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NOUN iajs-2510 28 38 definition definition NOUN iajs-2510 28 39 1.4 1.4 NUM iajs-2510 29 1 [ [ X iajs-2510 29 2 5 5 NUM iajs-2510 29 3 ] ] PUNCT iajs-2510 29 4 : : PUNCT iajs-2510 29 5 let let VERB iajs-2510 29 6 𝐴 𝐴 PROPN iajs-2510 29 7 , , PUNCT iajs-2510 29 8 𝐴 𝐴 PROPN iajs-2510 29 9 are be AUX iajs-2510 29 10 f f NOUN iajs-2510 29 11 - - PUNCT iajs-2510 29 12 sets set NOUN iajs-2510 29 13 in in ADP iajs-2510 29 14 s s NOUN iajs-2510 29 15 , , PUNCT iajs-2510 29 16 then then ADV iajs-2510 29 17 : : PUNCT iajs-2510 29 18 1 1 X iajs-2510 29 19 . . X iajs-2510 29 20 𝐴 𝐴 PROPN iajs-2510 29 21 = = SYM iajs-2510 29 22 𝐴 𝐴 PROPN iajs-2510 30 1 if if SCONJ iajs-2510 31 1 and and CCONJ iajs-2510 31 2 only only ADV iajs-2510 31 3 if if SCONJ iajs-2510 31 4 𝐴 𝐴 PROPN iajs-2510 31 5 ( ( PUNCT iajs-2510 31 6 x x X iajs-2510 31 7 ) ) PUNCT iajs-2510 31 8 = = NOUN iajs-2510 31 9 𝐴 𝐴 PROPN iajs-2510 31 10 ( ( PUNCT iajs-2510 31 11 x x NOUN iajs-2510 31 12 ) ) PUNCT iajs-2510 31 13 , , PUNCT iajs-2510 31 14 ∀ ∀ X iajs-2510 31 15 x x SYM iajs-2510 31 16 ∈ ∈ NOUN iajs-2510 31 17 𝑆. 𝑆. PROPN iajs-2510 31 18 2 2 NUM iajs-2510 31 19 . . PUNCT iajs-2510 31 20 𝐴 𝐴 PROPN iajs-2510 31 21 ⊆ ⊆ NUM iajs-2510 31 22 𝐴2 𝐴2 NOUN iajs-2510 31 23 if if SCONJ iajs-2510 31 24 and and CCONJ iajs-2510 31 25 only only ADV iajs-2510 31 26 if if SCONJ iajs-2510 31 27 𝐴 𝐴 PROPN iajs-2510 31 28 ( ( PUNCT iajs-2510 31 29 x x X iajs-2510 31 30 ) ) PUNCT iajs-2510 31 31 𝐴 𝐴 PROPN iajs-2510 31 32 x x X iajs-2510 31 33 , , PUNCT iajs-2510 31 34 ∀ ∀ X iajs-2510 31 35 x x SYM iajs-2510 31 36 ∈ ∈ NOUN iajs-2510 31 37 𝑆. 𝑆. NOUN iajs-2510 31 38 if if SCONJ iajs-2510 31 39 𝐴 𝐴 PROPN iajs-2510 31 40 ⊂ ⊂ PROPN iajs-2510 31 41 𝐴 𝐴 PROPN iajs-2510 31 42 and and CCONJ iajs-2510 31 43 there there PRON iajs-2510 31 44 exists exist VERB iajs-2510 31 45 x x X iajs-2510 31 46 ∈ ∈ NOUN iajs-2510 31 47 s s VERB iajs-2510 31 48 such such ADJ iajs-2510 31 49 that that SCONJ iajs-2510 31 50 𝐴 𝐴 PROPN iajs-2510 31 51 ( ( PUNCT iajs-2510 31 52 x x NOUN iajs-2510 31 53 ) ) PUNCT iajs-2510 31 54 𝐴 𝐴 PROPN iajs-2510 31 55 ( ( PUNCT iajs-2510 31 56 x x NOUN iajs-2510 31 57 ) ) PUNCT iajs-2510 31 58 , , PUNCT iajs-2510 31 59 then then ADV iajs-2510 31 60 𝐴 𝐴 PROPN iajs-2510 31 61 is be AUX iajs-2510 31 62 called call VERB iajs-2510 31 63 a a DET iajs-2510 31 64 proper proper ADJ iajs-2510 31 65 fsubset fsubset NOUN iajs-2510 31 66 of of ADP iajs-2510 31 67 𝐴 𝐴 PROPN iajs-2510 31 68 . . PUNCT iajs-2510 32 1 3 3 X iajs-2510 32 2 . . NUM iajs-2510 32 3 x x SYM iajs-2510 33 1 ⊆ ⊆ NUM iajs-2510 33 2 a a DET iajs-2510 33 3 if if NOUN iajs-2510 33 4 and and CCONJ iajs-2510 33 5 only only ADV iajs-2510 33 6 x x X iajs-2510 33 7 𝑦 𝑦 DET iajs-2510 33 8 a a DET iajs-2510 33 9 𝑦 𝑦 NOUN iajs-2510 33 10 , , PUNCT iajs-2510 33 11 ∀ ∀ VERB iajs-2510 33 12 y y PROPN iajs-2510 33 13 ∈ ∈ PROPN iajs-2510 33 14 𝑆 𝑆 PROPN iajs-2510 33 15 𝑎𝑛𝑑 𝑎𝑛𝑑 NOUN iajs-2510 33 16 if if SCONJ iajs-2510 33 17 t t PROPN iajs-2510 33 18 0 0 NUM iajs-2510 33 19 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒𝑛 VERB iajs-2510 33 20 a a DET iajs-2510 33 21 x x X iajs-2510 33 22 t. t. NOUN iajs-2510 33 23 thus thus ADV iajs-2510 33 24 x x PROPN iajs-2510 33 25 ⊆ ⊆ NUM iajs-2510 33 26 a a DET iajs-2510 33 27 ( ( PUNCT iajs-2510 33 28 x x SYM iajs-2510 33 29 ∈ ∈ PROPN iajs-2510 33 30 𝐴 𝐴 PROPN iajs-2510 33 31 ) ) PUNCT iajs-2510 33 32 , , PUNCT iajs-2510 33 33 ( ( PUNCT iajs-2510 33 34 that that PRON iajs-2510 33 35 is be AUX iajs-2510 33 36 x x PART iajs-2510 33 37 ∈ ∈ PROPN iajs-2510 33 38 𝐴 𝐴 PROPN iajs-2510 33 39 if if SCONJ iajs-2510 33 40 and and CCONJ iajs-2510 33 41 only only ADV iajs-2510 33 42 if if SCONJ iajs-2510 33 43 x x PROPN iajs-2510 33 44 ⊆ ⊆ NUM iajs-2510 33 45 a a DET iajs-2510 33 46 ) ) PUNCT iajs-2510 33 47 definition definition NOUN iajs-2510 33 48 1.5 1.5 NUM iajs-2510 33 49 [ [ X iajs-2510 33 50 5 5 NUM iajs-2510 33 51 ] ] PUNCT iajs-2510 33 52 : : PUNCT iajs-2510 33 53 let let VERB iajs-2510 33 54 𝐴 𝐴 PROPN iajs-2510 33 55 , , PUNCT iajs-2510 33 56 𝐴 𝐴 PROPN iajs-2510 33 57 are be AUX iajs-2510 33 58 f f NOUN iajs-2510 33 59 - - PUNCT iajs-2510 33 60 sets set NOUN iajs-2510 33 61 in in ADP iajs-2510 33 62 s s PROPN iajs-2510 33 63 , , PUNCT iajs-2510 33 64 then then ADV iajs-2510 33 65 : : PUNCT iajs-2510 33 66 1 1 X iajs-2510 33 67 . . X iajs-2510 33 68 𝐴 𝐴 PROPN iajs-2510 33 69 ∪ ∪ PROPN iajs-2510 33 70 𝐴 𝐴 PROPN iajs-2510 33 71 ( ( PUNCT iajs-2510 33 72 x x NOUN iajs-2510 33 73 ) ) PUNCT iajs-2510 33 74 = = SYM iajs-2510 33 75 max max PROPN iajs-2510 33 76 𝐴 𝐴 PROPN iajs-2510 33 77 x x X iajs-2510 33 78 , , PUNCT iajs-2510 33 79 𝐴 𝐴 PROPN iajs-2510 33 80 x x X iajs-2510 33 81 , , PUNCT iajs-2510 33 82 ∀ ∀ X iajs-2510 33 83 x x SYM iajs-2510 33 84 ∈ ∈ NOUN iajs-2510 33 85 𝑆. 𝑆. PROPN iajs-2510 33 86 2 2 NUM iajs-2510 33 87 . . PUNCT iajs-2510 34 1 𝐴 𝐴 PROPN iajs-2510 34 2 ∩ ∩ NOUN iajs-2510 34 3 𝐴 𝐴 PROPN iajs-2510 34 4 x x PUNCT iajs-2510 34 5 min min PROPN iajs-2510 34 6 𝐴 𝐴 PROPN iajs-2510 34 7 x x SYM iajs-2510 34 8 , , PUNCT iajs-2510 34 9 𝐴 𝐴 PROPN iajs-2510 34 10 x x X iajs-2510 34 11 , , PUNCT iajs-2510 34 12 ∀ ∀ X iajs-2510 34 13 x x SYM iajs-2510 34 14 ∈ ∈ NOUN iajs-2510 34 15 𝑆. 𝑆. PROPN iajs-2510 34 16 𝐴 𝐴 PROPN iajs-2510 34 17 ∪ ∪ ADP iajs-2510 34 18 𝐴 𝐴 PROPN iajs-2510 34 19 𝑎𝑛𝑑 𝑎𝑛𝑑 PROPN iajs-2510 34 20 𝐴 𝐴 PROPN iajs-2510 34 21 ∩ ∩ NOUN iajs-2510 34 22 𝐴 𝐴 PROPN iajs-2510 34 23 are be AUX iajs-2510 34 24 f f NOUN iajs-2510 34 25 - - PUNCT iajs-2510 34 26 sets set NOUN iajs-2510 34 27 in in ADP iajs-2510 34 28 s. s. PROPN iajs-2510 34 29 in in ADP iajs-2510 34 30 general general ADJ iajs-2510 34 31 if if SCONJ iajs-2510 34 32 𝐴 𝐴 PROPN iajs-2510 34 33 , , PUNCT iajs-2510 34 34 𝛼 𝛼 PROPN iajs-2510 34 35 ∈ ∈ PROPN iajs-2510 34 36 λ λ NOUN iajs-2510 34 37 , , PUNCT iajs-2510 34 38 is be AUX iajs-2510 34 39 a a DET iajs-2510 34 40 family family NOUN iajs-2510 34 41 of of ADP iajs-2510 34 42 f f NOUN iajs-2510 34 43 - - PUNCT iajs-2510 34 44 sets set NOUN iajs-2510 34 45 in in ADP iajs-2510 34 46 s s NOUN iajs-2510 34 47 , , PUNCT iajs-2510 34 48 then then ADV iajs-2510 34 49 : : PUNCT iajs-2510 34 50 𝐴 𝐴 PROPN iajs-2510 34 51 ∈ ∈ PROPN iajs-2510 34 52 x x PUNCT iajs-2510 34 53 𝑖𝑛𝑓 𝑖𝑛𝑓 PUNCT iajs-2510 34 54 𝐴 𝐴 PROPN iajs-2510 34 55 x x X iajs-2510 34 56 , , PUNCT iajs-2510 34 57 𝛼 𝛼 PROPN iajs-2510 34 58 ∈ ∈ PROPN iajs-2510 34 59 λ λ X iajs-2510 34 60 , , PUNCT iajs-2510 34 61 𝑓𝑜𝑟 𝑓𝑜𝑟 ADV iajs-2510 34 62 𝑎𝑙𝑙 𝑎𝑙𝑙 VERB iajs-2510 34 63 𝑥 𝑥 PRON iajs-2510 34 64 ∈ ∈ PROPN iajs-2510 34 65 𝑆. 𝑆. PROPN iajs-2510 34 66 𝐴 𝐴 NOUN iajs-2510 34 67 ∈ ∈ NOUN iajs-2510 34 68 x x PUNCT iajs-2510 34 69 sup sup NOUN iajs-2510 34 70 𝐴 𝐴 PROPN iajs-2510 34 71 x x X iajs-2510 34 72 , , PUNCT iajs-2510 34 73 𝛼 𝛼 PROPN iajs-2510 34 74 ∈ ∈ PROPN iajs-2510 34 75 λ λ X iajs-2510 34 76 , , PUNCT iajs-2510 34 77 𝑓𝑜𝑟 𝑓𝑜𝑟 ADV iajs-2510 34 78 𝑎𝑙𝑙 𝑎𝑙𝑙 VERB iajs-2510 34 79 𝑥 𝑥 X iajs-2510 34 80 ∈ ∈ PROPN iajs-2510 34 81 𝑆. 𝑆. PROPN iajs-2510 34 82 now now ADV iajs-2510 34 83 , , PUNCT iajs-2510 34 84 we we PRON iajs-2510 34 85 give give VERB iajs-2510 34 86 the the DET iajs-2510 34 87 definition definition NOUN iajs-2510 34 88 of of ADP iajs-2510 34 89 level level NOUN iajs-2510 34 90 subset subset NOUN iajs-2510 34 91 , , PUNCT iajs-2510 34 92 which which PRON iajs-2510 34 93 is be AUX iajs-2510 34 94 a a DET iajs-2510 34 95 set set NOUN iajs-2510 34 96 between between ADP iajs-2510 34 97 f f NOUN iajs-2510 34 98 - - PUNCT iajs-2510 34 99 set set VERB iajs-2510 34 100 and and CCONJ iajs-2510 34 101 ordinary ordinary ADJ iajs-2510 34 102 set set NOUN iajs-2510 34 103 . . PUNCT iajs-2510 35 1 definition definition NOUN iajs-2510 35 2 1.6 1.6 NUM iajs-2510 36 1 [ [ X iajs-2510 36 2 6 6 NUM iajs-2510 36 3 ] ] PUNCT iajs-2510 36 4 : : PUNCT iajs-2510 36 5 let let VERB iajs-2510 36 6 α α PRON iajs-2510 36 7 be be AUX iajs-2510 36 8 a a DET iajs-2510 36 9 f f NOUN iajs-2510 36 10 - - PUNCT iajs-2510 36 11 set set VERB iajs-2510 36 12 in in ADP iajs-2510 36 13 s. s. PROPN iajs-2510 36 14 for for ADP iajs-2510 36 15 t t PROPN iajs-2510 36 16 ∈ ∈ PROPN iajs-2510 37 1 i i PRON iajs-2510 37 2 , , PUNCT iajs-2510 37 3 the the DET iajs-2510 37 4 set set NOUN iajs-2510 37 5 𝐴 𝐴 PROPN iajs-2510 37 6 𝑥 𝑥 X iajs-2510 37 7 ∈ ∈ PROPN iajs-2510 37 8 𝑆 𝑆 PROPN iajs-2510 37 9 , , PUNCT iajs-2510 37 10 α α PROPN iajs-2510 37 11 𝑥 𝑥 X iajs-2510 37 12 𝑡 𝑡 PROPN iajs-2510 37 13 is be AUX iajs-2510 37 14 called call VERB iajs-2510 37 15 level level NOUN iajs-2510 37 16 𝒔𝒖𝒃𝒔𝒆𝒕 𝒔𝒖𝒃𝒔𝒆𝒕 NOUN iajs-2510 37 17 𝒐𝒇 𝒐𝒇 PUNCT iajs-2510 37 18 𝐗 𝐗 NOUN iajs-2510 37 19 . . PUNCT iajs-2510 37 20 " " PUNCT iajs-2510 38 1 the the DET iajs-2510 38 2 following follow VERB iajs-2510 38 3 are be AUX iajs-2510 38 4 some some DET iajs-2510 38 5 properties property NOUN iajs-2510 38 6 of of ADP iajs-2510 38 7 the the DET iajs-2510 38 8 level level NOUN iajs-2510 38 9 subset subset NOUN iajs-2510 38 10 : : PUNCT iajs-2510 38 11 remark remark VERB iajs-2510 38 12 1.7 1.7 NUM iajs-2510 39 1 [ [ X iajs-2510 39 2 1 1 NUM iajs-2510 39 3 ] ] PUNCT iajs-2510 39 4 : : PUNCT iajs-2510 39 5 let let VERB iajs-2510 39 6 α α PRON iajs-2510 39 7 , , PUNCT iajs-2510 39 8 β β X iajs-2510 39 9 are be AUX iajs-2510 39 10 f f NOUN iajs-2510 39 11 - - PUNCT iajs-2510 39 12 subsets subset NOUN iajs-2510 39 13 of of ADP iajs-2510 39 14 s s PROPN iajs-2510 39 15 , , PUNCT iajs-2510 39 16 t t PROPN iajs-2510 39 17 ∈ ∈ PROPN iajs-2510 40 1 i i PRON iajs-2510 40 2 , , PUNCT iajs-2510 40 3 then then ADV iajs-2510 40 4 : : PUNCT iajs-2510 40 5 1 1 X iajs-2510 40 6 . . X iajs-2510 40 7 𝐴 𝐴 PROPN iajs-2510 40 8 ∩ ∩ NOUN iajs-2510 40 9 𝐵 𝐵 PROPN iajs-2510 40 10 𝐴 𝐴 PROPN iajs-2510 40 11 ∩ ∩ ADJ iajs-2510 40 12 𝐵 𝐵 NOUN iajs-2510 40 13 . . PUNCT iajs-2510 41 1 2 2 X iajs-2510 41 2 . . X iajs-2510 41 3 𝐴 𝐴 PROPN iajs-2510 41 4 ∪ ∪ AUX iajs-2510 41 5 𝐵 𝐵 PROPN iajs-2510 41 6 𝐴 𝐴 PROPN iajs-2510 41 7 ∪ ∪ VERB iajs-2510 41 8 𝐵 𝐵 NOUN iajs-2510 41 9 . . PUNCT iajs-2510 42 1 3 3 X iajs-2510 42 2 . . X iajs-2510 42 3 a a DET iajs-2510 42 4 = = SYM iajs-2510 42 5 b b NOUN iajs-2510 43 1 if if SCONJ iajs-2510 43 2 and and CCONJ iajs-2510 43 3 only only ADV iajs-2510 43 4 if if SCONJ iajs-2510 43 5 𝐴 𝐴 PROPN iajs-2510 43 6 𝐵 𝐵 PROPN iajs-2510 43 7 , , PUNCT iajs-2510 43 8 for for ADP iajs-2510 43 9 all all DET iajs-2510 43 10 t t NOUN iajs-2510 44 1 [ [ X iajs-2510 44 2 0,1 0,1 NUM iajs-2510 44 3 ] ] PUNCT iajs-2510 44 4 . . PUNCT iajs-2510 44 5     SPACE iajs-2510 45 1 67 67 NUM iajs-2510 45 2     SPACE iajs-2510 45 3 ibn ibn PROPN iajs-2510 45 4 al al PROPN iajs-2510 45 5 - - PUNCT iajs-2510 45 6 haitham haitham PROPN iajs-2510 45 7 jour jour X iajs-2510 45 8 . . PROPN iajs-2510 45 9 for for ADP iajs-2510 45 10 pure pure ADJ iajs-2510 45 11 & & CCONJ iajs-2510 45 12 appl appl PROPN iajs-2510 45 13 . . PUNCT iajs-2510 46 1 sci sci PROPN iajs-2510 46 2 . . PROPN iajs-2510 47 1 33 33 NUM iajs-2510 47 2 ( ( PUNCT iajs-2510 47 3 4 4 NUM iajs-2510 47 4 ) ) PUNCT iajs-2510 47 5 2020 2020 NUM iajs-2510 47 6 definition1.8 definition1.8 PROPN iajs-2510 47 7 [ [ X iajs-2510 47 8 7 7 NUM iajs-2510 47 9 ] ] PUNCT iajs-2510 47 10 : : PUNCT iajs-2510 47 11 let let VERB iajs-2510 47 12 f f PRON iajs-2510 47 13 be be AUX iajs-2510 47 14 a a DET iajs-2510 47 15 mapping mapping NOUN iajs-2510 47 16 from from ADP iajs-2510 47 17 a a DET iajs-2510 47 18 set set ADJ iajs-2510 47 19 ℳ ℳ NOUN iajs-2510 47 20 into into ADP iajs-2510 47 21 a a DET iajs-2510 47 22 set set ADJ iajs-2510 47 23 ℳ ℳ NOUN iajs-2510 47 24 , , PUNCT iajs-2510 47 25 let let VERB iajs-2510 47 26 a a PRON iajs-2510 47 27 be be AUX iajs-2510 47 28 a a DET iajs-2510 47 29 f f NOUN iajs-2510 47 30 - - PUNCT iajs-2510 47 31 set set VERB iajs-2510 47 32 in in ADP iajs-2510 47 33 ℳ ℳ PROPN iajs-2510 47 34 and and CCONJ iajs-2510 47 35 b b NOUN iajs-2510 47 36 be be AUX iajs-2510 47 37 a a DET iajs-2510 47 38 f f NOUN iajs-2510 47 39 - - PUNCT iajs-2510 47 40 set set VERB iajs-2510 47 41 in in ADP iajs-2510 47 42 ℳ ℳ PROPN iajs-2510 47 43 . . PUNCT iajs-2510 48 1 the the DET iajs-2510 48 2 image image NOUN iajs-2510 48 3 of of ADP iajs-2510 48 4 a a DET iajs-2510 48 5 denoted denote VERB iajs-2510 48 6 by by ADP iajs-2510 48 7 f f PROPN iajs-2510 48 8 ( ( PUNCT iajs-2510 48 9 a a NOUN iajs-2510 48 10 ) ) PUNCT iajs-2510 48 11 is be AUX iajs-2510 48 12 the the DET iajs-2510 48 13 f f NOUN iajs-2510 48 14 - - PUNCT iajs-2510 48 15 set set VERB iajs-2510 48 16 in in ADP iajs-2510 48 17 ℳ ℳ PROPN iajs-2510 48 18 defined define VERB iajs-2510 48 19 by by ADP iajs-2510 48 20 : : PUNCT iajs-2510 48 21 f f PROPN iajs-2510 48 22 α α PROPN iajs-2510 48 23 ( ( PUNCT iajs-2510 48 24 y y NOUN iajs-2510 48 25 = = PUNCT iajs-2510 48 26 𝑠𝑢𝑝 𝑠𝑢𝑝 NOUN iajs-2510 48 27 𝐴 𝐴 PROPN iajs-2510 48 28 𝑧 𝑧 PROPN iajs-2510 48 29 | | ADV iajs-2510 48 30 𝑧 𝑧 ADP iajs-2510 48 31 ∈ ∈ PROPN iajs-2510 48 32 𝑓 𝑓 PRON iajs-2510 48 33 y y NOUN iajs-2510 48 34 𝑖𝑓 𝑖𝑓 ADP iajs-2510 48 35 𝑓 𝑓 DET iajs-2510 48 36 𝑦 𝑦 NOUN iajs-2510 48 37 ∅ ∅ NOUN iajs-2510 48 38 , , PUNCT iajs-2510 48 39 𝑓𝑜𝑟 𝑓𝑜𝑟 ADV iajs-2510 48 40 𝑒𝑎𝑐ℎ 𝑒𝑎𝑐ℎ PROPN iajs-2510 48 41 y y PROPN iajs-2510 48 42 ∈ ∈ PROPN iajs-2510 48 43 ℳ ℳ PROPN iajs-2510 48 44 0 0 NUM iajs-2510 48 45 𝑜. 