN iajs-2475 204 52 w+(a∩h w+(a∩h X iajs-2475 204 53 ) ) PUNCT iajs-2475 204 54 and and CCONJ iajs-2475 204 55 w∩(a w∩(a VERB iajs-2475 204 56 ∩m)= ∩m)= PROPN iajs-2475 204 57 ( ( PUNCT iajs-2475 204 58 w∩h)∩a w∩h)∩a PROPN iajs-2475 204 59 . . PUNCT iajs-2475 205 1 since since SCONJ iajs-2475 205 2 ( ( PUNCT iajs-2475 205 3 w∩h)∩a w∩h)∩a INTJ iajs-2475 205 4 be be AUX iajs-2475 205 5 a a DET iajs-2475 205 6 submodule submodule NOUN iajs-2475 205 7 of of ADP iajs-2475 205 8 w∩h w∩h PROPN iajs-2475 205 9 and and CCONJ iajs-2475 205 10 w∩hβ‰ͺ w∩hβ‰ͺ NOUN iajs-2475 205 11 m. m. NOUN iajs-2475 205 12 so so ADV iajs-2475 205 13 , , PUNCT iajs-2475 205 14 w∩(a∩h w∩(a∩h X iajs-2475 205 15 ) ) PUNCT iajs-2475 205 16 β‰ͺ β‰ͺ PUNCT iajs-2475 205 17 m m NOUN iajs-2475 205 18 and and CCONJ iajs-2475 205 19 since since SCONJ iajs-2475 205 20 w∩(a∩h w∩(a∩h NOUN iajs-2475 205 21 ) ) PUNCT iajs-2475 205 22 be be AUX iajs-2475 205 23 a a DET iajs-2475 205 24 submodule submodule NOUN iajs-2475 205 25 in in ADP iajs-2475 205 26 a a PRON iajs-2475 205 27 and and CCONJ iajs-2475 205 28 a a PRON iajs-2475 205 29 is be AUX iajs-2475 205 30 a a DET iajs-2475 205 31 submodule submodule NOUN iajs-2475 205 32 in in ADP iajs-2475 205 33 m. m. NOUN iajs-2475 205 34 hence hence ADV iajs-2475 205 35 w∩(a∩h w∩(a∩h X iajs-2475 205 36 ) ) PUNCT iajs-2475 205 37 β‰ͺ β‰ͺ PUNCT iajs-2475 205 38 a a PRON iajs-2475 205 39 , , PUNCT iajs-2475 205 40 then then ADV iajs-2475 205 41 w w PROPN iajs-2475 205 42 has have VERB iajs-2475 205 43 a a DET iajs-2475 205 44 weakly weakly ADJ iajs-2475 205 45 supplement supplement NOUN iajs-2475 205 46 submodule submodule NOUN iajs-2475 205 47 in in ADP iajs-2475 205 48 a a PRON iajs-2475 205 49 , , PUNCT iajs-2475 205 50 then then ADV iajs-2475 205 51 by by ADP iajs-2475 205 52 hypothesis hypothesis NOUN iajs-2475 205 53 ( ( PUNCT iajs-2475 205 54 a∩h a∩h X iajs-2475 205 55 ) ) PUNCT iajs-2475 205 56 is be AUX iajs-2475 205 57 direct direct ADJ iajs-2475 205 58 summand summand NOUN iajs-2475 205 59 in in ADP iajs-2475 205 60 a a PRON iajs-2475 205 61 , , PUNCT iajs-2475 205 62 then then ADV iajs-2475 205 63 by by ADP iajs-2475 205 64 proposition proposition NOUN iajs-2475 205 65 ( ( PUNCT iajs-2475 205 66 3 3 NUM iajs-2475 205 67 ) ) PUNCT iajs-2475 205 68 , , PUNCT iajs-2475 205 69 a a PRON iajs-2475 205 70 is be AUX iajs-2475 205 71 βŠ•-s βŠ•-s ADJ iajs-2475 205 72 - - PUNCT iajs-2475 205 73 extending extend VERB iajs-2475 205 74 module module NOUN iajs-2475 205 75 . . PUNCT iajs-2475 206 1 proposition proposition NOUN iajs-2475 206 2 ( ( PUNCT iajs-2475 206 3 17 17 NUM iajs-2475 206 4 ) ) PUNCT iajs-2475 206 5 every every DET iajs-2475 206 6 fully fully ADV iajs-2475 206 7 invariant invariant ADJ iajs-2475 206 8 direct direct ADJ iajs-2475 206 9 summand summand NOUN iajs-2475 206 10 a a PRON iajs-2475 206 11 of of ADP iajs-2475 206 12 βŠ•-s βŠ•-s NOUN iajs-2475 206 13 - - PUNCT iajs-2475 206 14 extending extend VERB iajs-2475 206 15 module module NOUN