𝑜. NOUN iajs-2510 48 46 𝑤 𝑤 ADP iajs-2510 48 47 where where SCONJ iajs-2510 48 48 𝑓 𝑓 PRON iajs-2510 48 49 y y NOUN iajs-2510 48 50 x x PUNCT iajs-2510 48 51 ∶ ∶ VERB iajs-2510 48 52 𝑓 𝑓 PRON iajs-2510 48 53 x x X iajs-2510 48 54 y y PROPN iajs-2510 48 55 and and CCONJ iajs-2510 48 56 the the DET iajs-2510 48 57 inverse inverse NOUN iajs-2510 48 58 of of ADP iajs-2510 48 59 b(x b(x NOUN iajs-2510 48 60 ) ) PUNCT iajs-2510 48 61 , , PUNCT iajs-2510 48 62 denoted denote VERB iajs-2510 48 63 by by ADP iajs-2510 48 64 𝑓 𝑓 DET iajs-2510 48 65 b b NOUN iajs-2510 48 66 is be AUX iajs-2510 48 67 the the DET iajs-2510 48 68 f f NOUN iajs-2510 48 69 - - PUNCT iajs-2510 48 70 set set VERB iajs-2510 48 71 in in ADP iajs-2510 48 72 ℳ ℳ PROPN iajs-2510 48 73 defined define VERB iajs-2510 48 74 by by ADP iajs-2510 48 75 : : PUNCT iajs-2510 48 76 𝑓 𝑓 DET iajs-2510 48 77 β β X iajs-2510 48 78 β β X iajs-2510 48 79 𝑓 𝑓 X iajs-2510 48 80 x x X iajs-2510 48 81 , , PUNCT iajs-2510 48 82 for for ADP iajs-2510 48 83 all all DET iajs-2510 48 84 x x SYM iajs-2510 48 85 ∈ ∈ PROPN iajs-2510 48 86 ℳ ℳ PROPN iajs-2510 48 87 . . PUNCT iajs-2510 49 1 definition definition NOUN iajs-2510 49 2 1.9 1.9 NUM iajs-2510 50 1 [ [ X iajs-2510 50 2 8 8 NUM iajs-2510 50 3 ] ] PUNCT iajs-2510 50 4 : : PUNCT iajs-2510 50 5 let let VERB iajs-2510 50 6 f f PRON iajs-2510 50 7 be be AUX iajs-2510 50 8 a a DET iajs-2510 50 9 function function NOUN iajs-2510 50 10 from from ADP iajs-2510 50 11 a a DET iajs-2510 50 12 set set ADJ iajs-2510 50 13 ℳ ℳ NOUN iajs-2510 50 14 into into ADP iajs-2510 50 15 a a DET iajs-2510 50 16 set set ADJ iajs-2510 50 17 ℳ ℳ NOUN iajs-2510 50 18 . . 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PUNCT iajs-2510 52 1 proposition proposition NOUN iajs-2510 52 2 1.10 1.10 NUM iajs-2510 52 3 [ [ X iajs-2510 52 4 8 8 NUM iajs-2510 52 5 ] ] X iajs-2510 52 6 : : PUNCT iajs-2510 52 7 if if SCONJ iajs-2510 52 8 f f PROPN iajs-2510 52 9 is be AUX iajs-2510 52 10 a a DET iajs-2510 52 11 function function NOUN iajs-2510 52 12 defined define VERB iajs-2510 52 13 on on ADP iajs-2510 52 14 a a DET iajs-2510 52 15 set set ADJ iajs-2510 52 16 ℳ ℳ NOUN iajs-2510 52 17 , , PUNCT iajs-2510 52 18 𝐴 𝐴 PROPN iajs-2510 52 19 𝑎𝑛𝑑𝐴 𝑎𝑛𝑑𝐴 X iajs-2510 52 20 are be AUX iajs-2510 52 21 f f NOUN iajs-2510 52 22 - - PUNCT iajs-2510 52 23 subsets subset NOUN iajs-2510 52 24 of of ADP iajs-2510 52 25 ℳ ℳ PROPN iajs-2510 52 26 , , PUNCT iajs-2510 52 27 𝐵 𝐵 NOUN iajs-2510 52 28 𝑎𝑛𝑑 𝑎𝑛𝑑 ADJ iajs-2510 52 29 𝐵 𝐵 PROPN iajs-2510 52 30 are be AUX iajs-2510 52 31 fsubset fsubset VERB iajs-2510 52 32 of of ADP iajs-2510 52 33 f f PROPN iajs-2510 52 34 ( ( PUNCT iajs-2510 52 35 ℳ ℳ PROPN iajs-2510 52 36 ) ) PUNCT iajs-2510 52 37 . . PUNCT iajs-2510 53 1 the the DET iajs-2510 53 2 followings following NOUN iajs-2510 53 3 are be AUX iajs-2510 53 4 true true ADJ iajs-2510 53 5 : : PUNCT iajs-2510 53 6 1 1 X iajs-2510 53 7 . . X iajs-2510 53 8 𝐴 𝐴 PROPN iajs-2510 53 9 ⊆ ⊆ NUM iajs-2510 53 10 𝑓 𝑓 DET iajs-2510 53 11 𝑓 𝑓 DET iajs-2510 53 12 𝐴 𝐴 PROPN iajs-2510 53 13 . . PUNCT iajs-2510 54 1 2 2 X iajs-2510 54 2 . . X iajs-2510 54 3 𝐴 𝐴 PROPN iajs-2510 54 4 𝑓 𝑓 PROPN iajs-2510 54 5 𝑓 𝑓 PROPN iajs-2510 54 6 𝐴 𝐴 PROPN iajs-2510 54 7 , , PUNCT iajs-2510 54 8 whenever whenever SCONJ iajs-2510 54 9 𝐴 𝐴 PROPN iajs-2510 54 10 is be AUX iajs-2510 54 11 f f NOUN iajs-2510 54 12 - - PUNCT iajs-2510 54 13 invariant invariant ADJ iajs-2510 54 14 . . PUNCT iajs-2510 55 1 3 3 X iajs-2510 55 2 . . X iajs-2510 55 3 𝑓 𝑓 DET iajs-2510 55 4 𝑓 𝑓 DET iajs-2510 55 5 𝐵 𝐵 NOUN iajs-2510 55 6 𝐵 𝐵 NOUN iajs-2510 55 7 . . PUNCT iajs-2510 56 1 4 4 X iajs-2510 56 2 . . X iajs-2510 56 3 if if SCONJ iajs-2510 56 4 𝐴 𝐴 PROPN iajs-2510 56 5 ⊆ ⊆ NUM iajs-2510 56 6 𝐴 𝐴 PROPN iajs-2510 56 7 , , PUNCT iajs-2510 56 8 then then ADV iajs-2510 56 9 𝑓 𝑓 DET iajs-2510 56 10 𝐴 𝐴 PROPN iajs-2510 56 11 ⊆ ⊆ NUM iajs-2510 56 12 𝑓 𝑓 PRON iajs-2510 56 13 𝐴 𝐴 PROPN iajs-2510 56 14 . . PUNCT iajs-2510 57 1 5 5 X iajs-2510 57 2 . . X iajs-2510 57 3 if if SCONJ iajs-2510 57 4 𝐵 𝐵 PROPN iajs-2510 57 5 ⊆ ⊆ NUM iajs-2510 57 6 𝐵 𝐵 NOUN iajs-2510 57 7 , , PUNCT iajs-2510 57 8 then then ADV iajs-2510 57 9 𝑓 𝑓 DET iajs-2510 57 10 𝐵 𝐵 NOUN iajs-2510 57 11 ⊆ ⊆ PROPN iajs-2510 57 12 𝑓 𝑓 DET iajs-2510 57 13 𝐵 𝐵 NOUN iajs-2510 57 14 . . PUNCT iajs-2510 58 1 6 6 X iajs-2510 58 2 . . X iajs-2510 58 3 let let VERB iajs-2510 58 4 f f PRON iajs-2510 58 5 be be AUX iajs-2510 58 6 a a DET iajs-2510 58 7 function function NOUN iajs-2510 58 8 from from ADP iajs-2510 58 9 a a DET iajs-2510 58 10 set set ADJ iajs-2510 58 11 ℳ ℳ NOUN iajs-2510 58 12 into into ADP iajs-2510 58 13 n. n. 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PUNCT iajs-2510 66 1 if if SCONJ iajs-2510 66 2 𝐴 𝐴 PROPN iajs-2510 66 3 and and CCONJ iajs-2510 66 4 𝐴 𝐴 PROPN iajs-2510 66 5 are be AUX iajs-2510 66 6 two two NUM iajs-2510 66 7 f f NOUN iajs-2510 66 8 - - PUNCT iajs-2510 66 9 submodules submodule NOUN iajs-2510 66 10 of of ADP iajs-2510 66 11 x x X iajs-2510 66 12 and and CCONJ iajs-2510 66 13 x x NOUN iajs-2510 66 14 resp resp NOUN iajs-2510 66 15 . . PUNCT iajs-2510 66 16 , , PUNCT iajs-2510 66 17 then then ADV iajs-2510 66 18 : : PUNCT iajs-2510 66 19 1 1 X iajs-2510 66 20 . . X iajs-2510 66 21 𝑓 𝑓 DET iajs-2510 66 22 𝐴 𝐴 PROPN iajs-2510 66 23 is be AUX iajs-2510 66 24 a a DET iajs-2510 66 25 f f NOUN iajs-2510 66 26 - - PUNCT iajs-2510 66 27 submodule submodule NOUN iajs-2510 66 28 of of ADP iajs-2510 66 29 x x PROPN iajs-2510 66 30 . . PUNCT iajs-2510 67 1 2 2 X iajs-2510 67 2 . . 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PROPN iajs-2510 69 16 then then ADV iajs-2510 69 17 , , PUNCT iajs-2510 69 18 the the DET iajs-2510 69 19 level level NOUN iajs-2510 69 20 subset subset NOUN iajs-2510 69 21 𝐴 𝐴 PROPN iajs-2510 69 22 , , PUNCT iajs-2510 69 23 t t PROPN iajs-2510 69 24 ∈ ∈ PROPN iajs-2510 70 1 i i PRON iajs-2510 70 2 , , PUNCT iajs-2510 70 3 is be AUX iajs-2510 70 4 a a DET iajs-2510 70 5 submodule submodule NOUN iajs-2510 70 6 of of ADP iajs-2510 70 7 ℳ ℳ PROPN iajs-2510 70 8 iff iff PROPN iajs-2510 70 9 α α PROPN iajs-2510 70 10 is be AUX iajs-2510 70 11 ϝ-submodule ϝ-submodule NOUN iajs-2510 70 12 of of ADP iajs-2510 70 13 x. x. 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NOUN iajs-2510 80 78 " " PUNCT iajs-2510 80 79 definition definition NOUN iajs-2510 80 80 2.2 2.2 NUM iajs-2510 80 81 : : PUNCT iajs-2510 80 82 let let VERB iajs-2510 80 83 𝐴 𝐴 PROPN iajs-2510 80 84 be be AUX iajs-2510 80 85 f f NOUN iajs-2510 80 86 - - PUNCT iajs-2510 80 87 submodule submodule NOUN iajs-2510 80 88 of of ADP iajs-2510 80 89 f f NOUN iajs-2510 80 90 - - PUNCT iajs-2510 80 91 module module NOUN iajs-2510 80 92 x. x. NOUN iajs-2510 80 93 𝐴 𝐴 PROPN iajs-2510 80 94 is be AUX iajs-2510 80 95 a a DET iajs-2510 80 96 said say VERB iajs-2510 80 97 weak weak ADJ iajs-2510 80 98 essential essential ADJ iajs-2510 80 99 f f NOUN iajs-2510 80 100 - - PUNCT iajs-2510 80 101 submodule submodule NOUN iajs-2510 80 102 if if SCONJ iajs-2510 80 103 𝐴 𝐴 PROPN iajs-2510 80 104 ∩ ∩ NOUN iajs-2510 80 105 𝑆 𝑆 PROPN iajs-2510 80 106 0 0 NUM iajs-2510 80 107 , , PUNCT iajs-2510 80 108 for for ADP iajs-2510 80 109 each each DET iajs-2510 80 110 non non ADJ iajs-2510 80 111 - - ADJ iajs-2510 80 112 trivial trivial ADJ iajs-2510 80 113 semiprime semiprime NOUN iajs-2510 80 114 f f X iajs-2510 80 115 - - PUNCT iajs-2510 80 116 submodules submodule NOUN iajs-2510 80 117 of of ADP iajs-2510 80 118 x. x. NOUN iajs-2510 80 119 equivalently equivalently ADV iajs-2510 80 120 fsubmodule fsubmodule VERB iajs-2510 80 121 a a PRON iajs-2510 80 122 of of ADP iajs-2510 80 123 a a DET iajs-2510 80 124 f f NOUN iajs-2510 80 125 - - PUNCT iajs-2510 80 126 module module NOUN iajs-2510 80 127 x x PRON iajs-2510 80 128 is be AUX iajs-2510 80 129 called call VERB iajs-2510 80 130 weak weak ADJ iajs-2510 80 131 essential essential ADJ iajs-2510 80 132 f f NOUN iajs-2510 80 133 - - PUNCT iajs-2510 80 134 submodule submodule NOUN iajs-2510 80 135 if if SCONJ iajs-2510 80 136 a a DET iajs-2510 80 137 ∩ ∩ ADJ iajs-2510 80 138 𝑆 𝑆 PROPN iajs-2510 80 139 0 0 NUM iajs-2510 80 140 , , PUNCT iajs-2510 80 141 then then ADV iajs-2510 80 142 s s VERB iajs-2510 80 143 0 0 NUM iajs-2510 80 144 , , PUNCT iajs-2510 80 145 for for ADP iajs-2510 80 146 every every DET iajs-2510 80 147 semiprime semiprime NOUN iajs-2510 80 148 f f X iajs-2510 80 149 - - PUNCT iajs-2510 80 150 submodule submodule NOUN iajs-2510 80 151 of of ADP iajs-2510 80 152 x x PROPN iajs-2510 80 153 . . PUNCT iajs-2510 81 1 next next ADV iajs-2510 81 2 , , PUNCT iajs-2510 81 3 proposition proposition NOUN iajs-2510 81 4 is be AUX iajs-2510 81 5 a a DET iajs-2510 81 6 characterization characterization NOUN iajs-2510 81 7 of of ADP iajs-2510 81 8 a a DET iajs-2510 81 9 weak weak ADJ iajs-2510 81 10 essential essential ADJ iajs-2510 81 11 f f NOUN iajs-2510 81 12 - - PUNCT iajs-2510 81 13 submodule submodule NOUN iajs-2510 81 14 . . 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PUNCT iajs-2510 83 1 proof proof NOUN iajs-2510 83 2 : : PUNCT iajs-2510 83 3 suppose suppose VERB iajs-2510 83 4 that that SCONJ iajs-2510 83 5 non non ADJ iajs-2510 83 6 - - ADJ iajs-2510 83 7 trivial trivial ADJ iajs-2510 83 8 semiprime semiprime NOUN iajs-2510 83 9 f f X iajs-2510 83 10 - - PUNCT iajs-2510 83 11 submodule submodule NOUN iajs-2510 83 12 s s PROPN iajs-2510 83 13 of of ADP iajs-2510 83 14 x x NOUN iajs-2510 83 15 , , PUNCT iajs-2510 83 16 there there PRON iajs-2510 83 17 exists exist VERB iajs-2510 83 18 x x X iajs-2510 83 19 ⊆ ⊆ NUM iajs-2510 83 20 𝑆 𝑆 PROPN iajs-2510 83 21 and and CCONJ iajs-2510 83 22 r r NOUN iajs-2510 83 23 of of ADP iajs-2510 83 24 ℛ ℛ NOUN iajs-2510 83 25 such such ADJ iajs-2510 83 26 that that DET iajs-2510 83 27 0 0 NUM iajs-2510 83 28 x x SYM iajs-2510 83 29 r r NOUN iajs-2510 83 30 ⊆ ⊆ NUM iajs-2510 83 31 𝐴. 𝐴. NOUN iajs-2510 83 32 note note NOUN iajs-2510 83 33 that that SCONJ iajs-2510 83 34 x x PUNCT iajs-2510 83 35 r r NOUN iajs-2510 83 36 ⊆ ⊆ NUM iajs-2510 83 37 𝑆. 𝑆. NOUN iajs-2510 83 38 0 0 NUM iajs-2510 83 39 x x SYM iajs-2510 83 40 r r NOUN iajs-2510 83 41 ⊆ ⊆ NUM iajs-2510 83 42 𝐴 𝐴 PROPN iajs-2510 83 43 ∩ ∩ NOUN iajs-2510 83 44 𝐵. 𝐵. NOUN iajs-2510 83 45 thus thus ADV iajs-2510 83 46 a∩ a∩ PROPN iajs-2510 83 47 𝐵 𝐵 NOUN iajs-2510 83 48 0 0 NUM iajs-2510 83 49 , , PUNCT iajs-2510 83 50 that that PRON iajs-2510 83 51 is be AUX iajs-2510 83 52 a a DET iajs-2510 83 53 is be AUX iajs-2510 83 54 weak weak ADJ iajs-2510 83 55 essential essential ADJ iajs-2510 83 56 f f NOUN iajs-2510 83 57 - - PUNCT iajs-2510 83 58 submodule submodule NOUN iajs-2510 83 59 . . 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NOUN iajs-2510 84 28 thus thus ADV iajs-2510 84 29 , , PUNCT iajs-2510 84 30 there there PRON iajs-2510 84 31 exists exist VERB iajs-2510 84 32 0 0 NUM iajs-2510 84 33 x x SYM iajs-2510 84 34 ⊆ ⊆ NUM iajs-2510 84 35 𝐴 𝐴 PROPN iajs-2510 84 36 ∩ ∩ ADJ iajs-2510 84 37 𝑆 𝑆 PROPN iajs-2510 84 38 , , PUNCT iajs-2510 84 39 implying imply VERB iajs-2510 84 40 that that SCONJ iajs-2510 84 41 x x PROPN iajs-2510 84 42 ⊆ ⊆ NUM iajs-2510 84 43 𝐴 𝐴 PROPN iajs-2510 84 44 and and CCONJ iajs-2510 84 45 hence hence ADV iajs-2510 84 46 0 0 NUM iajs-2510 85 1 𝑥 𝑥 PRON iajs-2510 85 2 𝑟 𝑟 NOUN iajs-2510 85 3 ⊆ ⊆ NUM iajs-2510 85 4 𝐴 𝐴 PROPN iajs-2510 85 5 , , PUNCT iajs-2510 85 6 ∀ ∀ PUNCT iajs-2510 85 7 𝑡 𝑡 NOUN iajs-2510 85 8 ∈ ∈ NOUN iajs-2510 85 9 0,1 0,1 NUM iajs-2510 85 10 . . 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PUNCT iajs-2510 87 1 lemma lemma PROPN iajs-2510 87 2 2.4 2.4 NUM iajs-2510 87 3 : : PUNCT iajs-2510 87 4 let let VERB iajs-2510 87 5 a a PRON iajs-2510 87 6 be be AUX iajs-2510 87 7 a a DET iajs-2510 87 8 f f NOUN iajs-2510 87 9 - - PUNCT iajs-2510 87 10 submodule submodule NOUN iajs-2510 87 11 of of ADP iajs-2510 87 12 a a DET iajs-2510 87 13 f f NOUN iajs-2510 87 14 - - PUNCT iajs-2510 87 15 module module NOUN iajs-2510 87 16 x x PUNCT iajs-2510 87 17 if if SCONJ iajs-2510 87 18 𝐴 𝐴 PROPN iajs-2510 87 19 weak weak ADJ iajs-2510 87 20 essential essential ADJ iajs-2510 87 21 submodule submodule NOUN iajs-2510 87 22 of of ADP iajs-2510 87 23 x x PRON iajs-2510 87 24 , , PUNCT iajs-2510 87 25 ∀ ∀ PUNCT iajs-2510 87 26 𝑡 𝑡 PROPN iajs-2510 87 27 ∈ ∈ PROPN iajs-2510 87 28 i. i. NOUN iajs-2510 87 29 then then ADV iajs-2510 87 30 α α PROPN iajs-2510 87 31 is be AUX iajs-2510 87 32 weak weak ADJ iajs-2510 87 33 essential essential ADJ iajs-2510 87 34 f f NOUN iajs-2510 87 35 - - PUNCT iajs-2510 87 36 submodule submodule NOUN iajs-2510 87 37 in in ADP iajs-2510 87 38 x. x. NOUN iajs-2510 87 39 proof proof NOUN iajs-2510 87 40 : : PUNCT iajs-2510 87 41 assume assume VERB iajs-2510 87 42 β β X iajs-2510 87 43 a a DET iajs-2510 87 44 semiprime semiprime NOUN iajs-2510 87 45 f f X iajs-2510 87 46 - - PUNCT iajs-2510 87 47 submodule submodule NOUN iajs-2510 87 48 of of ADP iajs-2510 87 49 x x SYM iajs-2510 87 50 such such ADJ iajs-2510 87 51 that that DET iajs-2510 87 52 b b NOUN iajs-2510 87 53 0 0 NUM iajs-2510 87 54 , , PUNCT iajs-2510 87 55 since since SCONJ iajs-2510 87 56 b b NUM iajs-2510 87 57 semiprime semiprime NOUN iajs-2510 87 58 f f PROPN iajs-2510 87 59 - - PUNCT iajs-2510 87 60 submodule submodule NOUN iajs-2510 87 61 of of ADP iajs-2510 87 62 x x PRON iajs-2510 87 63 , , PUNCT iajs-2510 87 64 hence hence ADV iajs-2510 87 65 𝐵 𝐵 PROPN iajs-2510 87 66 semiprime semiprime NOUN iajs-2510 87 67 submodule submodule NOUN iajs-2510 87 68 of of ADP iajs-2510 87 69 x x PRON iajs-2510 87 70 , , PUNCT iajs-2510 87 71 ∀ ∀ PUNCT iajs-2510 87 72 𝑡 𝑡 NOUN iajs-2510 87 73 ∈ ∈ NOUN iajs-2510 87 74 0,1 0,1 NUM iajs-2510 87 75 , , PUNCT iajs-2510 87 76 see see VERB iajs-2510 87 77 [ [ X iajs-2510 87 78 14 14 NUM iajs-2510 87 79 , , PUNCT iajs-2510 87 80 theorem(2.4 theorem(2.4 NOUN iajs-2510 87 81 ) ) PUNCT iajs-2510 87 82 ] ] PUNCT iajs-2510 87 83 , , PUNCT iajs-2510 87 84 which which PRON iajs-2510 87 85 implies imply VERB iajs-2510 87 86 𝐴 𝐴 PROPN iajs-2510 87 87 ∩ ∩ ADJ iajs-2510 87 88 𝐵 𝐵 NOUN iajs-2510 87 89 0 0 NUM iajs-2510 87 90 , , PUNCT iajs-2510 87 91 since since SCONJ iajs-2510 87 92 𝐴 𝐴 PROPN iajs-2510 87 93 is be AUX iajs-2510 87 94 weak weak ADJ iajs-2510 87 95 essential essential ADJ iajs-2510 87 96 submodule submodule NOUN iajs-2510 87 97 and and CCONJ iajs-2510 87 98 𝐴 𝐴 PROPN iajs-2510 87 99 ∩ ∩ ADJ iajs-2510 87 100 𝐵 𝐵 PROPN iajs-2510 87 101 𝐴 𝐴 PROPN iajs-2510 87 102 ∩ ∩ ADJ iajs-2510 87 103 𝐵 𝐵 NOUN iajs-2510 87 104 0 0 NUM iajs-2510 87 105 , , PUNCT iajs-2510 87 106 hence hence ADV iajs-2510 87 107 a a DET iajs-2510 87 108 ∩ ∩ ADJ iajs-2510 87 109 𝐵 𝐵 NOUN iajs-2510 87 110 0 0 NUM iajs-2510 87 111 by by ADP iajs-2510 87 112 remark remark NOUN iajs-2510 87 113 ( ( PUNCT iajs-2510 87 114 1.7)(3 1.7)(3 NUM iajs-2510 87 115 ) ) PUNCT iajs-2510 87 116 . . PUNCT iajs-2510 88 1 thus thus ADV iajs-2510 88 2 , , PUNCT iajs-2510 88 3 a a PRON iajs-2510 88 4 is be AUX iajs-2510 88 5 a a DET iajs-2510 88 6 weak weak ADJ iajs-2510 88 7 essential essential ADJ iajs-2510 88 8 fsubmodule fsubmodule NOUN iajs-2510 88 9 of of ADP iajs-2510 88 10 x. x. NOUN iajs-2510 88 11 remark remark NOUN iajs-2510 88 12 2.5 2.5 NUM iajs-2510 88 13 : : PUNCT iajs-2510 88 14 every every DET iajs-2510 88 15 essential essential ADJ iajs-2510 88 16 f f X iajs-2510 88 17 - - PUNCT iajs-2510 88 18 submodule submodule NOUN iajs-2510 88 19 is be AUX iajs-2510 88 20 weak weak ADJ iajs-2510 88 21 essential essential ADJ iajs-2510 88 22 f f NOUN iajs-2510 88 23 - - PUNCT iajs-2510 88 24 submodule submodule NOUN iajs-2510 88 25 . . PUNCT iajs-2510 89 1 but but CCONJ iajs-2510 89 2 the the DET iajs-2510 89 3 converse converse NOUN iajs-2510 89 4 is be AUX iajs-2510 89 5 not not PART iajs-2510 89 6 true true ADJ iajs-2510 89 7 in in ADP iajs-2510 89 8 general general ADJ iajs-2510 89 9 , , PUNCT iajs-2510 89 10 for for ADP iajs-2510 89 11 example example NOUN iajs-2510 89 12 : : PUNCT iajs-2510 89 13 example example NOUN iajs-2510 89 14 : : PUNCT iajs-2510 89 15 let let VERB iajs-2510 89 16 ℳ ℳ NOUN iajs-2510 89 17 = = SYM iajs-2510 89 18 𝑍 𝑍 PROPN iajs-2510 89 19 as as ADP iajs-2510 89 20 z z NOUN iajs-2510 89 21 - - PUNCT iajs-2510 89 22 module module NOUN iajs-2510 89 23 . . PUNCT iajs-2510 90 1 define define VERB iajs-2510 90 2 x x NOUN iajs-2510 90 3 : : PUNCT iajs-2510 90 4 ℳ ℳ PROPN iajs-2510 90 5 ⟶ ⟶ NOUN iajs-2510 90 6 i i PROPN iajs-2510 90 7 , , PUNCT iajs-2510 90 8 by by ADP iajs-2510 90 9 : : PUNCT iajs-2510 90 10     SPACE iajs-2510 90 11 69 69 NUM iajs-2510 90 12     SPACE iajs-2510 90 13 ibn ibn PROPN iajs-2510 90 14 al al PROPN iajs-2510 90 15 - - PUNCT iajs-2510 90 16 haitham haitham PROPN iajs-2510 90 17 jour jour X iajs-2510 90 18 . . PROPN iajs-2510 91 1 for for ADP iajs-2510 91 2 pure pure ADJ iajs-2510 91 3 & & CCONJ iajs-2510 91 4 appl appl PROPN iajs-2510 91 5 . . PUNCT iajs-2510 92 1 sci sci PROPN iajs-2510 92 2 . . PROPN iajs-2510 93 1 33 33 NUM iajs-2510 93 2 ( ( PUNCT iajs-2510 93 3 4 4 NUM iajs-2510 93 4 ) ) PUNCT iajs-2510 93 5 2020 2020 NUM iajs-2510 94 1 x(a x(a NOUN iajs-2510 94 2 ) ) PUNCT iajs-2510 94 3 = = SYM iajs-2510 94 4 1 1 NUM iajs-2510 94 5 , , PUNCT iajs-2510 94 6 for for ADP iajs-2510 94 7 all all PRON iajs-2510 94 8 𝑎 𝑎 PRON iajs-2510 94 9 ∈ ∈ NOUN iajs-2510 94 10 𝑍 𝑍 NOUN iajs-2510 94 11 let let VERB iajs-2510 94 12 a a DET iajs-2510 94 13 : : PUNCT iajs-2510 94 14 ℳ ℳ PROPN iajs-2510 94 15 ⟶ ⟶ NOUN iajs-2510 94 16 i i PROPN iajs-2510 94 17 , , PUNCT iajs-2510 94 18 define define VERB iajs-2510 94 19 by by ADP iajs-2510 94 20 : : PUNCT iajs-2510 94 21 a(x a(x PROPN iajs-2510 94 22 ) ) PUNCT iajs-2510 94 23 = = SYM iajs-2510 95 1 1 1 NUM iajs-2510 95 2 𝑖𝑓 𝑖𝑓 ADP iajs-2510 95 3 x x SYM iajs-2510 95 4 0 0 NUM iajs-2510 95 5 1 1 NUM iajs-2510 95 6 2 2 NUM iajs-2510 95 7 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 95 8 x x SYM iajs-2510 95 9 ∈ ∈ PROPN iajs-2510 95 10 9 9 NUM iajs-2510 95 11 0 0 NUM iajs-2510 95 12 0 0 NUM iajs-2510 95 13 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 95 14 it it PRON iajs-2510 95 15 is be AUX iajs-2510 95 16 clear clear ADJ iajs-2510 95 17 that that SCONJ iajs-2510 95 18 a a DET iajs-2510 95 19 f f X iajs-2510 95 20 - - PUNCT iajs-2510 95 21 submodule submodule NOUN iajs-2510 95 22 of of ADP iajs-2510 95 23 x x PROPN iajs-2510 95 24 , , PUNCT iajs-2510 95 25 𝐴 𝐴 PROPN iajs-2510 95 26 9 9 NUM iajs-2510 95 27 is be AUX iajs-2510 95 28 weak weak ADJ iajs-2510 95 29 essential essential ADJ iajs-2510 95 30 by by ADP iajs-2510 95 31 [ [ X iajs-2510 95 32 4 4 NUM iajs-2510 95 33 , , PUNCT iajs-2510 95 34 remarks(1.5 remarks(1.5 NOUN iajs-2510 95 35 ) ) PUNCT iajs-2510 95 36 ] ] PUNCT iajs-2510 95 37 , , PUNCT iajs-2510 95 38 then then ADV iajs-2510 95 39 a a PRON iajs-2510 95 40 is be AUX iajs-2510 95 41 weak weak ADJ iajs-2510 95 42 essential essential ADJ iajs-2510 95 43 f f NOUN iajs-2510 95 44 - - PUNCT iajs-2510 95 45 submodule submodule NOUN iajs-2510 95 46 by by ADP iajs-2510 95 47 lemma(2.4 lemma(2.4 NOUN iajs-2510 95 48 ) ) PUNCT iajs-2510 95 49 . . PUNCT iajs-2510 96 1 let let VERB iajs-2510 96 2 b b NOUN iajs-2510 96 3 : : PUNCT iajs-2510 96 4 ℳ ℳ PROPN iajs-2510 96 5 ⟶ ⟶ NOUN iajs-2510 96 6 i i PRON iajs-2510 96 7 , , PUNCT iajs-2510 96 8 as as SCONJ iajs-2510 96 9 defined define VERB iajs-2510 96 10 by by ADP iajs-2510 96 11 : : PUNCT iajs-2510 96 12 b(x b(x NOUN iajs-2510 96 13 ) ) PUNCT iajs-2510 96 14 = = SYM iajs-2510 96 15 1 1 NUM iajs-2510 96 16 𝑖𝑓 𝑖𝑓 ADP iajs-2510 96 17 x x SYM iajs-2510 96 18 0 0 NUM iajs-2510 96 19 1 1 NUM iajs-2510 96 20 2 2 NUM iajs-2510 96 21 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 96 22 x x SYM iajs-2510 96 23 ∈ ∈ PROPN iajs-2510 96 24 4 4 NUM iajs-2510 96 25 0 0 NUM iajs-2510 96 26 0 0 NUM iajs-2510 96 27 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 96 28 it it PRON iajs-2510 96 29 is be AUX iajs-2510 96 30 clear clear ADJ iajs-2510 96 31 that that SCONJ iajs-2510 96 32 b b X iajs-2510 96 33 f f X iajs-2510 96 34 - - PUNCT iajs-2510 96 35 submodule submodule NOUN iajs-2510 96 36 of of ADP iajs-2510 96 37 x. x. PROPN iajs-2510 96 38 a a PRON iajs-2510 96 39 is be AUX iajs-2510 96 40 not not PART iajs-2510 96 41 essential essential ADJ iajs-2510 96 42 , , PUNCT iajs-2510 96 43 since since SCONJ iajs-2510 96 44 α α PRON iajs-2510 96 45 ∩ ∩ ADJ iajs-2510 96 46 𝐵 𝐵 NOUN iajs-2510 96 47 x x SYM iajs-2510 96 48 1 1 NUM iajs-2510 96 49 𝑖𝑓 𝑖𝑓 ADP iajs-2510 96 50 x x SYM iajs-2510 96 51 0 0 NUM iajs-2510 96 52 0 0 NUM iajs-2510 96 53 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 VERB iajs-2510 96 54 a a DET iajs-2510 96 55 ∩ ∩ ADJ iajs-2510 96 56 𝐵 𝐵 NOUN iajs-2510 96 57 0 0 NUM iajs-2510 96 58 and and CCONJ iajs-2510 96 59 b b NOUN iajs-2510 96 60 0 0 NUM iajs-2510 96 61 ; ; PUNCT iajs-2510 96 62 therefore therefore ADV iajs-2510 96 63 a a PRON iajs-2510 96 64 is be AUX iajs-2510 96 65 not not PART iajs-2510 96 66 essential essential ADJ iajs-2510 96 67 f f NOUN iajs-2510 96 68 - - PUNCT iajs-2510 96 69 submodule submodule NOUN iajs-2510 96 70 . . PUNCT iajs-2510 97 1 remark remark VERB iajs-2510 97 2 2.6 2.6 NUM iajs-2510 97 3 : : PUNCT iajs-2510 98 1 the the DET iajs-2510 98 2 converse converse NOUN iajs-2510 98 3 of of ADP iajs-2510 98 4 lemma lemma PROPN iajs-2510 98 5 ( ( PUNCT iajs-2510 98 6 2.4 2.4 NUM iajs-2510 98 7 ) ) PUNCT iajs-2510 98 8 is be AUX iajs-2510 98 9 not not PART iajs-2510 98 10 true true ADJ iajs-2510 98 11 in in ADP iajs-2510 98 12 general general ADJ iajs-2510 98 13 . . PUNCT iajs-2510 98 14 example example NOUN iajs-2510 98 15 2.7 2.7 NUM iajs-2510 98 16 : : PUNCT iajs-2510 98 17 let let VERB iajs-2510 98 18 ℳ ℳ NOUN iajs-2510 98 19 = = SYM iajs-2510 98 20 𝑍 𝑍 PROPN iajs-2510 98 21 as as ADP iajs-2510 98 22 z z NOUN iajs-2510 98 23 - - PUNCT iajs-2510 98 