iajs-2475 206 16 m m VERB iajs-2475 206 17 is be AUX iajs-2475 206 18 βŠ•-sextending βŠ•-sextende VERB iajs-2475 206 19 . . PUNCT iajs-2475 207 1 proof proof NOUN iajs-2475 207 2 let let VERB iajs-2475 207 3 a a PRON iajs-2475 207 4 be be AUX iajs-2475 207 5 a a DET iajs-2475 207 6 direct direct ADJ iajs-2475 207 7 summand summand NOUN iajs-2475 207 8 fully fully ADV iajs-2475 207 9 invariant invariant ADJ iajs-2475 207 10 of of ADP iajs-2475 207 11 βŠ•-s βŠ•-s NOUN iajs-2475 207 12 - - PUNCT iajs-2475 207 13 extending extend VERB iajs-2475 207 14 module module NOUN iajs-2475 207 15 m m NOUN iajs-2475 207 16 and and CCONJ iajs-2475 207 17 let let VERB iajs-2475 207 18 l l NOUN iajs-2475 207 19 be be AUX iajs-2475 207 20 a a DET iajs-2475 207 21 closed closed ADJ iajs-2475 207 22 submodule submodule NOUN iajs-2475 207 23 in in ADP iajs-2475 207 24 a a PRON iajs-2475 207 25 , , PUNCT iajs-2475 207 26 so so CCONJ iajs-2475 207 27 l l NOUN iajs-2475 207 28 is be AUX iajs-2475 207 29 closed close VERB iajs-2475 207 30 in in ADP iajs-2475 207 31 m. m. NOUN iajs-2475 207 32 but but CCONJ iajs-2475 207 33 m m NOUN iajs-2475 207 34 isβŠ•-s isβŠ•-s NOUN iajs-2475 207 35 - - PUNCT iajs-2475 207 36 extending extend VERB iajs-2475 207 37 . . PUNCT iajs-2475 208 1 then then ADV iajs-2475 208 2 l l PROPN iajs-2475 208 3 has have VERB iajs-2475 208 4 a a DET iajs-2475 208 5 weakly weakly ADJ iajs-2475 208 6 87 87 NUM iajs-2475 208 7 ibn ibn PROPN iajs-2475 208 8 al al PROPN iajs-2475 208 9 - - PUNCT iajs-2475 208 10 haitham haitham PROPN iajs-2475 208 11 jour jour X iajs-2475 208 12 . . PROPN iajs-2475 209 1 for for ADP iajs-2475 209 2 pure pure ADJ iajs-2475 209 3 & & CCONJ iajs-2475 209 4 appl appl PROPN iajs-2475 209 5 . . PUNCT iajs-2475 210 1 sci sci PROPN iajs-2475 210 2 . . PROPN iajs-2475 211 1 33 33 NUM iajs-2475 211 2 ( ( PUNCT iajs-2475 211 3 3 3 NUM iajs-2475 211 4 ) ) PUNCT iajs-2475 211 5 2020 2020 NUM iajs-2475 211 6 supplement supplement VERB iajs-2475 211 7 w w NOUN iajs-2475 211 8 which which PRON iajs-2475 211 9 is be AUX iajs-2475 211 10 direct direct ADJ iajs-2475 211 11 summand summand NOUN iajs-2475 211 12 in in ADP iajs-2475 211 13 m m PROPN iajs-2475 211 14 , , PUNCT iajs-2475 211 15 so so ADV iajs-2475 211 16 m m NOUN iajs-2475 211 17 = = NOUN iajs-2475 211 18 wβŠ•w wβŠ•w NOUN iajs-2475 211 19 ' ' PUNCT iajs-2475 211 20 , , PUNCT iajs-2475 211 21 m m PROPN iajs-2475 211 22 = = NOUN iajs-2475 211 23 w+l w+l NUM iajs-2475 211 24 and and CCONJ iajs-2475 211 25 w∩lβ‰ͺm w∩lβ‰ͺm PROPN iajs-2475 211 26 . . PUNCT iajs-2475 212 1 now now ADV iajs-2475 212 2 a a DET iajs-2475 212 3 = = NOUN iajs-2475 212 4 a∩m a∩m ADJ iajs-2475 212 5 = = SYM iajs-2475 212 6 a∩(w+l a∩(w+l NOUN iajs-2475 212 7 ) ) PUNCT iajs-2475 212 8 by by ADP iajs-2475 212 9 ( ( PUNCT iajs-2475 212 10 modular modular ADJ iajs-2475 212 11 law law NOUN iajs-2475 212 12 ) ) PUNCT iajs-2475 212 13 a a DET iajs-2475 212 14 = = X iajs-2475 212 15 l+(a∩w l+(a∩w NOUN iajs-2475 212 16 ) ) PUNCT iajs-2475 212 17 , , PUNCT iajs-2475 212 18 but but CCONJ iajs-2475 212 19 a=(a∩ a=(a∩ PROPN iajs-2475 212 20 π‘Š π‘Š PROPN iajs-2475 212 21 ) ) PUNCT iajs-2475 212 22 βŠ• βŠ• PROPN iajs-2475 212 23 ( ( PUNCT iajs-2475 212 24 𝐴 𝐴 PROPN iajs-2475 212 25 ∩ ∩ NOUN iajs-2475 212 26 π‘Šβ€² π‘Šβ€² NOUN iajs-2475 212 27 ) ) PUNCT iajs-2475 212 28 . . PUNCT iajs-2475 213 1 so so ADV iajs-2475 213 2 a∩ a∩ PROPN iajs-2475 213 3 π‘Š π‘Š PROPN iajs-2475 213 4 is be AUX iajs-2475 213 5 direct direct ADJ iajs-2475 213 6 summand summand NOUN iajs-2475 213 7 in in ADP iajs-2475 213 8 a a PRON iajs-2475 213 9 and and CCONJ iajs-2475 213 10 l∩ l∩ ADJ iajs-2475 213 11 ( ( PUNCT iajs-2475 213 12 𝐴 𝐴 PROPN iajs-2475 213 13 ∩ ∩ NOUN iajs-2475 213 14 π‘Š)= π‘Š)= PART iajs-2475 213 15 w∩lβ‰ͺm w∩lβ‰ͺm NOUN iajs-2475 213 16 , , PUNCT iajs-2475 213 17 since since SCONJ iajs-2475 213 18 a a PRON iajs-2475 213 19 is be AUX iajs-2475 213 20 direct direct ADJ iajs-2475 213 21 summand summand NOUN iajs-2475 213 22 in in ADP iajs-2475 213 23 m. m. NOUN iajs-2475 213 24 hence hence ADV iajs-2475 213 25 a∩ a∩ PROPN iajs-2475 213 26 π‘Š π‘Š PROPN iajs-2475 213 27 is be AUX iajs-2475 213 28 weakly weakly ADJ iajs-2475 213 29 supplement supplement NOUN iajs-2475 213 30 of of ADP iajs-2475 213 31 l l NOUN iajs-2475 213 32 in in ADP iajs-2475 213 33 n. n. NOUN iajs-2475 213 34 thus thus ADV iajs-2475 213 35 a a DET iajs-2475 213 36 is be AUX iajs-2475 213 37 βŠ•-s βŠ•-s NOUN iajs-2475 213 38 - - PUNCT iajs-2475 213 39 extending extending NOUN iajs-2475 213 40 . . PUNCT iajs-2475 214 1 recall recall NOUN iajs-2475 214 2 that that SCONJ iajs-2475 214 3 , , PUNCT iajs-2475 214 4 if if SCONJ iajs-2475 214 5 for for ADP iajs-2475 214 6 each each DET iajs-2475 214 7 closed closed ADJ iajs-2475 214 8 submodules submodule NOUN iajs-2475 214 9 f f NOUN iajs-2475 214 10 , , PUNCT iajs-2475 214 11 g g PROPN iajs-2475 214 12 and and CCONJ iajs-2475 214 13 s s X iajs-2475 214 14 in in ADP iajs-2475 214 15 a a DET iajs-2475 214 16 module module NOUN iajs-2475 214 17 m m VERB iajs-2475 214 18 such such ADJ iajs-2475 214 19 that that SCONJ iajs-2475 214 20 f∩(g+s)= f∩(g+s)= PUNCT iajs-2475 214 21 f∩g+f∩s f∩g+f∩s PROPN iajs-2475 214 22 , , PUNCT iajs-2475 214 23 then then ADV iajs-2475 214 24 a a DET iajs-2475 214 25 module module NOUN iajs-2475 214 26 m m VERB iajs-2475 214 27 is be AUX iajs-2475 214 28 said say VERB iajs-2475 214 29 to to PART iajs-2475 214 30 be be AUX iajs-2475 214 31 local local ADJ iajs-2475 214 32 distributive distributive ADJ iajs-2475 214 33 [ [ X iajs-2475 214 34 10 10 NUM iajs-2475 214 35 ] ] PUNCT iajs-2475 214 36 . . PUNCT iajs-2475 215 1 a a DET iajs-2475 215 2 direct direct ADJ iajs-2475 215 3 sum sum NOUN iajs-2475 215 4 of of ADP iajs-2475 215 5 βŠ•-s βŠ•-s ADJ iajs-2475 215 6 - - PUNCT iajs-2475 215 7 extending extend VERB iajs-2475 215 8 module module NOUN iajs-2475 215 9 need need VERB iajs-2475 215 10 not not PART iajs-2475 215 11 necessary necessary ADJ iajs-2475 