24 module module NOUN iajs-2510 98 25 . . PUNCT iajs-2510 99 1 define define VERB iajs-2510 99 2 x x NOUN iajs-2510 99 3 : : PUNCT iajs-2510 99 4 ℳ ℳ PROPN iajs-2510 99 5 ⟶ ⟶ NOUN iajs-2510 99 6 i i PROPN iajs-2510 99 7 , , PUNCT iajs-2510 99 8 a a DET iajs-2510 99 9 : : PUNCT iajs-2510 99 10 ℳ ℳ PROPN iajs-2510 99 11 ⟶ ⟶ NOUN iajs-2510 99 12 i i PRON iajs-2510 99 13 by by ADP iajs-2510 99 14 : : PUNCT iajs-2510 99 15 x(a x(a PROPN iajs-2510 99 16 ) ) PUNCT iajs-2510 99 17 = = SYM iajs-2510 100 1 1 1 NUM iajs-2510 100 2 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 100 3 𝑎 𝑎 ADJ iajs-2510 100 4 0 0 NUM iajs-2510 100 5 1 1 NUM iajs-2510 100 6 2 2 NUM iajs-2510 100 7 𝑖𝑓 𝑖𝑓 ADP iajs-2510 100 8 𝑎 𝑎 PRON iajs-2510 100 9 2,4 2,4 NUM iajs-2510 100 10 0 0 NUM iajs-2510 100 11 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 100 12 , , PUNCT iajs-2510 100 13 a(a a(a PROPN iajs-2510 100 14 ) ) PUNCT iajs-2510 100 15 = = PUNCT iajs-2510 101 1 1 1 NUM iajs-2510 101 2 𝑖𝑓 𝑖𝑓 X iajs-2510 101 3 𝑎 𝑎 ADJ iajs-2510 101 4 0 0 NUM iajs-2510 101 5 1 1 NUM iajs-2510 101 6 3 3 NUM iajs-2510 101 7 𝑖𝑓 𝑖𝑓 ADP iajs-2510 101 8 𝑎 𝑎 PRON iajs-2510 101 9 2,4 2,4 NUM iajs-2510 101 10 0 0 NUM iajs-2510 101 11 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 101 12 a a PRON iajs-2510 101 13 is be AUX iajs-2510 101 14 an an DET iajs-2510 101 15 essential essential ADJ iajs-2510 101 16 f f NOUN iajs-2510 101 17 - - PUNCT iajs-2510 101 18 submodule submodule NOUN iajs-2510 101 19 , , PUNCT iajs-2510 101 20 then then ADV iajs-2510 101 21 a a PRON iajs-2510 101 22 is be AUX iajs-2510 101 23 weak weak ADJ iajs-2510 101 24 essential essential ADJ iajs-2510 101 25 by by ADP iajs-2510 101 26 remark remark NOUN iajs-2510 101 27 ( ( PUNCT iajs-2510 101 28 2.5 2.5 NUM iajs-2510 101 29 ) ) PUNCT iajs-2510 101 30 , , PUNCT iajs-2510 101 31 but but CCONJ iajs-2510 101 32 𝐴 𝐴 PROPN iajs-2510 101 33 0 0 NUM iajs-2510 101 34 is be AUX iajs-2510 101 35 not not PART iajs-2510 101 36 essential essential ADJ iajs-2510 101 37 see see NOUN iajs-2510 101 38 [ [ X iajs-2510 101 39 15 15 NUM iajs-2510 101 40 , , PUNCT iajs-2510 101 41 remark remark NOUN iajs-2510 101 42 ( ( PUNCT iajs-2510 101 43 2.1 2.1 NUM iajs-2510 101 44 ) ) PUNCT iajs-2510 101 45 ] ] PUNCT iajs-2510 101 46 . . PUNCT iajs-2510 102 1 also also ADV iajs-2510 102 2 𝐴 𝐴 PROPN iajs-2510 102 3 is be AUX iajs-2510 102 4 not not PART iajs-2510 102 5 weak weak ADJ iajs-2510 102 6 essential essential ADJ iajs-2510 102 7 , , PUNCT iajs-2510 102 8 since since SCONJ iajs-2510 102 9 𝐴 𝐴 PROPN iajs-2510 102 10 ∩ ∩ NOUN iajs-2510 102 11 𝑆 𝑆 PROPN iajs-2510 102 12 0 0 NUM iajs-2510 102 13 , , PUNCT iajs-2510 102 14 where where SCONJ iajs-2510 102 15 s s VERB iajs-2510 102 16 any any DET iajs-2510 102 17 semiprime semiprime NOUN iajs-2510 102 18 submodule submodule NOUN iajs-2510 102 19 . . PUNCT iajs-2510 103 1 therefore therefore ADV iajs-2510 103 2 𝐴 𝐴 PROPN iajs-2510 103 3 is be AUX iajs-2510 103 4 not not PART iajs-2510 103 5 weak weak ADJ iajs-2510 103 6 essential essential ADJ iajs-2510 103 7 of of ADP iajs-2510 103 8 x x X iajs-2510 103 9 . . PUNCT iajs-2510 104 1 proposition proposition NOUN iajs-2510 104 2 2.8 2.8 NUM iajs-2510 104 3 : : PUNCT iajs-2510 104 4 let let VERB iajs-2510 104 5 α α PRON iajs-2510 104 6 be be AUX iajs-2510 104 7 a a DET iajs-2510 104 8 f f NOUN iajs-2510 104 9 - - PUNCT iajs-2510 104 10 submodule submodule NOUN iajs-2510 104 11 of of ADP iajs-2510 104 12 a a DET iajs-2510 104 13 f f NOUN iajs-2510 104 14 - - PUNCT iajs-2510 104 15 module module NOUN iajs-2510 104 16 x x NOUN iajs-2510 104 17 , , PUNCT iajs-2510 104 18 then then ADV iajs-2510 104 19 α α PROPN iajs-2510 104 20 is be AUX iajs-2510 104 21 weak weak ADJ iajs-2510 104 22 essential essential ADJ iajs-2510 104 23 in in SCONJ iajs-2510 104 24 x x PROPN iajs-2510 104 25 iff iff PROPN iajs-2510 104 26 𝐴∗ 𝐴∗ PROPN iajs-2510 104 27 is be AUX iajs-2510 104 28 weak weak ADJ iajs-2510 104 29 essential essential ADJ iajs-2510 104 30 submodule submodule NOUN iajs-2510 104 31 in in ADP iajs-2510 104 32 x∗. x∗. ADJ iajs-2510 104 33 proof proof NOUN iajs-2510 104 34 : : PUNCT iajs-2510 105 1 let let VERB iajs-2510 105 2 𝐴∗ 𝐴∗ NOUN iajs-2510 105 3 is be AUX iajs-2510 105 4 a a DET iajs-2510 105 5 weak weak ADJ iajs-2510 105 6 essential essential ADJ iajs-2510 105 7 submodule submodule NOUN iajs-2510 105 8 in in ADP iajs-2510 105 9 x∗. x∗. NOUN iajs-2510 105 10 to to PART iajs-2510 105 11 show show VERB iajs-2510 105 12 a a PRON iajs-2510 105 13 is be AUX iajs-2510 105 14 weak weak ADJ iajs-2510 105 15 essential essential ADJ iajs-2510 105 16 f f NOUN iajs-2510 105 17 - - PUNCT iajs-2510 105 18 submodule submodule NOUN iajs-2510 105 19 in in ADP iajs-2510 105 20 x x PROPN iajs-2510 105 21 . . PUNCT iajs-2510 106 1 assume assume VERB iajs-2510 106 2 that that SCONJ iajs-2510 106 3 s s VERB iajs-2510 106 4 is be AUX iajs-2510 106 5 semiprime semiprime NOUN iajs-2510 106 6 f f X iajs-2510 106 7 - - PUNCT iajs-2510 106 8 submodule submodule NOUN iajs-2510 106 9 of of ADP iajs-2510 106 10 x x X iajs-2510 106 11 and and CCONJ iajs-2510 106 12 a a DET iajs-2510 106 13 ∩ ∩ ADJ iajs-2510 106 14 𝑆 𝑆 PROPN iajs-2510 106 15 0 0 NUM iajs-2510 106 16 , , PUNCT iajs-2510 106 17 then then ADV iajs-2510 106 18 𝐴 𝐴 PROPN iajs-2510 106 19 ∩ ∩ NOUN iajs-2510 106 20 𝑆 𝑆 PROPN iajs-2510 106 21 ∗ ∗ NOUN iajs-2510 106 22 0 0 NUM iajs-2510 106 23 , , PUNCT iajs-2510 106 24 implies imply VERB iajs-2510 106 25 that that SCONJ iajs-2510 106 26 𝐴∗ 𝐴∗ NUM iajs-2510 106 27 ∩ ∩ NOUN iajs-2510 106 28 𝑆∗ 𝑆∗ X iajs-2510 106 29 0 0 PUNCT iajs-2510 106 30 . . PUNCT iajs-2510 107 1 but but CCONJ iajs-2510 107 2 s s NOUN iajs-2510 107 3 is be AUX iajs-2510 107 4 semiprime semiprime NOUN iajs-2510 107 5 f f X iajs-2510 107 6 - - PUNCT iajs-2510 107 7 submodule submodule NOUN iajs-2510 107 8 , , PUNCT iajs-2510 107 9 then then ADV iajs-2510 107 10 𝑆 𝑆 PROPN iajs-2510 107 11 is be AUX iajs-2510 107 12 semiprime semiprime NOUN iajs-2510 107 13 see see VERB iajs-2510 107 14 [ [ X iajs-2510 107 15 14 14 NUM iajs-2510 107 16 , , PUNCT iajs-2510 107 17 theorem theorem ADJ iajs-2510 107 18 ( ( PUNCT iajs-2510 107 19 2.4 2.4 NUM iajs-2510 107 20 ) ) PUNCT iajs-2510 107 21 ] ] PUNCT iajs-2510 107 22 , , PUNCT iajs-2510 107 23 so so ADV iajs-2510 107 24 𝑆∗ 𝑆∗ PROPN iajs-2510 107 25 is be AUX iajs-2510 107 26 semiprime semiprime NOUN iajs-2510 107 27 , , PUNCT iajs-2510 107 28 hence hence ADV iajs-2510 107 29 𝑆∗ 𝑆∗ X iajs-2510 107 30 0 0 PUNCT iajs-2510 108 1 , , PUNCT iajs-2510 108 2 so so SCONJ iajs-2510 108 3 s s VERB iajs-2510 108 4 = = NOUN iajs-2510 108 5 0 0 NUM iajs-2510 108 6 . . PUNCT iajs-2510 109 1 thus thus ADV iajs-2510 109 2 , , PUNCT iajs-2510 109 3 a a PRON iajs-2510 109 4 is be AUX iajs-2510 109 5 weak weak ADJ iajs-2510 109 6 essential essential ADJ iajs-2510 109 7 f f NOUN iajs-2510 109 8 - - PUNCT iajs-2510 109 9 submodule submodule NOUN iajs-2510 109 10 in in ADP iajs-2510 109 11 x. x. NOUN iajs-2510 109 12 conversely conversely ADV iajs-2510 109 13 , , PUNCT iajs-2510 109 14 let let VERB iajs-2510 109 15 a a PRON iajs-2510 109 16 is be AUX iajs-2510 109 17 a a DET iajs-2510 109 18 weak weak ADJ iajs-2510 109 19 essential essential ADJ iajs-2510 109 20 f f NOUN iajs-2510 109 21 - - PUNCT iajs-2510 109 22 submodule submodule NOUN iajs-2510 109 23 in in ADP iajs-2510 109 24 x x NOUN iajs-2510 109 25 , , PUNCT iajs-2510 109 26 we we PRON iajs-2510 109 27 have have VERB iajs-2510 109 28 to to PART iajs-2510 109 29 show show VERB iajs-2510 109 30 that that SCONJ iajs-2510 109 31 𝐴∗ 𝐴∗ NOUN iajs-2510 109 32 is be AUX iajs-2510 109 33 weak weak ADJ iajs-2510 109 34 essential essential ADJ iajs-2510 109 35 submodule submodule NOUN iajs-2510 109 36 in in ADP iajs-2510 109 37 x∗. x∗. PROPN iajs-2510 110 1 let let VERB iajs-2510 110 2 n n PRON iajs-2510 110 3 is be AUX iajs-2510 110 4 semiprime semiprime NOUN iajs-2510 110 5 submodule submodule NOUN iajs-2510 110 6 of of ADP iajs-2510 110 7 x∗ x∗ PROPN iajs-2510 110 8 and and CCONJ iajs-2510 110 9 𝐴∗ 𝐴∗ NOUN iajs-2510 110 10 ∩ ∩ NOUN iajs-2510 110 11 𝑁 𝑁 PROPN iajs-2510 110 12 0 0 NUM iajs-2510 110 13 , , PUNCT iajs-2510 110 14 we we PRON iajs-2510 110 15 must must AUX iajs-2510 110 16 prove prove VERB iajs-2510 110 17 n n PRON iajs-2510 110 18 = = SYM iajs-2510 110 19 ( ( PUNCT iajs-2510 110 20 0 0 NUM iajs-2510 110 21 ) ) PUNCT iajs-2510 110 22 . . PUNCT iajs-2510 111 1 define define VERB iajs-2510 111 2 b b NOUN iajs-2510 111 3 : : PUNCT iajs-2510 111 4 ℳ ℳ PROPN iajs-2510 111 5 ⟶ ⟶ NOUN iajs-2510 111 6 i i PRON iajs-2510 111 7 by by ADP iajs-2510 111 8 : : PUNCT iajs-2510 111 9 b(x b(x NOUN iajs-2510 111 10 ) ) PUNCT iajs-2510 111 11 = = SYM iajs-2510 112 1 1 1 NUM iajs-2510 112 2 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 112 3 x x SYM iajs-2510 112 4 ∈ ∈ PROPN iajs-2510 112 5 𝑁 𝑁 NOUN iajs-2510 112 6 0 0 NUM iajs-2510 112 7 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 112 8 it it PRON iajs-2510 112 9 is be AUX iajs-2510 112 10 clear clear ADJ iajs-2510 112 11 that that SCONJ iajs-2510 112 12 b b X iajs-2510 112 13 f f X iajs-2510 112 14 - - PUNCT iajs-2510 112 15 submodule submodule NOUN iajs-2510 112 16 of of ADP iajs-2510 112 17 x x PROPN iajs-2510 112 18 , , PUNCT iajs-2510 112 19 𝐵∗ 𝐵∗ PROPN iajs-2510 112 20 𝑁 𝑁 PROPN iajs-2510 112 21 , , PUNCT iajs-2510 112 22 so so ADJ iajs-2510 112 23 𝐴∗ 𝐴∗ NUM iajs-2510 112 24 ∩ ∩ ADJ iajs-2510 112 25 𝐵∗ 𝐵∗ PROPN iajs-2510 112 26 0 0 NUM iajs-2510 112 27 , , PUNCT iajs-2510 112 28 then then ADV iajs-2510 112 29 𝐴 𝐴 PROPN iajs-2510 112 30 ∩ ∩ ADJ iajs-2510 112 31 𝐵 𝐵 NOUN iajs-2510 112 32 ∗ ∗ NOUN iajs-2510 112 33 0 0 NUM iajs-2510 112 34 , , PUNCT iajs-2510 112 35 hence hence ADV iajs-2510 112 36 by by ADP iajs-2510 112 37 remark(1.7)(3 remark(1.7)(3 NOUN iajs-2510 112 38 ) ) PUNCT iajs-2510 112 39 , , PUNCT iajs-2510 112 40 a∩ a∩ PROPN iajs-2510 112 41 𝐵 𝐵 PROPN iajs-2510 112 42 0 0 PUNCT iajs-2510 112 43 and and CCONJ iajs-2510 112 44 b b X iajs-2510 112 45 = = NOUN iajs-2510 112 46 0 0 NUM iajs-2510 112 47 , , PUNCT iajs-2510 112 48 since since SCONJ iajs-2510 112 49 a a PRON iajs-2510 112 50 is be AUX iajs-2510 112 51 weak weak ADJ iajs-2510 112 52 essential essential ADJ iajs-2510 112 53 f f NOUN iajs-2510 112 54 - - PUNCT iajs-2510 112 55 submodule submodule NOUN iajs-2510 112 56 in in ADP iajs-2510 112 57 x x NOUN iajs-2510 112 58 , , PUNCT iajs-2510 112 59 so so ADV iajs-2510 112 60 𝐵∗ 𝐵∗ NOUN iajs-2510 112 61 0 0 NUM iajs-2510 112 62 ; ; PUNCT iajs-2510 112 63 therefore therefore ADV iajs-2510 112 64     SPACE iajs-2510 112 65 70 70 NUM iajs-2510 112 66     SPACE iajs-2510 112 67 ibn ibn PROPN iajs-2510 112 68 al al PROPN iajs-2510 112 69 - - PUNCT iajs-2510 112 70 haitham haitham PROPN iajs-2510 112 71 jour jour X iajs-2510 112 72 . . PROPN iajs-2510 113 1 for for ADP iajs-2510 113 2 pure pure ADJ iajs-2510 113 3 & & CCONJ iajs-2510 113 4 appl appl PROPN iajs-2510 113 5 . . PUNCT iajs-2510 114 1 sci sci PROPN iajs-2510 114 2 . . PROPN iajs-2510 115 1 33 33 NUM iajs-2510 115 2 ( ( PUNCT iajs-2510 115 3 4 4 NUM iajs-2510 115 4 ) ) PUNCT iajs-2510 115 5 2020 2020 NUM iajs-2510 116 1 n n NOUN iajs-2510 116 2 = = SYM iajs-2510 116 3 ( ( PUNCT iajs-2510 116 4 0 0 NUM iajs-2510 116 5 ) ) PUNCT iajs-2510 116 6 . . PUNCT iajs-2510 117 1 thus thus ADV iajs-2510 117 2 𝐴∗ 𝐴∗ NUM iajs-2510 117 3 is be AUX iajs-2510 117 4 weak weak ADJ iajs-2510 117 5 essential essential ADJ iajs-2510 117 6 submodule submodule NOUN iajs-2510 117 7 in in ADP iajs-2510 117 8 x∗. x∗. PROPN iajs-2510 117 9 remarks remark NOUN iajs-2510 117 10 2.9 2.9 NUM iajs-2510 117 11 : : SYM iajs-2510 117 12 1 1 X iajs-2510 117 13 . . X iajs-2510 117 14 let let VERB iajs-2510 117 15 α α PRON iajs-2510 117 16 , , PUNCT iajs-2510 117 17 β β X iajs-2510 117 18 are be AUX iajs-2510 117 19 f f NOUN iajs-2510 117 20 - - PUNCT iajs-2510 117 21 submodules submodule NOUN iajs-2510 117 22 of of ADP iajs-2510 117 23 x x SYM iajs-2510 117 24 such such ADJ iajs-2510 117 25 that that SCONJ iajs-2510 117 26 a a DET iajs-2510 117 27 ⊆ ⊆ NUM iajs-2510 117 28 𝐵 𝐵 NOUN iajs-2510 117 29 and and CCONJ iajs-2510 117 30 β β X iajs-2510 117 31 is be AUX iajs-2510 117 32 weak weak ADJ iajs-2510 117 33 essential essential ADJ iajs-2510 117 34 f f NOUN iajs-2510 117 35 - - PUNCT iajs-2510 117 36 submodule submodule NOUN iajs-2510 117 37 of of ADP iajs-2510 117 38 x x PROPN iajs-2510 117 39 , , PUNCT iajs-2510 117 40 then then ADV iajs-2510 117 41 a a DET iajs-2510 117 42 need need NOUN iajs-2510 117 43 not not PART iajs-2510 117 44 be be AUX iajs-2510 117 45 weak weak ADJ iajs-2510 117 46 essential essential ADJ iajs-2510 117 47 f f NOUN iajs-2510 117 48 - - PUNCT iajs-2510 117 49 submodule submodule NOUN iajs-2510 117 50 for for ADP iajs-2510 117 51 example example NOUN iajs-2510 117 52 : : PUNCT iajs-2510 117 53 let let VERB iajs-2510 117 54 ℳ ℳ PRON iajs-2510 117 55 be be AUX iajs-2510 117 56 as as ADP iajs-2510 117 57 z z NOUN iajs-2510 117 58 - - PUNCT iajs-2510 117 59 module module NOUN iajs-2510 117 60 𝑍 𝑍 NOUN iajs-2510 117 61 . . PUNCT iajs-2510 118 1 let let VERB iajs-2510 118 2 x x PRON iajs-2510 118 3 : : PUNCT iajs-2510 118 4 ℳ ℳ PROPN iajs-2510 118 5 ⟶ ⟶ NOUN iajs-2510 118 6 i i PROPN iajs-2510 118 7 , , PUNCT iajs-2510 118 8 define define VERB iajs-2510 118 9 by by ADP iajs-2510 118 10 : : PUNCT iajs-2510 118 11 x(a x(a PROPN iajs-2510 118 12 ) ) PUNCT iajs-2510 118 13 = = SYM iajs-2510 118 14 1 1 NUM iajs-2510 118 15 , , PUNCT iajs-2510 118 16 for for ADP iajs-2510 118 17 all all PRON iajs-2510 118 18 𝑎 𝑎 DET iajs-2510 118 19 ∈ ∈ NOUN iajs-2510 118 20 𝑍 𝑍 NOUN iajs-2510 118 21 . . PUNCT iajs-2510 119 1 define define VERB iajs-2510 119 2 a a DET iajs-2510 119 3 : : PUNCT iajs-2510 119 4 ℳ ℳ PROPN iajs-2510 119 5 ⟶ ⟶ NOUN iajs-2510 119 6 i i PROPN iajs-2510 119 7 , , PUNCT iajs-2510 119 8 b b X iajs-2510 119 9 : : PUNCT iajs-2510 119 10 ℳ ℳ PROPN iajs-2510 119 11 ⟶ ⟶ NOUN iajs-2510 119 12 i i PRON iajs-2510 119 13 by by ADP iajs-2510 119 14 : : PUNCT iajs-2510 119 15 a(x a(x PROPN iajs-2510 119 16 ) ) PUNCT iajs-2510 119 17 = = SYM iajs-2510 119 18 1 1 NUM iajs-2510 119 19 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 119 20 x x SYM iajs-2510 119 21 ∈ ∈ PROPN iajs-2510 119 22 18 18 NUM iajs-2510 119 23 0 0 NUM iajs-2510 119 24 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 119 25 , , PUNCT iajs-2510 119 26 b(x b(x NOUN iajs-2510 119 27 ) ) PUNCT iajs-2510 119 28 = = SYM iajs-2510 120 1 1 1 NUM iajs-2510 120 2 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 120 3 x x SYM iajs-2510 120 4 ∈ ∈ PROPN iajs-2510 120 5 2 2 NUM iajs-2510 120 6 0 0 NUM iajs-2510 120 7 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 120 8 it it PRON iajs-2510 120 9 is be AUX iajs-2510 120 10 clear clear ADJ iajs-2510 120 11 that that SCONJ iajs-2510 120 12 x x SYM iajs-2510 120 13 𝑍 𝑍 NOUN iajs-2510 120 14 and and CCONJ iajs-2510 120 15 a a DET iajs-2510 120 16 , , PUNCT iajs-2510 120 17 b b NOUN iajs-2510 120 18 are be AUX iajs-2510 120 19 f f NOUN iajs-2510 120 20 - - PUNCT iajs-2510 120 21 submodules submodule NOUN iajs-2510 120 22 of of ADP iajs-2510 120 23 x x X iajs-2510 120 24 . . PUNCT iajs-2510 121 1 𝐵 𝐵 NOUN iajs-2510 121 2 a a DET iajs-2510 121 3 weak weak ADJ iajs-2510 121 4 essential essential ADJ iajs-2510 121 5 submodule submodule NOUN iajs-2510 121 6 in in ADP iajs-2510 121 7 x x PART iajs-2510 121 8 see see VERB iajs-2510 121 9 [ [ X iajs-2510 121 10 4 4 NUM iajs-2510 121 11 , , PUNCT iajs-2510 121 12 remarks(1.5 remarks(1.5 NOUN iajs-2510 121 13 ) ) PUNCT iajs-2510 121 14 ] ] PUNCT iajs-2510 121 15 . . PUNCT iajs-2510 122 1 thus thus ADV iajs-2510 122 2 b b X iajs-2510 122 3 is be AUX iajs-2510 122 4 weak weak ADJ iajs-2510 122 5 essential essential ADJ iajs-2510 122 6 fsubmodule fsubmodule NOUN iajs-2510 122 7 of of ADP iajs-2510 122 8 x x PUNCT iajs-2510 122 9 by by ADP iajs-2510 122 10 lemma lemma PROPN iajs-2510 122 11 ( ( PUNCT iajs-2510 122 12 2.4 2.4 NUM iajs-2510 122 13 ) ) PUNCT iajs-2510 122 14 . . PUNCT iajs-2510 123 1 let let VERB iajs-2510 124 1 c c NOUN iajs-2510 124 2 : : PUNCT iajs-2510 124 3 ℳ ℳ PROPN iajs-2510 124 4 ⟶ ⟶ NOUN iajs-2510 124 5 i i PRON iajs-2510 124 6 , , PUNCT iajs-2510 124 7 as as SCONJ iajs-2510 124 8 defined define VERB iajs-2510 124 9 by by ADP iajs-2510 124 10 : : PUNCT iajs-2510 124 11 c(x c(x NOUN iajs-2510 124 12 ) ) PUNCT iajs-2510 124 13 = = SYM iajs-2510 125 1 1 1 NUM iajs-2510 125 2 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 125 3 x x SYM iajs-2510 125 4 ∈ ∈ PROPN iajs-2510 125 5 12 12 NUM iajs-2510 125 6 0 0 NUM iajs-2510 125 7 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 125 8 , , PUNCT iajs-2510 125 9 where where SCONJ iajs-2510 125 10 c c NOUN iajs-2510 125 11 semiprime semiprime NOUN iajs-2510 125 12 f f X iajs-2510 125 13 - - PUNCT iajs-2510 125 14 submodule submodule NOUN iajs-2510 125 15 c c PROPN iajs-2510 125 16 12 12 NUM iajs-2510 125 17 , , PUNCT iajs-2510 125 18 is be AUX iajs-2510 125 19 semiprime semiprime NOUN iajs-2510 125 20 submodule submodule NOUN iajs-2510 125 21 of of ADP iajs-2510 125 22 x x SYM iajs-2510 125 23 ( ( PUNCT iajs-2510 125 24 ∀ ∀ X iajs-2510 125 25 𝑡 𝑡 NOUN iajs-2510 125 26 0 0 NUM iajs-2510 125 27 . . PUNCT iajs-2510 126 1 but but CCONJ iajs-2510 126 2 a a DET iajs-2510 126 3 ∩ ∩ ADJ iajs-2510 126 4 𝐶 𝐶 PROPN iajs-2510 126 5 0 0 NUM iajs-2510 126 6 , , PUNCT iajs-2510 126 7 therefore therefore ADV iajs-2510 126 8 α α PROPN iajs-2510 126 9 is be AUX iajs-2510 126 10 not not PART iajs-2510 126 11 weak weak ADJ iajs-2510 126 12 essential essential ADJ iajs-2510 126 13 f f NOUN iajs-2510 126 14 - - PUNCT iajs-2510 126 15 submodule submodule NOUN iajs-2510 126 16 of of ADP iajs-2510 126 17 x x PROPN iajs-2510 126 18 . . PUNCT iajs-2510 127 1 2 2 X iajs-2510 127 2 . . X iajs-2510 127 3 let let VERB iajs-2510 127 4 a a DET iajs-2510 127 5 , , PUNCT iajs-2510 127 6 b b NOUN iajs-2510 127 7 are be AUX iajs-2510 127 8 f f NOUN iajs-2510 127 9 - - PUNCT iajs-2510 127 10 submodule submodule NOUN iajs-2510 127 11 such such ADJ iajs-2510 127 12 that that SCONJ iajs-2510 127 13 a a DET iajs-2510 127 14 ⊆ ⊆ NUM iajs-2510 127 15 𝐵. 𝐵. NOUN iajs-2510 127 16 if if SCONJ iajs-2510 127 17 a a PRON iajs-2510 127 18 is be AUX iajs-2510 127 19 weak weak ADJ iajs-2510 127 20 essential essential ADJ iajs-2510 127 21 f f NOUN iajs-2510 127 22 - - PUNCT iajs-2510 127 23 submodule submodule NOUN iajs-2510 127 24 in in ADP iajs-2510 127 25 x x PUNCT iajs-2510 127 26 implying imply VERB iajs-2510 127 27 β β NOUN iajs-2510 127 28 is be AUX iajs-2510 127 29 a a DET iajs-2510 127 30 weak weak ADJ iajs-2510 127 31 essential essential ADJ iajs-2510 127 32 f f NOUN iajs-2510 127 33 - - PUNCT iajs-2510 127 34 submodule submodule NOUN iajs-2510 127 35 of of ADP iajs-2510 127 36 x. x. NOUN iajs-2510 127 37 proof proof NOUN iajs-2510 127 38 : : PUNCT iajs-2510 127 39 assume assume VERB iajs-2510 127 40 that that SCONJ iajs-2510 127 41 b b X iajs-2510 127 42 ∩ ∩ NOUN iajs-2510 127 43 𝑆 𝑆 PROPN iajs-2510 127 44 0 0 NUM iajs-2510 127 45 , , PUNCT iajs-2510 127 46 for for ADP iajs-2510 127 47 some some DET iajs-2510 127 48 semi semi ADJ iajs-2510 127 49 - - ADJ iajs-2510 127 50 prime prime ADJ iajs-2510 127 51 f f NOUN iajs-2510 127 52 - - PUNCT iajs-2510 127 53 submodule submodule NOUN iajs-2510 127 54 s s PROPN iajs-2510 127 55 of of ADP iajs-2510 127 56 x x NOUN iajs-2510 127 57 , , PUNCT iajs-2510 127 58 then then ADV iajs-2510 127 59 a a DET iajs-2510 127 60 ∩ ∩ ADJ iajs-2510 127 61 𝑆 𝑆 PROPN iajs-2510 127 62 0 0 NUM iajs-2510 127 63 . . PUNCT iajs-2510 128 1 but but CCONJ iajs-2510 128 2 a a PRON iajs-2510 128 3 is be AUX iajs-2510 128 4 weak weak ADJ iajs-2510 128 5 essential essential ADJ iajs-2510 128 6 f f NOUN iajs-2510 128 7 - - PUNCT iajs-2510 128 8 submodule submodule NOUN iajs-2510 128 9 , , PUNCT iajs-2510 128 10 hence hence ADV iajs-2510 128 11 s s PART iajs-2510 128 12 = = NOUN iajs-2510 128 13 0 0 PROPN iajs-2510 128 14 . . PUNCT iajs-2510 129 1 that that PRON iajs-2510 129 2 is be AUX iajs-2510 129 3 b b NOUN iajs-2510 129 4 is be AUX iajs-2510 129 5 weak weak ADJ iajs-2510 129 6 essential essential ADJ iajs-2510 129 7 f f NOUN iajs-2510 129 8 - - PUNCT iajs-2510 129 9 submodule submodule NOUN iajs-2510 129 10 of of ADP iajs-2510 129 11 x x PROPN iajs-2510 129 12 . . PUNCT iajs-2510 130 1 3 3 X iajs-2510 130 2 . . X iajs-2510 130 3 let let VERB iajs-2510 130 4 a a DET iajs-2510 130 5 , , PUNCT iajs-2510 130 6 b b NOUN iajs-2510 130 7 be be AUX iajs-2510 130 8 are be AUX iajs-2510 130 9 f f NOUN iajs-2510 130 10 - - PUNCT iajs-2510 130 11 submodules submodule NOUN iajs-2510 130 12 of of ADP iajs-2510 130 13 f f NOUN iajs-2510 130 14 - - PUNCT iajs-2510 130 15 module module NOUN iajs-2510 130 16 x x PUNCT iajs-2510 130 17 if if SCONJ iajs-2510 130 18 a a DET iajs-2510 130 19 ∩ ∩ NOUN iajs-2510 130 20 𝐵 𝐵 NOUN iajs-2510 130 21 a a DET iajs-2510 130 22 weak weak ADJ iajs-2510 130 23 essential essential ADJ iajs-2510 130 24 f f NOUN iajs-2510 130 25 - - PUNCT iajs-2510 130 26 submodule submodule NOUN iajs-2510 130 27 of of ADP iajs-2510 130 28 x x PRON iajs-2510 130 29 , , PUNCT iajs-2510 130 30 then then ADV iajs-2510 130 31 both both PRON iajs-2510 130 32 of of ADP iajs-2510 130 33 a a PRON iajs-2510 130 34 and and CCONJ iajs-2510 130 35 b b NOUN iajs-2510 130 36 are be AUX iajs-2510 130 37 weak weak ADJ iajs-2510 130 38 essential essential ADJ iajs-2510 130 39 f f NOUN iajs-2510 130 40 - - PUNCT iajs-2510 130 41 submodules submodule NOUN iajs-2510 130 42 of of ADP iajs-2510 130 43 x. x. NOUN iajs-2510 130 44 proof proof NOUN iajs-2510 130 45 : : PUNCT iajs-2510 130 46 it it PRON iajs-2510 130 47 is be AUX iajs-2510 130 48 clear clear ADJ iajs-2510 130 49 by by ADP iajs-2510 130 50 ( ( PUNCT iajs-2510 130 51 2 2 NUM iajs-2510 130 52 ) ) PUNCT iajs-2510 130 53 . . PUNCT iajs-2510 131 1 note note VERB iajs-2510 131 2 that that SCONJ iajs-2510 131 3 , , PUNCT iajs-2510 131 4 the the DET iajs-2510 131 5 converse converse NOUN iajs-2510 131 6 is be AUX iajs-2510 131 7 not not PART iajs-2510 131 8 true true ADJ iajs-2510 131 9 in in ADP iajs-2510 131 10 general general ADJ iajs-2510 131 11 , , PUNCT iajs-2510 131 12 for for ADP iajs-2510 131 13 example example NOUN iajs-2510 131 14 : : PUNCT iajs-2510 131 15 example example NOUN iajs-2510 131 16 : : PUNCT iajs-2510 131 17 let let VERB iajs-2510 131 18 ℳ ℳ PRON iajs-2510 131 19 be be AUX iajs-2510 131 20 𝑍 𝑍 NOUN iajs-2510 131 21 as as ADP iajs-2510 131 22 z z NOUN iajs-2510 131 23 - - PUNCT iajs-2510 131 24 module module NOUN iajs-2510 131 25 . . PUNCT iajs-2510 132 1 define define VERB iajs-2510 132 2 x x NOUN iajs-2510 132 3 : : PUNCT iajs-2510 132 4 ℳ ℳ PROPN iajs-2510 132 5 ⟶ ⟶ NOUN iajs-2510 132 6 i i PRON iajs-2510 132 7 by by ADP iajs-2510 132 8 : : PUNCT iajs-2510 132 9 x(a x(a PROPN iajs-2510 132 10 ) ) PUNCT iajs-2510 132 11 = = SYM iajs-2510 132 12 1 1 NUM iajs-2510 132 13 , , PUNCT iajs-2510 132 14 for for ADP iajs-2510 132 15 all all PRON iajs-2510 132 16 𝑎 𝑎 DET iajs-2510 132 17 ∈ ∈ NOUN iajs-2510 132 18 𝑍 𝑍 NOUN iajs-2510 132 19 . . PUNCT iajs-2510 133 1 let let VERB iajs-2510 133 2 a a DET iajs-2510 133 3 : : PUNCT iajs-2510 133 4 ℳ ℳ PROPN iajs-2510 133 5 ⟶ ⟶ NOUN iajs-2510 133 6 i i PROPN iajs-2510 133 7 , , PUNCT iajs-2510 133 8 b b NOUN iajs-2510 133 9 : : PUNCT