215 12 βŠ•-s βŠ•-s ADJ iajs-2475 215 13 - - PUNCT iajs-2475 215 14 extending extending NOUN iajs-2475 215 15 . . PUNCT iajs-2475 216 1 in in ADP iajs-2475 216 2 fact fact NOUN iajs-2475 216 3 , , PUNCT iajs-2475 216 4 for for ADP iajs-2475 216 5 example example NOUN iajs-2475 216 6 . . PUNCT iajs-2475 217 1 m m PUNCT iajs-2475 218 1 = = NOUN iajs-2475 218 2 z⨁z2 z⨁z2 NOUN iajs-2475 218 3 as as ADP iajs-2475 218 4 z z NOUN iajs-2475 218 5 - - PUNCT iajs-2475 218 6 module module NOUN iajs-2475 218 7 is be AUX iajs-2475 218 8 not not PART iajs-2475 218 9 βŠ•-s βŠ•-s ADJ iajs-2475 218 10 - - PUNCT iajs-2475 218 11 extending extend VERB iajs-2475 218 12 ( ( PUNCT iajs-2475 218 13 since since SCONJ iajs-2475 218 14 it it PRON iajs-2475 218 15 is be AUX iajs-2475 218 16 not not PART iajs-2475 218 17 weakly weakly ADJ iajs-2475 218 18 supplement supplement ADJ iajs-2475 218 19 extending extend VERB iajs-2475 218 20 module module NOUN iajs-2475 218 21 , , PUNCT iajs-2475 218 22 while while SCONJ iajs-2475 218 23 z z PROPN iajs-2475 218 24 and and CCONJ iajs-2475 218 25 z2 z2 PROPN iajs-2475 218 26 are be AUX iajs-2475 218 27 βŠ•-s βŠ•-s ADJ iajs-2475 218 28 - - PUNCT iajs-2475 218 29 extending extending NOUN iajs-2475 218 30 . . PUNCT iajs-2475 219 1 we we PRON iajs-2475 219 2 give give VERB iajs-2475 219 3 the the DET iajs-2475 219 4 condition condition NOUN iajs-2475 219 5 so so SCONJ iajs-2475 219 6 we we PRON iajs-2475 219 7 get get VERB iajs-2475 219 8 the the DET iajs-2475 219 9 proposition proposition NOUN iajs-2475 219 10 : : PUNCT iajs-2475 219 11 proposition proposition NOUN iajs-2475 219 12 ( ( PUNCT iajs-2475 219 13 18 18 NUM iajs-2475 219 14 ) ) PUNCT iajs-2475 219 15 let let VERB iajs-2475 219 16 m=𝑀1⨁ m=𝑀1⨁ NOUN iajs-2475 219 17 𝑀2 𝑀2 NOUN iajs-2475 219 18 where where SCONJ iajs-2475 219 19 𝑀1and 𝑀1and PROPN iajs-2475 219 20 𝑀2 𝑀2 PROPN iajs-2475 219 21 are be AUX iajs-2475 219 22 βŠ•-s βŠ•-s ADV iajs-2475 219 23 - - PUNCT iajs-2475 219 24 extending extend VERB iajs-2475 219 25 such such ADJ iajs-2475 219 26 that that SCONJ iajs-2475 219 27 m m PROPN iajs-2475 219 28 is be AUX iajs-2475 219 29 local local ADJ iajs-2475 219 30 distributive distributive ADJ iajs-2475 219 31 module module NOUN iajs-2475 219 32 . . PUNCT iajs-2475 220 1 then then ADV iajs-2475 220 2 m m PROPN iajs-2475 220 3 is be AUX iajs-2475 220 4 βŠ•-s βŠ•-s ADJ iajs-2475 220 5 - - PUNCT iajs-2475 220 6 extending extend VERB iajs-2475 220 7 module module NOUN iajs-2475 220 8 . . PUNCT iajs-2475 221 1 proof proof NOUN iajs-2475 221 2 let let VERB iajs-2475 221 3 f f PRON iajs-2475 221 4 be be AUX iajs-2475 221 5 a a DET iajs-2475 221 6 closed closed ADJ iajs-2475 221 7 submodule submodule NOUN iajs-2475 221 8 in in ADP iajs-2475 221 9 m. m. NOUN iajs-2475 221 10 to to PART iajs-2475 221 11 prove prove VERB iajs-2475 221 12 fβˆ©π‘€π‘– fβˆ©π‘€π‘– PROPN iajs-2475 221 13 is be AUX iajs-2475 221 14 closed close VERB iajs-2475 221 15 in in ADP iajs-2475 221 16 𝑀𝑖 𝑀𝑖 PROPN iajs-2475 