iajs-2510 133 10 ℳ ℳ PROPN iajs-2510 133 11 ⟶ ⟶ NOUN iajs-2510 133 12 i i PROPN iajs-2510 133 13 , , PUNCT iajs-2510 133 14 define define VERB iajs-2510 133 15 by by ADP iajs-2510 133 16 : : PUNCT iajs-2510 133 17 a(x a(x PROPN iajs-2510 133 18 ) ) PUNCT iajs-2510 133 19 = = SYM iajs-2510 134 1 1 1 NUM iajs-2510 134 2 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 134 3 x x SYM iajs-2510 134 4 ∈ ∈ PROPN iajs-2510 134 5 12 12 NUM iajs-2510 134 6 0 0 NUM iajs-2510 134 7 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 NOUN iajs-2510 134 8 , , PUNCT iajs-2510 134 9 b(x b(x NOUN iajs-2510 134 10 ) ) PUNCT iajs-2510 134 11 = = SYM iajs-2510 134 12 1 1 NUM iajs-2510 134 13 𝑖𝑓 𝑖𝑓 NOUN iajs-2510 134 14 x x SYM iajs-2510 134 15 ∈ ∈ PROPN iajs-2510 134 16 18 18 NUM iajs-2510 134 17 0 0 NUM iajs-2510 134 18 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 VERB iajs-2510 134 19 clearly clearly ADV iajs-2510 134 20 a a DET iajs-2510 134 21 , , PUNCT iajs-2510 134 22 b b NOUN iajs-2510 134 23 are be AUX iajs-2510 134 24 f f NOUN iajs-2510 134 25 - - PUNCT iajs-2510 134 26 submodules submodule NOUN iajs-2510 134 27 of of ADP iajs-2510 134 28 χ χ NOUN iajs-2510 134 29 , , PUNCT iajs-2510 134 30 𝐴 𝐴 PROPN iajs-2510 134 31 12 12 NUM iajs-2510 134 32 , , PUNCT iajs-2510 134 33 𝐵 𝐵 NOUN iajs-2510 134 34 18 18 NUM iajs-2510 134 35 , , PUNCT iajs-2510 134 36 ∀ ∀ NUM iajs-2510 134 37 𝑡 𝑡 PART iajs-2510 134 38 ∈ ∈ NOUN iajs-2510 134 39 0,1 0,1 NUM iajs-2510 134 40 are be AUX iajs-2510 134 41 weak weak ADJ iajs-2510 134 42 essential essential ADJ iajs-2510 134 43 submodules submodule NOUN iajs-2510 134 44 of of ADP iajs-2510 134 45 x x PUNCT iajs-2510 134 46 by by ADP iajs-2510 134 47 [ [ X iajs-2510 134 48 4 4 NUM iajs-2510 134 49 , , PUNCT iajs-2510 134 50 remark(1.5 remark(1.5 NOUN iajs-2510 134 51 ) ) PUNCT iajs-2510 134 52 ] ] PUNCT iajs-2510 134 53 . . PUNCT iajs-2510 135 1 hence hence ADV iajs-2510 135 2 a a X iajs-2510 135 3 , , PUNCT iajs-2510 135 4 β β X iajs-2510 135 5 are be AUX iajs-2510 135 6 weak weak ADJ iajs-2510 135 7 essential essential ADJ iajs-2510 135 8 f f NOUN iajs-2510 135 9 - - PUNCT iajs-2510 135 10 submodules submodule NOUN iajs-2510 135 11 of of ADP iajs-2510 135 12 x x PRON iajs-2510 135 13 ; ; PUNCT iajs-2510 135 14 see see VERB iajs-2510 135 15 lemma(2.4 lemma(2.4 NOUN iajs-2510 135 16 ) ) PUNCT iajs-2510 135 17 . . PUNCT iajs-2510 136 1 but but CCONJ iajs-2510 136 2 a a DET iajs-2510 136 3 ∩ ∩ ADJ iajs-2510 136 4 𝐵 𝐵 NOUN iajs-2510 136 5 0 0 NUM iajs-2510 136 6 ; ; PUNCT iajs-2510 136 7 that that PRON iajs-2510 136 8 is be AUX iajs-2510 136 9 a a DET iajs-2510 136 10 ∩ ∩ ADJ iajs-2510 136 11 b b NOUN iajs-2510 136 12 is be AUX iajs-2510 136 13 not not PART iajs-2510 136 14 weak weak ADJ iajs-2510 136 15 essential essential ADJ iajs-2510 136 16 f f NOUN iajs-2510 136 17 - - PUNCT iajs-2510 136 18 submodule submodule NOUN iajs-2510 136 19 of of ADP iajs-2510 136 20 x x X iajs-2510 136 21 . . PUNCT iajs-2510 137 1 under under ADP iajs-2510 137 2 some some DET iajs-2510 137 3 conditions condition NOUN iajs-2510 137 4 the the DET iajs-2510 137 5 converse converse NOUN iajs-2510 137 6 ( ( PUNCT iajs-2510 137 7 3 3 X iajs-2510 137 8 ) ) PUNCT iajs-2510 137 9 will will AUX iajs-2510 137 10 be be AUX iajs-2510 137 11 true true ADJ iajs-2510 137 12 as as ADP iajs-2510 137 13 in in ADP iajs-2510 137 14 the the DET iajs-2510 137 15 following follow VERB iajs-2510 137 16 proposition proposition NOUN iajs-2510 137 17 . . PUNCT iajs-2510 138 1 proposition proposition NOUN iajs-2510 138 2 2.10 2.10 NUM iajs-2510 138 3 : : PUNCT iajs-2510 138 4 let let VERB iajs-2510 138 5 a a DET iajs-2510 138 6 , , PUNCT iajs-2510 138 7 b b NOUN iajs-2510 138 8 are be AUX iajs-2510 138 9 f f NOUN iajs-2510 138 10 - - PUNCT iajs-2510 138 11 submodules submodule NOUN iajs-2510 138 12 of of ADP iajs-2510 138 13 f f NOUN iajs-2510 138 14 - - PUNCT iajs-2510 138 15 module module NOUN iajs-2510 138 16 x x NOUN iajs-2510 138 17 such such ADJ iajs-2510 138 18 that that SCONJ iajs-2510 138 19 a a PRON iajs-2510 138 20 is be AUX iajs-2510 138 21 an an DET iajs-2510 138 22 essential essential ADJ iajs-2510 138 23 f f NOUN iajs-2510 138 24 - - PUNCT iajs-2510 138 25 submodule submodule NOUN iajs-2510 138 26 , , PUNCT iajs-2510 138 27 b b X iajs-2510 138 28 weak weak ADJ iajs-2510 138 29 essential essential ADJ iajs-2510 138 30 f f NOUN iajs-2510 138 31 - - PUNCT iajs-2510 138 32 submodule submodule NOUN iajs-2510 138 33 , , PUNCT iajs-2510 138 34 then then ADV iajs-2510 138 35 a a DET iajs-2510 138 36 ∩ ∩ ADJ iajs-2510 138 37 𝐵 𝐵 NOUN iajs-2510 138 38 is be AUX iajs-2510 138 39 a a DET iajs-2510 138 40 weak weak ADJ iajs-2510 138 41 essential essential ADJ iajs-2510 138 42 f f NOUN iajs-2510 138 43 - - PUNCT iajs-2510 138 44 submodule submodule NOUN iajs-2510 138 45 of of ADP iajs-2510 138 46 x. x. NOUN iajs-2510 138 47 proof proof NOUN iajs-2510 138 48 : : PUNCT iajs-2510 138 49     SPACE iajs-2510 138 50 71 71 NUM iajs-2510 138 51     SPACE iajs-2510 138 52 ibn ibn PROPN iajs-2510 138 53 al al PROPN iajs-2510 138 54 - - PUNCT iajs-2510 138 55 haitham haitham PROPN iajs-2510 138 56 jour jour X iajs-2510 138 57 . . PROPN iajs-2510 139 1 for for ADP iajs-2510 139 2 pure pure ADJ iajs-2510 139 3 & & CCONJ iajs-2510 139 4 appl appl PROPN iajs-2510 139 5 . . PUNCT iajs-2510 140 1 sci sci PROPN iajs-2510 140 2 . . PROPN iajs-2510 141 1 33 33 NUM iajs-2510 141 2 ( ( PUNCT iajs-2510 141 3 4 4 NUM iajs-2510 141 4 ) ) PUNCT iajs-2510 141 5 2020 2020 NUM iajs-2510 141 6 suppose suppose VERB iajs-2510 141 7 s s VERB iajs-2510 141 8 is be AUX iajs-2510 141 9 a a DET iajs-2510 141 10 non non ADJ iajs-2510 141 11 - - ADJ iajs-2510 141 12 trivial trivial ADJ iajs-2510 141 13 semiprime semiprime NOUN iajs-2510 141 14 f f X iajs-2510 141 15 - - PUNCT iajs-2510 141 16 submodule submodule NOUN iajs-2510 141 17 of of ADP iajs-2510 141 18 x x PRON iajs-2510 141 19 , , PUNCT iajs-2510 141 20 but but CCONJ iajs-2510 141 21 b b NOUN iajs-2510 141 22 is be AUX iajs-2510 141 23 weak weak ADJ iajs-2510 141 24 essential essential ADJ iajs-2510 141 25 fsubmodule fsubmodule NOUN iajs-2510 141 26 of of ADP iajs-2510 141 27 x x PRON iajs-2510 141 28 , , PUNCT iajs-2510 141 29 hence hence ADV iajs-2510 141 30 b b NOUN iajs-2510 141 31 ∩ ∩ ADJ iajs-2510 141 32 𝑆 𝑆 PROPN iajs-2510 141 33 0 0 NUM iajs-2510 141 34 . . PUNCT iajs-2510 142 1 so so ADV iajs-2510 142 2 a a PRON iajs-2510 142 3 is be AUX iajs-2510 142 4 an an DET iajs-2510 142 5 essential essential ADJ iajs-2510 142 6 f f NOUN iajs-2510 142 7 - - PUNCT iajs-2510 142 8 submodule submodule NOUN iajs-2510 142 9 of of ADP iajs-2510 142 10 x x PUNCT iajs-2510 143 1 and and CCONJ iajs-2510 143 2 we we PRON iajs-2510 143 3 have have VERB iajs-2510 143 4 a a DET iajs-2510 143 5 ∩ ∩ ADJ iajs-2510 143 6 𝐵 𝐵 NOUN iajs-2510 143 7 ∩ ∩ NOUN iajs-2510 143 8 𝑆 𝑆 PROPN iajs-2510 143 9 a a DET iajs-2510 143 10 ∩ ∩ ADJ iajs-2510 143 11 𝐵 𝐵 NOUN iajs-2510 143 12 ∩ ∩ NOUN iajs-2510 143 13 𝑆 𝑆 PROPN iajs-2510 143 14 0 0 NUM iajs-2510 143 15 , , PUNCT iajs-2510 143 16 hence hence ADV iajs-2510 143 17 , , PUNCT iajs-2510 143 18 a a DET iajs-2510 143 19 ∩ ∩ ADJ iajs-2510 143 20 b b NOUN iajs-2510 143 21 is be AUX iajs-2510 143 22 weak weak ADJ iajs-2510 143 23 essential essential ADJ iajs-2510 143 24 f f NOUN iajs-2510 143 25 - - PUNCT iajs-2510 143 26 submodule submodule NOUN iajs-2510 143 27 of of ADP iajs-2510 143 28 x. x. PROPN iajs-2510 143 29 lemma lemma PROPN iajs-2510 144 1 2.11 2.11 NUM iajs-2510 144 2 : : PUNCT iajs-2510 144 3 if if SCONJ iajs-2510 144 4 s s NOUN iajs-2510 144 5 is be AUX iajs-2510 144 6 a a DET iajs-2510 144 7 semiprime semiprime NOUN iajs-2510 144 8 f f NOUN iajs-2510 144 9 - - PUNCT iajs-2510 144 10 submodule submodule NOUN iajs-2510 144 11 of of ADP iajs-2510 144 12 f f NOUN iajs-2510 144 13 - - PUNCT iajs-2510 144 14 module module NOUN iajs-2510 144 15 x x NOUN iajs-2510 144 16 , , PUNCT iajs-2510 144 17 β β X iajs-2510 144 18 be be AUX iajs-2510 144 19 a a DET iajs-2510 144 20 f f NOUN iajs-2510 144 21 - - PUNCT iajs-2510 144 22 submodule submodule NOUN iajs-2510 144 23 of of ADP iajs-2510 144 24 x x SYM iajs-2510 144 25 such such ADJ iajs-2510 144 26 that that PRON iajs-2510 144 27 b b X iajs-2510 144 28 ⊈ ⊈ PROPN iajs-2510 144 29 s s PART iajs-2510 144 30 , , PUNCT iajs-2510 144 31 then then ADV iajs-2510 144 32 s s VERB iajs-2510 144 33 ∩ ∩ NOUN iajs-2510 144 34 b b PROPN iajs-2510 144 35 is be AUX iajs-2510 144 36 semiprime semiprime NOUN iajs-2510 144 37 f f X iajs-2510 144 38 - - PUNCT iajs-2510 144 39 submodule submodule NOUN iajs-2510 144 40 in in ADP iajs-2510 144 41 b. b. PROPN iajs-2510 144 42 proof proof NOUN iajs-2510 144 43 : : PUNCT iajs-2510 144 44 let let VERB iajs-2510 144 45 s s PRON iajs-2510 144 46 be be AUX iajs-2510 144 47 a a DET iajs-2510 144 48 semiprime semiprime NOUN iajs-2510 144 49 f f NOUN iajs-2510 144 50 - - PUNCT iajs-2510 144 51 submodule submodule NOUN iajs-2510 144 52 of of ADP iajs-2510 144 53 x x NOUN iajs-2510 144 54 , , PUNCT iajs-2510 144 55 then then ADV iajs-2510 144 56 by by ADP iajs-2510 144 57 [ [ X iajs-2510 144 58 14,theorem(2.4 14,theorem(2.4 NUM iajs-2510 144 59 ) ) PUNCT iajs-2510 144 60 ] ] PUNCT iajs-2510 144 61 , , PUNCT iajs-2510 144 62 𝑆 𝑆 PROPN iajs-2510 144 63 semiprime semiprime NOUN iajs-2510 144 64 submodule submodule NOUN iajs-2510 144 65 and and CCONJ iajs-2510 144 66 𝐵 𝐵 NOUN iajs-2510 144 67 submodule submodule NOUN iajs-2510 144 68 of of ADP iajs-2510 144 69 x x PRON iajs-2510 144 70 ; ; PUNCT iajs-2510 144 71 see see VERB iajs-2510 144 72 proposition proposition NOUN iajs-2510 144 73 ( ( PUNCT iajs-2510 144 74 1.14 1.14 NUM iajs-2510 144 75 ) ) PUNCT iajs-2510 144 76 such such ADJ iajs-2510 144 77 that that SCONJ iajs-2510 144 78 𝐵 𝐵 NOUN iajs-2510 144 79 ⊈ ⊈ PROPN iajs-2510 144 80 𝑋 𝑋 PROPN iajs-2510 144 81 , , PUNCT iajs-2510 144 82 then then ADV iajs-2510 144 83 by by ADP iajs-2510 144 84 [ [ PUNCT iajs-2510 144 85 13 13 NUM iajs-2510 144 86 , , PUNCT iajs-2510 144 87 proposition(1.11 proposition(1.11 NOUN iajs-2510 144 88 ) ) PUNCT iajs-2510 144 89 ] ] PUNCT iajs-2510 144 90 , , PUNCT iajs-2510 144 91 𝑆 𝑆 PROPN iajs-2510 144 92 ∩ ∩ ADJ iajs-2510 144 93 𝐵 𝐵 PROPN iajs-2510 144 94 𝑆 𝑆 PROPN iajs-2510 144 95 ∩ ∩ ADJ iajs-2510 144 96 𝐵 𝐵 NOUN iajs-2510 144 97 ; ; PUNCT iajs-2510 144 98 see see VERB iajs-2510 144 99 proposition proposition NOUN iajs-2510 144 100 ( ( PUNCT iajs-2510 144 101 1.7)(1 1.7)(1 NUM iajs-2510 144 102 ) ) PUNCT iajs-2510 144 103 is be AUX iajs-2510 144 104 a a DET iajs-2510 144 105 semiprime semiprime NOUN iajs-2510 144 106 submodule submodule NOUN iajs-2510 144 107 in in ADP iajs-2510 144 108 𝐵 𝐵 PROPN iajs-2510 144 109 , , PUNCT iajs-2510 144 110 therefore therefore ADV iajs-2510 144 111 s s VERB iajs-2510 144 112 ∩ ∩ NOUN iajs-2510 144 113 b b NOUN iajs-2510 144 114 is be AUX iajs-2510 144 115 a a DET iajs-2510 144 116 semiprime semiprime NOUN iajs-2510 144 117 f f NOUN iajs-2510 144 118 - - PUNCT iajs-2510 144 119 submodule submodule NOUN iajs-2510 144 120 in in ADP iajs-2510 144 121 b b PROPN iajs-2510 144 122 ; ; PUNCT iajs-2510 144 123 see see VERB iajs-2510 144 124 [ [ X iajs-2510 144 125 14 14 NUM iajs-2510 144 126 , , PUNCT iajs-2510 144 127 theorem(2.4 theorem(2.4 NOUN iajs-2510 144 128 ) ) PUNCT iajs-2510 144 129 ] ] PUNCT iajs-2510 144 130 . . PUNCT iajs-2510 145 1 in in ADP iajs-2510 145 2 the the DET iajs-2510 145 3 following follow VERB iajs-2510 145 4 proposition proposition NOUN iajs-2510 145 5 , , PUNCT iajs-2510 145 6 we we PRON iajs-2510 145 7 prove prove VERB iajs-2510 145 8 the the DET iajs-2510 145 9 transitive transitive ADJ iajs-2510 145 10 property property NOUN iajs-2510 145 11 for for ADP iajs-2510 145 12 non non ADJ iajs-2510 145 13 - - ADJ iajs-2510 145 14 trivial trivial ADJ iajs-2510 145 15 fsubmodule fsubmodule NOUN iajs-2510 145 16 . . PUNCT iajs-2510 146 1 proposition proposition NOUN iajs-2510 146 2 2.12 2.12 NUM iajs-2510 146 3 : : PUNCT iajs-2510 146 4 let let VERB iajs-2510 146 5 a a DET iajs-2510 146 6 , , PUNCT iajs-2510 146 7 b b PROPN iajs-2510 146 8 be be AUX iajs-2510 146 9 a a DET iajs-2510 146 10 non non ADJ iajs-2510 146 11 - - ADJ iajs-2510 146 12 trivial trivial ADJ iajs-2510 146 13 f f NOUN iajs-2510 146 14 - - PUNCT iajs-2510 146 15 submodules submodule NOUN iajs-2510 146 16 of of ADP iajs-2510 146 17 f f NOUN iajs-2510 146 18 - - PUNCT iajs-2510 146 19 module module NOUN iajs-2510 146 20 x x NOUN iajs-2510 146 21 such such ADJ iajs-2510 146 22 that that SCONJ iajs-2510 146 23 a a DET iajs-2510 146 24 ⊆ ⊆ NUM iajs-2510 146 25 b. b. NOUN iajs-2510 146 26 if if SCONJ iajs-2510 146 27 a a PRON iajs-2510 146 28 is be AUX iajs-2510 146 29 a a DET iajs-2510 146 30 weak weak ADJ iajs-2510 146 31 essential essential ADJ iajs-2510 146 32 f f NOUN iajs-2510 146 33 - - PUNCT iajs-2510 146 34 submodule submodule NOUN iajs-2510 146 35 in in ADP iajs-2510 146 36 b b PROPN iajs-2510 146 37 and and CCONJ iajs-2510 146 38 b b PROPN iajs-2510 146 39 is be AUX iajs-2510 146 40 a a DET iajs-2510 146 41 weak weak ADJ iajs-2510 146 42 essential essential ADJ iajs-2510 146 43 ϝ-submodule ϝ-submodule NOUN iajs-2510 146 44 in in ADP iajs-2510 146 45 x x PUNCT iajs-2510 146 46 implying imply VERB iajs-2510 146 47 a a PRON iajs-2510 146 48 is be AUX iajs-2510 146 49 a a DET iajs-2510 146 50 weak weak ADJ iajs-2510 146 51 essential essential ADJ iajs-2510 146 52 f f NOUN iajs-2510 146 53 - - PUNCT iajs-2510 146 54 submodule submodule NOUN iajs-2510 146 55 in in ADP iajs-2510 146 56 x. x. NOUN iajs-2510 146 57 proof proof NOUN iajs-2510 146 58 : : PUNCT iajs-2510 146 59 assume assume VERB iajs-2510 146 60 that that SCONJ iajs-2510 146 61 s s VERB iajs-2510 146 62 is be AUX iajs-2510 146 63 a a DET iajs-2510 146 64 semiprime semiprime NOUN iajs-2510 146 65 f f NOUN iajs-2510 146 66 - - PUNCT iajs-2510 146 67 submodule submodule NOUN iajs-2510 146 68 in in ADP iajs-2510 146 69 x x PROPN iajs-2510 146 70 , , PUNCT iajs-2510 146 71 such such ADJ iajs-2510 146 72 that that SCONJ iajs-2510 146 73 a a DET iajs-2510 146 74 ∩ ∩ NOUN iajs-2510 146 75 s s PART iajs-2510 146 76 = = NOUN iajs-2510 146 77 0 0 NUM iajs-2510 146 78 . . PUNCT iajs-2510 147 1 note note VERB iajs-2510 147 2 that that SCONJ iajs-2510 147 3 0 0 PUNCT iajs-2510 148 1 a a DET iajs-2510 148 2 ∩ ∩ NOUN iajs-2510 148 3 s s PART iajs-2510 148 4 = = X iajs-2510 148 5 ( ( PUNCT iajs-2510 148 6 a a DET iajs-2510 148 7 ∩ ∩ ADJ iajs-2510 148 8 s s NOUN iajs-2510 148 9 ) ) PUNCT iajs-2510 148 10 ∩ ∩ NOUN iajs-2510 148 11 b b X iajs-2510 148 12 = = SYM iajs-2510 148 13 a a DET iajs-2510 148 14 ∩ ∩ NOUN iajs-2510 148 15 ( ( PUNCT iajs-2510 148 16 s s NOUN iajs-2510 148 17 ∩ ∩ NOUN iajs-2510 148 18 β β NOUN iajs-2510 148 19 ) ) PUNCT iajs-2510 148 20 . . PUNCT iajs-2510 149 1 but but CCONJ iajs-2510 149 2 s s VERB iajs-2510 149 3 is be AUX iajs-2510 149 4 a a DET iajs-2510 149 5 semi semi ADJ iajs-2510 149 6 - - ADJ iajs-2510 149 7 prime prime ADJ iajs-2510 149 8 f f NOUN iajs-2510 149 9 - - PUNCT iajs-2510 149 10 submodule submodule NOUN iajs-2510 149 11 of of ADP iajs-2510 149 12 x x PRON iajs-2510 149 13 , , PUNCT iajs-2510 149 14 so so ADV iajs-2510 149 15 we we PRON iajs-2510 149 16 have have VERB iajs-2510 149 17 two two NUM iajs-2510 149 18 cases case NOUN iajs-2510 149 19 . . PUNCT iajs-2510 150 1 if if SCONJ iajs-2510 150 2 b b PROPN iajs-2510 150 3 ⊆ ⊆ NUM iajs-2510 150 4 s s NOUN iajs-2510 150 5 , , PUNCT iajs-2510 150 6 then then ADV iajs-2510 150 7 0 0 NUM iajs-2510 150 8 a a DET iajs-2510 150 9 ∩ ∩ NOUN iajs-2510 150 10 ( ( PUNCT iajs-2510 150 11 s s X iajs-2510 150 12 ∩ ∩ ADJ iajs-2510 150 13 b b NOUN iajs-2510 150 14 ) ) PUNCT iajs-2510 150 15 = = SYM iajs-2510 150 16 a a DET iajs-2510 150 17 ∩ ∩ ADJ iajs-2510 150 18 b. b. NOUN iajs-2510 150 19 hence hence ADV iajs-2510 150 20 , , PUNCT iajs-2510 150 21 a a DET iajs-2510 150 22 ∩ ∩ ADJ iajs-2510 150 23 b b NOUN iajs-2510 150 24 = = SYM iajs-2510 150 25 0 0 NUM iajs-2510 150 26 , , PUNCT iajs-2510 150 27 but but CCONJ iajs-2510 150 28 a a DET iajs-2510 150 29 ⊆ ⊆ NUM iajs-2510 150 30 b b NOUN iajs-2510 150 31 so so SCONJ iajs-2510 150 32 a a DET iajs-2510 150 33 ∩ ∩ ADJ iajs-2510 150 34 b b X iajs-2510 150 35 = = SYM iajs-2510 150 36 a a PRON iajs-2510 150 37 implies imply VERB iajs-2510 150 38 a a DET iajs-2510 150 39 = = SYM iajs-2510 150 40 0 0 NUM iajs-2510 150 41 which which PRON iajs-2510 150 42 is be AUX iajs-2510 150 43 a a DET iajs-2510 150 44 contradiction contradiction NOUN iajs-2510 150 45 with with ADP iajs-2510 150 46 our our PRON iajs-2510 150 47 assumption assumption NOUN iajs-2510 150 48 . . PUNCT iajs-2510 151 1 thus thus ADV iajs-2510 151 2 b b X iajs-2510 151 3 ⊈ ⊈ PROPN iajs-2510 151 4 s s X iajs-2510 151 5 and and CCONJ iajs-2510 151 6 by by ADP iajs-2510 151 7 lemma lemma PROPN iajs-2510 151 8 ( ( PUNCT iajs-2510 151 9 2.11 2.11 NUM iajs-2510 151 10 ) ) PUNCT iajs-2510 151 11 , , PUNCT iajs-2510 151 12 s s PART iajs-2510 151 13 ∩ ∩ NOUN iajs-2510 151 14 b b NOUN iajs-2510 151 15 is be AUX iajs-2510 151 16 a a DET iajs-2510 151 17 semiprime semiprime NOUN iajs-2510 151 18 f f NOUN iajs-2510 151 19 - - PUNCT iajs-2510 151 20 submodule submodule NOUN iajs-2510 151 21 in in ADP iajs-2510 151 22 b. b. PROPN iajs-2510 151 23 since since SCONJ iajs-2510 151 24 a a PRON iajs-2510 151 25 is be AUX iajs-2510 151 26 a a DET iajs-2510 151 27 weak weak ADJ iajs-2510 151 28 essential essential ADJ iajs-2510 151 29 fsubmodule fsubmodule NOUN iajs-2510 151 30 in in ADP iajs-2510 151 31 b b PROPN iajs-2510 151 32 , , PUNCT iajs-2510 151 33 therefore therefore ADV iajs-2510 151 34 s s VERB iajs-2510 151 35 ∩ ∩ ADJ iajs-2510 151 36 b b NOUN iajs-2510 151 37 = = SYM iajs-2510 151 38 0 0 NUM iajs-2510 151 39 and and CCONJ iajs-2510 151 40 since since SCONJ iajs-2510 151 41 b b PROPN iajs-2510 151 42 is be AUX iajs-2510 151 43 a a DET iajs-2510 151 44 weak weak ADJ iajs-2510 151 45 essential essential ADJ iajs-2510 151 46 f f NOUN iajs-2510 151 47 - - PUNCT iajs-2510 151 48 submodule submodule NOUN iajs-2510 151 49 in in ADP iajs-2510 151 50 x x NOUN iajs-2510 151 51 , , PUNCT iajs-2510 151 52 then then ADV iajs-2510 151 53 s s VERB iajs-2510 151 54 = = NOUN iajs-2510 151 55 0 0 NUM iajs-2510 151 56 , , PUNCT iajs-2510 151 57 then then ADV iajs-2510 151 58 a a PRON iajs-2510 151 59 is be AUX iajs-2510 151 60 a a DET iajs-2510 151 61 weak weak ADJ iajs-2510 151 62 essential essential ADJ iajs-2510 151 63 f f NOUN iajs-2510 151 64 - - PUNCT iajs-2510 151 65 submodule submodule NOUN iajs-2510 151 66 in in ADP iajs-2510 151 67 x. x. NOUN iajs-2510 151 68 now now ADV iajs-2510 151 69 , , PUNCT iajs-2510 151 70 we we PRON iajs-2510 151 71 study study VERB iajs-2510 151 72 a a DET iajs-2510 151 73 homomorphic homomorphic ADJ iajs-2510 151 74 image image NOUN iajs-2510 151 75 of of ADP iajs-2510 151 76 a a DET iajs-2510 151 77 weak weak ADJ iajs-2510 151 78 essential essential ADJ iajs-2510 151 79 f f NOUN iajs-2510 151 80 - - PUNCT iajs-2510 151 81 submodule submodule NOUN iajs-2510 151 82 . . PUNCT iajs-2510 152 1 proposition proposition NOUN iajs-2510 152 2 2.13 2.13 NUM iajs-2510 152 3 : : PUNCT iajs-2510 152 4 let let VERB iajs-2510 152 5 x x PRON iajs-2510 152 6 , , PUNCT iajs-2510 152 7 x x PUNCT iajs-2510 152 8 be be AUX iajs-2510 152 9 f f NOUN iajs-2510 152 10 - - PUNCT iajs-2510 152 11 modules module NOUN iajs-2510 152 12 of of ADP iajs-2510 152 13 an an DET iajs-2510 152 14 ℛ-module ℛ-module PROPN iajs-2510 152 15 ℳ ℳ PROPN iajs-2510 152 16 and and CCONJ iajs-2510 152 17 ℳ ℳ NOUN iajs-2510 152 18 resp resp NOUN iajs-2510 152 19 . . PUNCT iajs-2510 153 1 and and CCONJ iajs-2510 153 2 f f X iajs-2510 153 3 : : PUNCT iajs-2510 154 1 x x SYM iajs-2510 154 2 ⟶ ⟶ NOUN iajs-2510 154 3 x x AUX iajs-2510 154 4 be be AUX iajs-2510 154 5 f f NOUN iajs-2510 154 6 - - PUNCT iajs-2510 154 7 epimorphism epimorphism NOUN iajs-2510 154 8 . . PUNCT iajs-2510 155 1 if if SCONJ iajs-2510 155 2 𝐴 𝐴 PROPN iajs-2510 155 3 is be AUX iajs-2510 155 4 a a DET iajs-2510 155 5 weak weak ADJ iajs-2510 155 6 essential essential ADJ iajs-2510 155 7 f f NOUN iajs-2510 155 8 - - PUNCT iajs-2510 155 9 submodule submodule NOUN iajs-2510 155 10 of of ADP iajs-2510 155 11 x x SYM iajs-2510 155 12 such such ADJ iajs-2510 155 13 that that SCONJ iajs-2510 155 14 𝐴 𝐴 PROPN iajs-2510 155 15 is be AUX iajs-2510 155 16 f f NOUN iajs-2510 155 17 - - PUNCT iajs-2510 155 18 invariant invariant ADJ iajs-2510 155 19 , , PUNCT iajs-2510 155 20 then then ADV iajs-2510 155 21 f f PROPN iajs-2510 155 22 ( ( PUNCT iajs-2510 155 23 𝐴 𝐴 PROPN iajs-2510 155 24 ) ) PUNCT iajs-2510 155 25 is be AUX iajs-2510 155 26 a a DET iajs-2510 155 27 weak weak ADJ iajs-2510 155 28 essential essential ADJ iajs-2510 155 29 f f NOUN iajs-2510 155 30 - - PUNCT iajs-2510 155 31 submodule submodule NOUN iajs-2510 155 32 of of ADP iajs-2510 155 33 x x X iajs-2510 155 34 . . PUNCT iajs-2510 156 1 proof proof NOUN iajs-2510 156 2 : : PUNCT iajs-2510 156 3 to to PART iajs-2510 156 4 show show VERB iajs-2510 156 5 f f PROPN iajs-2510 156 6 ( ( PUNCT iajs-2510 156 7 𝐴 𝐴 PROPN iajs-2510 156 8 ) ) PUNCT iajs-2510 156 9 is be AUX iajs-2510 156 10 a a DET iajs-2510 156 11 weak weak ADJ iajs-2510 156 12 essential essential ADJ iajs-2510 156 13 f f NOUN iajs-2510 156 14 - - PUNCT iajs-2510 156 15 submodule submodule NOUN iajs-2510 156 16 of of ADP iajs-2510 156 17 x x PRON iajs-2510 156 18 , , PUNCT iajs-2510 156 19 since since SCONJ iajs-2510 156 20 𝐴 𝐴 PROPN iajs-2510 156 21 is be AUX iajs-2510 156 22 a a DET iajs-2510 156 23 f f NOUN iajs-2510 156 24 - - PUNCT iajs-2510 156 25 submodule submodule NOUN iajs-2510 156 26 of of ADP iajs-2510 156 27 x x SYM iajs-2510 156 28 , , PUNCT iajs-2510 156 29 then then ADV iajs-2510 156 30 f f PROPN iajs-2510 156 31 ( ( PUNCT iajs-2510 156 32 𝐴 𝐴 PROPN iajs-2510 156 33 ) ) PUNCT iajs-2510 156 34 is be AUX iajs-2510 156 35 a a DET iajs-2510 156 36 f f NOUN iajs-2510 156 37 - - PUNCT iajs-2510 156 38 submodule submodule NOUN iajs-2510 156 39 of of ADP iajs-2510 156 40 x x PUNCT iajs-2510 156 41 by by ADP iajs-2510 156 42 proposition proposition NOUN iajs-2510 156 43 ( ( PUNCT iajs-2510 156 44 1.13)(1).now 1.13)(1).now PRON iajs-2510 156 45 suppose suppose VERB iajs-2510 156 46 that that SCONJ iajs-2510 156 47 s s VERB iajs-2510 156 48 semiprime semiprime NOUN iajs-2510 156 49 fsubmodule fsubmodule NOUN iajs-2510 156 50 of of ADP iajs-2510 156 51 x x SYM iajs-2510 156 52 such such ADJ iajs-2510 156 53 that that SCONJ iajs-2510 156 54 f f PROPN iajs-2510 156 55 ( ( PUNCT iajs-2510 156 56 𝐴 𝐴 PROPN iajs-2510 156 57 ) ) PUNCT iajs-2510 156 58 ∩ ∩ PROPN iajs-2510 156 59 𝑆 𝑆 PROPN iajs-2510 156 60 0 0 NUM iajs-2510 156 61 ; ; PUNCT iajs-2510 156 62 therefore therefore ADV iajs-2510 156 63 𝑓 𝑓 X iajs-2510 156 64 ( ( PUNCT iajs-2510 156 65 f f PROPN iajs-2510 156 66 ( ( PUNCT iajs-2510 156 67 𝐴 𝐴 PROPN iajs-2510 156 68 ) ) PUNCT iajs-2510 156 69 ∩ ∩ PROPN iajs-2510 156 70 𝑆 𝑆 PROPN iajs-2510 156 71 𝑓 𝑓 PRON iajs-2510 156 72 0 0 NUM iajs-2510 156 73 , , PUNCT iajs-2510 156 74 then then ADV iajs-2510 156 75 𝑓 𝑓 X iajs-2510 156 76 f f X iajs-2510 156 77 ( ( PUNCT iajs-2510 156 78 𝐴 𝐴 PROPN iajs-2510 156 79 ) ) PUNCT iajs-2510 156 80 ∩ ∩ NOUN iajs-2510 156 81 𝑓 𝑓 PROPN iajs-2510 156 82 𝑆 𝑆 PROPN iajs-2510 156 83 0 0 NUM iajs-2510 156 84 , , PUNCT