221 17 , , PUNCT iajs-2475 221 18 since since SCONJ iajs-2475 221 19 m m PROPN iajs-2475 221 20 is be AUX iajs-2475 221 21 local local ADJ iajs-2475 221 22 distributive distributive ADJ iajs-2475 221 23 module module NOUN iajs-2475 221 24 , , PUNCT iajs-2475 221 25 then then ADV iajs-2475 221 26 we we PRON iajs-2475 221 27 have have AUX iajs-2475 221 28 f=((fβˆ©π‘€1 f=((fβˆ©π‘€1 NUM iajs-2475 221 29 ) ) PUNCT iajs-2475 221 30 ⨁ ⨁ PROPN iajs-2475 221 31 ( ( PUNCT iajs-2475 221 32 fβˆ©π‘€2 fβˆ©π‘€2 PROPN iajs-2475 221 33 ) ) PUNCT iajs-2475 221 34 ) ) PUNCT iajs-2475 221 35 . . PUNCT iajs-2475 222 1 hence hence ADV iajs-2475 222 2 fβˆ©π‘€1 fβˆ©π‘€1 PROPN iajs-2475 222 3 is be AUX iajs-2475 222 4 closed close VERB iajs-2475 222 5 in in ADP iajs-2475 222 6 𝑀1 𝑀1 PROPN iajs-2475 222 7 and and CCONJ iajs-2475 222 8 fβˆ©π‘€2 fβˆ©π‘€2 PROPN iajs-2475 222 9 is be AUX iajs-2475 222 10 closed close VERB iajs-2475 222 11 in in ADP iajs-2475 222 12 𝑀2 𝑀2 PROPN iajs-2475 222 13 . . PUNCT iajs-2475 223 1 but but CCONJ iajs-2475 223 2 𝑀1 𝑀1 PROPN iajs-2475 223 3 and and CCONJ iajs-2475 223 4 𝑀2 𝑀2 PROPN iajs-2475 223 5 are be AUX iajs-2475 223 6 βŠ•-s βŠ•-s ADJ iajs-2475 223 7 - - PUNCT iajs-2475 223 8 extending extend VERB iajs-2475 223 9 module module NOUN iajs-2475 223 10 . . PUNCT iajs-2475 224 1 then then ADV iajs-2475 224 2 there there PRON iajs-2475 224 3 exists exist VERB iajs-2475 224 4 a a DET iajs-2475 224 5 weakly weakly ADJ iajs-2475 224 6 supplement supplement NOUN iajs-2475 224 7 submodule submodule NOUN iajs-2475 224 8 𝐺1 𝐺1 NOUN iajs-2475 224 9 of of ADP iajs-2475 224 10 𝑀1 𝑀1 PROPN iajs-2475 224 11 , , PUNCT iajs-2475 224 12 𝐺2 𝐺2 NOUN iajs-2475 224 13 of of ADP iajs-2475 224 14 𝑀2 𝑀2 PROPN iajs-2475 224 15 and and CCONJ iajs-2475 224 16 𝐺1 𝐺1 PROPN iajs-2475 224 17 , , PUNCT iajs-2475 224 18 𝐺2 𝐺2 NOUN iajs-2475 224 19 are be AUX iajs-2475 224 20 direct direct ADJ iajs-2475 224 21 summand summand NOUN iajs-2475 224 22 such such ADJ iajs-2475 224 23 that that PRON iajs-2475 224 24 𝐺1+(fβˆ©π‘€1)= 𝐺1+(fβˆ©π‘€1)= VERB iajs-2475 224 25 𝑀1 𝑀1 ADV iajs-2475 224 26 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2475 224 27 𝐺2 𝐺2 ADV iajs-2475 225 1 + + CCONJ iajs-2475 225 2 ( ( PUNCT iajs-2475 225 3 𝐹 𝐹 PROPN iajs-2475 225 4 ∩ ∩ ADJ iajs-2475 225 5 𝑀2 𝑀2 PROPN iajs-2475 225 6 ) ) PUNCT iajs-2475 226 1 = = SYM iajs-2475 226 2 𝑀2 𝑀2 PROPN iajs-2475 226 3 , , PUNCT iajs-2475 226 4 𝐺1 𝐺1 NOUN iajs-2475 226 5 ∩ ∩ X iajs-2475 226 6 ( ( PUNCT iajs-2475 226 7 𝐹 𝐹 PROPN iajs-2475 226 8 ∩ ∩ NOUN iajs-2475 226 9 𝑀1 𝑀1 NOUN iajs-2475 226 10 ) ) PUNCT iajs-2475 226 11 = = SYM iajs-2475 226 12 ( ( PUNCT iajs-2475 226 13 𝐺1∩f 𝐺1∩f NUM iajs-2475 226 14 ) ) PUNCT iajs-2475 226 15 β‰ͺ β‰ͺ ADJ iajs-2475 226 16 𝑀1 𝑀1 NOUN iajs-2475 226 17 and and CCONJ iajs-2475 226 18 𝐺2∩(fβˆ©π‘€2)= 𝐺2∩(fβˆ©π‘€2)= NOUN iajs-2475 226 19 ( ( PUNCT iajs-2475 226 20 𝐺2∩f)β‰ͺ 𝐺2∩f)β‰ͺ X iajs-2475 226 21 𝑀2 𝑀2 NOUN iajs-2475 226 22 . . PUNCT iajs-2475 227 1 now now ADV iajs-2475 227 2 m= m= X iajs-2475 227 3 𝑀1⨁𝑀2 𝑀1⨁𝑀2 ADJ iajs-2475 227 4 = = SYM iajs-2475 227 5 ( ( PUNCT iajs-2475 227 6 𝐺1+(fβˆ©π‘€1 𝐺1+(fβˆ©π‘€1 PROPN iajs-2475 227 7 ) ) PUNCT iajs-2475 227 8 ) ) PUNCT iajs-2475 227 9 ⨁ ⨁ PROPN iajs-2475 227 10 ( ( PUNCT iajs-2475 227 11 𝐺2+(fβˆ©π‘€2))=(𝐺1 𝐺2+(fβˆ©π‘€2))=(𝐺1 NOUN iajs-2475 227 12 ⨁𝐺2)+f ⨁𝐺2)+f NOUN iajs-2475 227 13 . . PUNCT iajs-2475 228 1 then then ADV iajs-2475 228 2 m m VERB iajs-2475 228 3 = = SYM iajs-2475 228 4 ( ( PUNCT iajs-2475 228 5 𝐺1⨁ 𝐺1⨁ NOUN iajs-2475 228 6 𝐺2 𝐺2 ADJ iajs-2475 228 7 ) ) PUNCT iajs-2475 229 1 + + CCONJ iajs-2475 229 2 f f PROPN iajs-2475 229 3 and and CCONJ iajs-2475 229 4 ( ( PUNCT iajs-2475 229 5 𝐺1⨁𝐺2)∩f 𝐺1⨁𝐺2)∩f PROPN iajs-2475 229 6 = = SYM iajs-2475 229 7 ( ( PUNCT iajs-2475 229 8 𝐺1∩f)⨁ 𝐺1∩f)⨁ PRON iajs-2475 229 9 ( ( PUNCT iajs-2475 229 10 𝐺2∩f)β‰ͺ(fβˆ©π‘€1 𝐺2∩f)β‰ͺ(fβˆ©π‘€1 ADJ iajs-2475 229 11 ) ) PUNCT iajs-2475 229 12 ⨁(fβˆ©π‘€2 ⨁(fβˆ©π‘€2 PROPN iajs-2475 229 13 ) ) PUNCT iajs-2475 229 14 β‰ͺ β‰ͺ PUNCT iajs-2475 230 1 𝑀1⨁ 𝑀1⨁ PROPN iajs-2475 230 2 𝑀2 𝑀2 PROPN iajs-2475 230 3 β‰ͺ β‰ͺ PUNCT iajs-2475 230 4 m. m. NOUN iajs-2475 230 5 we we PRON iajs-2475 230 6 have have VERB iajs-2475 230 7 𝐺1 𝐺1 NOUN iajs-2475 230 8 ⨁𝐺2 ⨁𝐺2 X iajs-2475 230 9 is be AUX iajs-2475 230 10 direct direct ADJ iajs-2475 230 11 summand summand NOUN iajs-2475 230 12 of of ADP iajs-2475 230 13 m. m. NOUN iajs-2475 230 14 then then ADV iajs-2475 230 15 m m VERB iajs-2475 230 16 is be AUX iajs-2475 230 17 βŠ•-s βŠ•-s ADJ iajs-2475 230 18 - - PUNCT iajs-2475 230 19 extending extend VERB iajs-2475 230 20 module module NOUN iajs-2475 230 21 . . PUNCT iajs-2475 231 1 recall recall VERB iajs-2475 231 2 that that PRON iajs-2475 231 3 , , PUNCT iajs-2475 231 4 a a DET iajs-2475 231 5 module module NOUN iajs-2475 231 6 m m VERB iajs-2475 231 7 is be AUX iajs-2475 231 8 distributive distributive ADJ iajs-2475 231 9 if if SCONJ iajs-2475 231 10 for for ADP iajs-2475 231 11 any any DET iajs-2475 231 12 submodule submodule NOUN iajs-2475 231 13 f f NOUN iajs-2475 231 14 , , PUNCT iajs-2475 231 15 g g PROPN iajs-2475 231 16 and and CCONJ iajs-2475 231 17 s s PROPN iajs-2475 231 18 in in ADP iajs-2475 231 19 m m PRON iajs-2475 231 20 such such ADJ iajs-2475 231 21 that that DET iajs-2475 231 22 f∩(g+s)= f∩(g+s)= PUNCT iajs-2475 231 23 f∩g+f∩s f∩g+f∩s ADV iajs-2475 231 24 . . PUNCT iajs-2475 232 1 following follow VERB iajs-2475 232 2 [ [ X iajs-2475 232 3 10 10 NUM iajs-2475 232 4 ] ] PUNCT iajs-2475 232 5 . . PUNCT iajs-2475 233 1 every every DET iajs-2475 233 2 distributive distributive ADJ iajs-2475 233 3 module module NOUN iajs-2475 233 4 is be AUX iajs-2475 233 5 local local ADJ iajs-2475 233 6 distributive distributive ADJ iajs-2475 233 7 . . PUNCT iajs-2475 234 1 thus thus ADV iajs-2475 234 2 , , PUNCT iajs-2475 234 3 we we PRON iajs-2475 234 4 have have VERB iajs-2475 234 5 this this DET iajs-2475 234 6 corollary corollary ADJ iajs-2475 234 7 : : PUNCT iajs-2475 234 8 corollary corollary ADJ iajs-2475 234 9 ( ( PUNCT iajs-2475 234 10 19 19 NUM iajs-2475 234 11 ) ) PUNCT iajs-2475 234 12 let let VERB iajs-2475 234 13 m=𝑀1⨁ m=𝑀1⨁ NOUN iajs-2475 234 14 𝑀2 𝑀2 NOUN iajs-2475 234 15 where where SCONJ iajs-2475 234 16 𝑀1and 𝑀1and PROPN iajs-2475 234 17 𝑀2 𝑀2 PROPN iajs-2475 234 18 are be AUX iajs-2475 234 19 βŠ•-s βŠ•-s ADJ iajs-2475 234 20 - - PUNCT iajs-2475 234 21 extending extend VERB iajs-2475 234 22 module module NOUN iajs-2475 234 23 such such ADJ iajs-2475 234 24 that that SCONJ iajs-2475 234 25 m m PROPN iajs-2475 234 26 is be AUX iajs-2475 234 27 a a DET iajs-2475 234 28 distributive distributive ADJ iajs-2475 234 29 module module NOUN iajs-2475 234 30 . . PUNCT iajs-2475 235 1 then then ADV iajs-2475 235 2 m m PROPN iajs-2475 235 3 is be AUX iajs-2475 235 4 βŠ•-s βŠ•-s ADJ iajs-2475 235 5 - - PUNCT iajs-2475 235 6 extending extending NOUN iajs-2475 235 7 . . PUNCT iajs-2475 236 1 2 2 X iajs-2475 236 2 . . X iajs-2475 236 3 conclusions conclusion NOUN iajs-2475 236 4 we we PRON iajs-2475 236 5 proved prove VERB iajs-2475 236 6 the the DET iajs-2475 236 7 following follow VERB iajs-2475 236 8 results result NOUN iajs-2475 236 9 : : PUNCT iajs-2475 237 1 1 1 X iajs-2475 237 2 . . X iajs-2475 238 1 a a DET iajs-2475 238 2 module module NOUN iajs-2475 238 3 m m VERB iajs-2475 238 4 is be AUX iajs-2475 238 5 βŠ•-s βŠ•-s ADV iajs-2475 238 6 - - PUNCT iajs-2475 238 7 extending extend VERB iajs-2475 238 8 if if SCONJ iajs-2475 239 1 and and CCONJ iajs-2475 239 2 only only ADV iajs-2475 239 3 if if SCONJ iajs-2475 239 4 each each DET iajs-2475 239 5 submodule submodule NOUN iajs-2475 239 6 in in ADP iajs-2475 239 7 m m PROPN iajs-2475 239 8 is be AUX iajs-2475 239 9 essential essential ADJ iajs-2475 239 10 in in ADP iajs-2475 239 11 submodule submodule NOUN iajs-2475 239 12 has have VERB iajs-2475 239 13 a a DET iajs-2475 239 14 weakly weakly ADJ iajs-2475 239 15 supplement supplement NOUN iajs-2475 239 16 which which PRON iajs-2475 239 17 is be AUX iajs-2475 239 18 direct direct ADJ iajs-2475 239 19 summand summand NOUN iajs-2475 239 20 . . PUNCT iajs-2475 240 1 2 2 X iajs-2475 240 2 . . X iajs-2475 240 3 a a DET iajs-2475 240 4 module module NOUN iajs-2475 240 5 m m VERB iajs-2475 240 6 is be AUX iajs-2475 240 7 βŠ•-s βŠ•-s ADV iajs-2475 240 8 - - PUNCT iajs-2475 240 9 extending extend VERB iajs-2475 240 10 if if SCONJ iajs-2475 241 1 and and CCONJ iajs-2475 241 2 only only ADV iajs-2475 241 3 if if SCONJ iajs-2475 241 4 each each DET iajs-2475 241 5 closed closed ADJ iajs-2475 241 6 submodule submodule NOUN iajs-2475 241 7 in in ADP iajs-2475 241 8 m m PROPN iajs-2475 241 9 has have VERB iajs-2475 241 10 a a DET iajs-2475 241 11 ( ( PUNCT iajs-2475 241 12 weakly weakly ADJ iajs-2475 241 13 ) ) PUNCT iajs-2475 241 14 supplement supplement NOUN iajs-2475 241 15 which which PRON iajs-2475 241 16 is be AUX iajs-2475 241 17 direct direct ADJ iajs-2475 241 18 summand summand NOUN iajs-2475 241 19 . . PUNCT iajs-2475 242 1 3 3 X iajs-2475 242 2 . . X iajs-2475 242 3 a a DET iajs-2475 242 4 module module NOUN iajs-2475 242 5 m m VERB iajs-2475 242 6 is be AUX iajs-2475 242 7 βŠ•-s βŠ•-s ADV iajs-2475 242 8 - - PUNCT iajs-2475 242 9 extending extend VERB iajs-2475 242 10 if if SCONJ iajs-2475 243 1 and and CCONJ iajs-2475 243 2 only only ADV iajs-2475 243 3 if if SCONJ iajs-2475 243 4 m m NOUN iajs-2475 243 5 is be AUX iajs-2475 243 6 closed close VERB iajs-2475 243 7 ⨁-supplemented ⨁-supplemente VERB iajs-2475 243 8 . . PUNCT iajs-2475 244 1 4 4 X iajs-2475 244 2 . . X iajs-2475 244 3 we we PRON iajs-2475 244 4 obtain obtain VERB iajs-2475 244 5 the the DET iajs-2475 244 6 following following ADJ iajs-2475 244 7 result result NOUN iajs-2475 244 8 : : PUNCT iajs-2475 244 9 extending extend VERB iajs-2475 244 10 modules module NOUN iajs-2475 244 11 β†šβƒ—βƒ—βƒ— β†šβƒ—βƒ—βƒ— ADJ iajs-2475 244 12 βŠ•-s βŠ•-s NOUN iajs-2475 244 13 - - PUNCT iajs-2475 244 14 extending extend VERB iajs-2475 244 15 modules module NOUN iajs-2475 244 16 β†šβƒ—βƒ—βƒ— β†šβƒ—βƒ—βƒ— ADJ iajs-2475 244 17 weakly weakly ADJ iajs-2475 244 18 supplement supplement NOUN iajs-2475 244 19 extending extend VERB iajs-2475 244 20 modules module NOUN iajs-2475 244 21 . . PUNCT iajs-2475 245 1 88 88 NUM iajs-2475 245 2 ibn ibn PROPN iajs-2475 245 3 al al PROPN iajs-2475 245 4 - - PUNCT iajs-2475 245 5 haitham haitham PROPN iajs-2475 245 6 jour jour X iajs-2475 245 7 . . PROPN iajs-2475 245 8 for for ADP iajs-2475 245 9 pure pure ADJ iajs-2475 245 10 & & CCONJ iajs-2475 245 11 appl appl PROPN iajs-2475 245 12 . . PUNCT iajs-2475 246 1 sci sci PROPN iajs-2475 246 2 . . PROPN iajs-2475 247 1 33 33 NUM iajs-2475 247 2 ( ( PUNCT iajs-2475 247 3 3 3 NUM iajs-2475 247 4 ) ) PUNCT iajs-2475 247 5 2020 2020 NUM iajs-2475 247 6 references reference NOUN iajs-2475 247 7 1 1 NUM iajs-2475 247 8 . . PUNCT iajs-2475 248 1 al al PROPN iajs-2475 248 2 - - PUNCT iajs-2475 248 3 rubaye rubaye PROPN iajs-2475 248 4 , , PUNCT iajs-2475 248 5 a.a a.a PROPN iajs-2475 248 6 . . PROPN iajs-2475 248 7 ; ; PUNCT iajs-2475 248 8 al al PROPN iajs-2475 248 9 - - PUNCT iajs-2475 248 10 saadi saadi PROPN iajs-2475 248 11 , , PUNCT iajs-2475 248 12 s.a s.a PROPN iajs-2475 248 13 . . PROPN iajs-2475 248 14 weakly weakly ADJ iajs-2475 248 15 supplement supplement NOUN iajs-2475 248 16 extending extend VERB iajs-2475 248 17 modules module NOUN iajs-2475 248 18 , , PUNCT iajs-2475 248 19 to to PART iajs-2475 248 20 appear appear VERB iajs-2475 248 21 . . PUNCT iajs-2475 249 1 2 2 X iajs-2475 249 2 . . X iajs-2475 249 3 clark clark PROPN iajs-2475 249 4 , , PUNCT iajs-2475 249 5 j. j. PROPN iajs-2475 249 6 ; ; PUNCT iajs-2475 249 7 lamp lamp PROPN iajs-2475 249 8 , , PUNCT iajs-2475 249 9 c. c. PROPN iajs-2475 249 10 ; ; PUNCT iajs-2475 249 11 vanaja vanaja PROPN iajs-2475 249 12 , , PUNCT iajs-2475 249 13 n. n. NOUN iajs-2475 249 14 ; ; PUNCT iajs-2475 249 15 wisbauer wisbauer NOUN iajs-2475 249 16 , , PUNCT iajs-2475 249 17 r. r. 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