iajs-2510 156 85 see see VERB iajs-2510 156 86 proposition proposition NOUN iajs-2510 156 87 ( ( PUNCT iajs-2510 156 88 1.10)(2 1.10)(2 NUM iajs-2510 156 89 ) ) PUNCT iajs-2510 156 90 . . PUNCT iajs-2510 157 1 but but CCONJ iajs-2510 157 2 𝐴 𝐴 PROPN iajs-2510 157 3 is be AUX iajs-2510 157 4 f f ADJ iajs-2510 157 5 - - PUNCT iajs-2510 157 6 invariant invariant ADJ iajs-2510 157 7 implying implying NOUN iajs-2510 157 8 that that SCONJ iajs-2510 157 9 𝐴 𝐴 PROPN iajs-2510 157 10 ∩ ∩ NOUN iajs-2510 157 11 𝑓 𝑓 X iajs-2510 157 12 ( ( PUNCT iajs-2510 157 13 s s NOUN iajs-2510 157 14 ) ) PUNCT iajs-2510 157 15 0 0 NUM iajs-2510 157 16 , , PUNCT iajs-2510 157 17 and and CCONJ iajs-2510 157 18 𝑓 𝑓 DET iajs-2510 157 19 𝑆 𝑆 PROPN iajs-2510 157 20 0 0 NUM iajs-2510 157 21 , , PUNCT iajs-2510 157 22 since since SCONJ iajs-2510 157 23 𝐴 𝐴 PROPN iajs-2510 157 24 is be AUX iajs-2510 157 25 weak weak ADJ iajs-2510 157 26 essential essential ADJ iajs-2510 157 27 ϝ-submodule ϝ-submodule NOUN iajs-2510 157 28 and and CCONJ iajs-2510 157 29 𝑓 𝑓 DET iajs-2510 157 30 𝑆 𝑆 PROPN iajs-2510 157 31 fsubmodule fsubmodule NOUN iajs-2510 157 32 of of ADP iajs-2510 157 33 x x PUNCT iajs-2510 157 34 by by ADP iajs-2510 157 35 proposition proposition NOUN iajs-2510 157 36 ( ( PUNCT iajs-2510 157 37 1.13)(2 1.13)(2 NUM iajs-2510 157 38 ) ) PUNCT iajs-2510 157 39 . . PUNCT iajs-2510 158 1 f f PROPN iajs-2510 158 2 ( ( PUNCT iajs-2510 158 3 𝑓 𝑓 DET iajs-2510 158 4 𝑆 𝑆 PROPN iajs-2510 158 5 𝑓 𝑓 PRON iajs-2510 158 6 0 0 NUM iajs-2510 158 7 , , PUNCT iajs-2510 158 8 then then ADV iajs-2510 158 9 s s VERB iajs-2510 158 10 = = NOUN iajs-2510 158 11 0 0 NUM iajs-2510 158 12 , , PUNCT iajs-2510 158 13 by by ADP iajs-2510 158 14 proposition proposition NOUN iajs-2510 158 15 ( ( PUNCT iajs-2510 158 16 1.10)(3 1.10)(3 NUM iajs-2510 158 17 ) ) PUNCT iajs-2510 158 18 . . PUNCT iajs-2510 159 1 that that PRON iajs-2510 159 2 is be AUX iajs-2510 159 3 f f PROPN iajs-2510 159 4 ( ( PUNCT iajs-2510 159 5 𝐴 𝐴 PROPN iajs-2510 159 6 ) ) PUNCT iajs-2510 159 7 is be AUX iajs-2510 159 8 a a DET iajs-2510 159 9 weak weak ADJ iajs-2510 159 10 essential essential ADJ iajs-2510 159 11 f f NOUN iajs-2510 159 12 - - PUNCT iajs-2510 159 13 submodule submodule NOUN iajs-2510 159 14 . . PUNCT iajs-2510 160 1 now now ADV iajs-2510 160 2 , , PUNCT iajs-2510 160 3 we we PRON iajs-2510 160 4 consider consider VERB iajs-2510 160 5 the the DET iajs-2510 160 6 inverse inverse ADJ iajs-2510 160 7 image image NOUN iajs-2510 160 8 of of ADP iajs-2510 160 9 a a DET iajs-2510 160 10 weak weak ADJ iajs-2510 160 11 f f NOUN iajs-2510 160 12 - - PUNCT iajs-2510 160 13 submodule submodule NOUN iajs-2510 160 14 . . PUNCT iajs-2510 160 15     SPACE iajs-2510 161 1 72 72 NUM iajs-2510 161 2     SPACE iajs-2510 161 3 ibn ibn PROPN iajs-2510 161 4 al al PROPN iajs-2510 161 5 - - PUNCT iajs-2510 161 6 haitham haitham PROPN iajs-2510 161 7 jour jour X iajs-2510 161 8 . . PROPN iajs-2510 161 9 for for ADP iajs-2510 161 10 pure pure ADJ iajs-2510 161 11 & & CCONJ iajs-2510 161 12 appl appl PROPN iajs-2510 161 13 . . PUNCT iajs-2510 162 1 sci sci PROPN iajs-2510 162 2 . . PROPN iajs-2510 163 1 33 33 NUM iajs-2510 163 2 ( ( PUNCT iajs-2510 163 3 4 4 NUM iajs-2510 163 4 ) ) PUNCT iajs-2510 163 5 2020 2020 NUM iajs-2510 163 6 proposition proposition NOUN iajs-2510 163 7 2.14 2.14 NUM iajs-2510 163 8 : : PUNCT iajs-2510 163 9 let let VERB iajs-2510 163 10 x x PRON iajs-2510 163 11 , , PUNCT iajs-2510 163 12 x x X iajs-2510 163 13 are be AUX iajs-2510 163 14 f f NOUN iajs-2510 163 15 - - PUNCT iajs-2510 163 16 modules module NOUN iajs-2510 163 17 of of ADP iajs-2510 163 18 an an DET iajs-2510 163 19 ℛ-module ℛ-module PROPN iajs-2510 163 20 ℳ ℳ PROPN iajs-2510 163 21 and and CCONJ iajs-2510 163 22 ℳ ℳ NOUN iajs-2510 163 23 resp resp NOUN iajs-2510 163 24 . . PUNCT iajs-2510 164 1 and and CCONJ iajs-2510 164 2 f f X iajs-2510 164 3 : : PUNCT iajs-2510 165 1 x x SYM iajs-2510 165 2 ⟶ ⟶ NOUN iajs-2510 165 3 x x AUX iajs-2510 165 4 be be AUX iajs-2510 165 5 f f NOUN iajs-2510 165 6 - - PUNCT iajs-2510 165 7 epimorphism epimorphism NOUN iajs-2510 165 8 . . PUNCT iajs-2510 166 1 if if SCONJ iajs-2510 166 2 𝐴 𝐴 PROPN iajs-2510 166 3 is be AUX iajs-2510 166 4 weak weak ADJ iajs-2510 166 5 essential essential ADJ iajs-2510 166 6 f f NOUN iajs-2510 166 7 - - PUNCT iajs-2510 166 8 submodule submodule NOUN iajs-2510 166 9 of of ADP iajs-2510 166 10 x x SYM iajs-2510 166 11 , , PUNCT iajs-2510 166 12 then then ADV iajs-2510 166 13 𝑓 𝑓 X iajs-2510 166 14 ( ( PUNCT iajs-2510 166 15 𝐴 𝐴 PROPN iajs-2510 166 16 ) ) PUNCT iajs-2510 166 17 is be AUX iajs-2510 166 18 a a DET iajs-2510 166 19 weak weak ADJ iajs-2510 166 20 essential essential ADJ iajs-2510 166 21 f f NOUN iajs-2510 166 22 - - PUNCT iajs-2510 166 23 submodule submodule NOUN iajs-2510 166 24 of of ADP iajs-2510 166 25 x x X iajs-2510 166 26 . . PUNCT iajs-2510 167 1 proof proof NOUN iajs-2510 167 2 : : PUNCT iajs-2510 167 3 since since SCONJ iajs-2510 167 4 𝐴 𝐴 PROPN iajs-2510 167 5 f f PROPN iajs-2510 167 6 - - PUNCT iajs-2510 167 7 submodule submodule NOUN iajs-2510 167 8 of of ADP iajs-2510 167 9 x x SYM iajs-2510 167 10 , , PUNCT iajs-2510 167 11 then then ADV iajs-2510 167 12 𝑓 𝑓 DET iajs-2510 167 13 𝐴 𝐴 PROPN iajs-2510 167 14 is be AUX iajs-2510 167 15 f f NOUN iajs-2510 167 16 - - PUNCT iajs-2510 167 17 submodule submodule NOUN iajs-2510 167 18 of of ADP iajs-2510 167 19 x x SYM iajs-2510 167 20 see see VERB iajs-2510 167 21 proposition(1.13)(2).now proposition(1.13)(2).now NOUN iajs-2510 167 22 suppose suppose VERB iajs-2510 167 23 s s NOUN iajs-2510 167 24 is be AUX iajs-2510 167 25 semiprime semiprime NOUN iajs-2510 167 26 f f X iajs-2510 167 27 - - PUNCT iajs-2510 167 28 submodule submodule NOUN iajs-2510 167 29 of of ADP iajs-2510 167 30 x x PROPN iajs-2510 167 31 , , PUNCT iajs-2510 167 32 such such ADJ iajs-2510 167 33 that that SCONJ iajs-2510 167 34 𝑓 𝑓 DET iajs-2510 167 35 𝐴 𝐴 PROPN iajs-2510 167 36 ∩ ∩ ADJ iajs-2510 167 37 𝑆 𝑆 PROPN iajs-2510 167 38 0 0 NUM iajs-2510 167 39 , , PUNCT iajs-2510 167 40 hence hence ADV iajs-2510 167 41 f f PROPN iajs-2510 167 42 ( ( PUNCT iajs-2510 167 43 𝑓 𝑓 DET iajs-2510 167 44 𝐴 𝐴 PROPN iajs-2510 167 45 ∩ ∩ NOUN iajs-2510 167 46 𝑆 𝑆 PROPN iajs-2510 167 47 𝑓 𝑓 DET iajs-2510 167 48 0 0 NUM iajs-2510 167 49 , , PUNCT iajs-2510 167 50 implies imply VERB iajs-2510 167 51 that that SCONJ iajs-2510 167 52 f f PROPN iajs-2510 167 53 ( ( PUNCT iajs-2510 167 54 𝑓 𝑓 PROPN iajs-2510 167 55 𝐴 𝐴 PROPN iajs-2510 167 56 ∩ ∩ NOUN iajs-2510 167 57 𝑓 𝑓 PRON iajs-2510 167 58 𝑆 𝑆 PROPN iajs-2510 167 59 𝑓 𝑓 DET iajs-2510 167 60 0 0 NUM iajs-2510 167 61 see see NOUN iajs-2510 167 62 proposition proposition NOUN iajs-2510 167 63 ( ( PUNCT iajs-2510 167 64 1.10)(6 1.10)(6 NUM iajs-2510 167 65 ) ) PUNCT iajs-2510 167 66 . . PUNCT iajs-2510 168 1 𝐴 𝐴 PROPN iajs-2510 168 2 ∩ ∩ VERB iajs-2510 168 3 𝑓 𝑓 PROPN iajs-2510 168 4 𝑆 𝑆 PROPN iajs-2510 168 5 0 0 NUM iajs-2510 168 6 ( ( PUNCT iajs-2510 168 7 since since SCONJ iajs-2510 168 8 𝐴 𝐴 PROPN iajs-2510 168 9 is be AUX iajs-2510 168 10 f f NOUN iajs-2510 168 11 - - PUNCT iajs-2510 168 12 invariant invariant ADJ iajs-2510 168 13 and and CCONJ iajs-2510 168 14 f f PROPN iajs-2510 168 15 is be AUX iajs-2510 168 16 epimorphism epimorphism NOUN iajs-2510 168 17 ) ) PUNCT iajs-2510 168 18 , , PUNCT iajs-2510 168 19 then then ADV iajs-2510 168 20 𝑓 𝑓 X iajs-2510 168 21 𝑓 𝑓 DET iajs-2510 168 22 𝑆 𝑆 PROPN iajs-2510 168 23 𝑓 𝑓 PROPN iajs-2510 168 24 0 0 NUM iajs-2510 168 25 ) ) PUNCT iajs-2510 168 26 , , PUNCT iajs-2510 168 27 implies imply VERB iajs-2510 168 28 that that SCONJ iajs-2510 168 29 s s VERB iajs-2510 168 30 = = NOUN iajs-2510 168 31 0 0 NUM iajs-2510 168 32 , , PUNCT iajs-2510 168 33 since since SCONJ iajs-2510 168 34 every every DET iajs-2510 168 35 f f PROPN iajs-2510 168 36 - - PUNCT iajs-2510 168 37 submodule submodule NOUN iajs-2510 168 38 of of ADP iajs-2510 168 39 x x SYM iajs-2510 168 40 is be AUX iajs-2510 168 41 f f NOUN iajs-2510 168 42 - - PUNCT iajs-2510 168 43 invariant invariant ADJ iajs-2510 168 44 , , PUNCT iajs-2510 168 45 implies imply VERB iajs-2510 168 46 𝑓 𝑓 PRON iajs-2510 168 47 𝐴 𝐴 PROPN iajs-2510 168 48 is be AUX iajs-2510 168 49 weak weak ADJ iajs-2510 168 50 essential essential ADJ iajs-2510 168 51 f f NOUN iajs-2510 168 52 - - PUNCT iajs-2510 168 53 submodule submodule NOUN iajs-2510 168 54 of of ADP iajs-2510 168 55 x x PROPN iajs-2510 168 56 . . PUNCT iajs-2510 169 1 reference reference NOUN iajs-2510 169 2 1 1 NUM iajs-2510 169 3 . . PUNCT iajs-2510 170 1 zadeh zadeh PROPN iajs-2510 170 2 , , PUNCT iajs-2510 170 3 l.a l.a PROPN iajs-2510 170 4 . . PROPN iajs-2510 170 5 fuzzy fuzzy ADJ iajs-2510 170 6 sets set NOUN iajs-2510 170 7 . . PUNCT iajs-2510 171 1 information information NOUN iajs-2510 171 2 and and CCONJ iajs-2510 171 3 control control NOUN iajs-2510 171 4 . . PUNCT iajs-2510 172 1 1965 1965 NUM iajs-2510 172 2 , , PUNCT iajs-2510 172 3 8 8 NUM iajs-2510 172 4 , , PUNCT iajs-2510 172 5 338 338 NUM iajs-2510 172 6 - - SYM iajs-2510 172 7 353 353 NUM iajs-2510 172 8 . . NOUN iajs-2510 173 1 2 2 NUM iajs-2510 173 2 . . X iajs-2510 173 3 negoita negoita PROPN iajs-2510 173 4 , , PUNCT iajs-2510 173 5 c. c. PROPN iajs-2510 173 6 v. v. PROPN iajs-2510 173 7 ; ; PUNCT iajs-2510 173 8 ralescu ralescu NOUN iajs-2510 173 9 , , PUNCT iajs-2510 173 10 d. d. PROPN iajs-2510 173 11 a. a. NOUN iajs-2510 173 12 applications application NOUN iajs-2510 173 13 of of ADP iajs-2510 173 14 fuzzy fuzzy ADJ iajs-2510 173 15 sets set NOUN iajs-2510 173 16 and and CCONJ iajs-2510 173 17 system system NOUN iajs-2510 173 18 analysis analysis NOUN iajs-2510 173 19 . . PUNCT iajs-2510 174 1 ( ( PUNCT iajs-2510 174 2 birkhous birkhous ADJ iajs-2510 174 3 basel basel PROPN iajs-2510 174 4 ) ) PUNCT iajs-2510 174 5 , , PUNCT iajs-2510 174 6 1975 1975 NUM iajs-2510 174 7 . . PUNCT iajs-2510 175 1 3 3 X iajs-2510 175 2 . . X iajs-2510 175 3 mashinchi mashinchi PROPN iajs-2510 175 4 , , PUNCT iajs-2510 175 5 m. m. NOUN iajs-2510 175 6 ; ; PUNCT iajs-2510 175 7 zahedi zahedi PROPN iajs-2510 175 8 , , PUNCT iajs-2510 175 9 m. m. NOUN iajs-2510 175 10 m. m. NOUN iajs-2510 175 11 on on ADP iajs-2510 175 12 l l ADJ iajs-2510 175 13 - - ADJ iajs-2510 175 14 fuzzy fuzzy ADJ iajs-2510 175 15 primary primary ADJ iajs-2510 175 16 submodule submodule NOUN iajs-2510 175 17 . . PUNCT iajs-2510 176 1 fuzzy fuzzy ADJ iajs-2510 176 2 sets set NOUN iajs-2510 176 3 and and CCONJ iajs-2510 176 4 systems system NOUN iajs-2510 176 5 . . PUNCT iajs-2510 177 1 1992 1992 NUM iajs-2510 177 2 , , PUNCT iajs-2510 177 3 49 49 NUM iajs-2510 177 4 , , PUNCT iajs-2510 177 5 231 231 NUM iajs-2510 177 6 - - SYM iajs-2510 177 7 236 236 NUM iajs-2510 177 8 . . NOUN iajs-2510 178 1 4 4 NUM iajs-2510 178 2 . . X iajs-2510 178 3 mona mona PROPN iajs-2510 178 4 , , PUNCT iajs-2510 178 5 a. a. NOUN iajs-2510 178 6 a. a. NOUN iajs-2510 178 7 weak weak ADJ iajs-2510 178 8 essential essential ADJ iajs-2510 178 9 submodules submodule NOUN iajs-2510 178 10 . . PUNCT iajs-2510 179 1 um um INTJ iajs-2510 179 2 - - PUNCT iajs-2510 179 3 salama salama NOUN iajs-2510 179 4 , , PUNCT iajs-2510 179 5 j. j. PROPN iajs-2510 179 6 2009 2009 NUM iajs-2510 179 7 , , PUNCT iajs-2510 179 8 6,1 6,1 NUM iajs-2510 179 9 , , PUNCT iajs-2510 179 10 214 214 NUM iajs-2510 179 11 - - SYM iajs-2510 179 12 221 221 NUM iajs-2510 179 13 . . PUNCT iajs-2510 180 1 5 5 NUM iajs-2510 180 2 . . X iajs-2510 180 3 zahedi zahedi PROPN iajs-2510 180 4 , , PUNCT iajs-2510 180 5 m. m. NOUN iajs-2510 180 6 m. m. NOUN iajs-2510 180 7 on on ADP iajs-2510 180 8 l l ADJ iajs-2510 180 9 - - ADJ iajs-2510 180 10 fuzzy fuzzy ADJ iajs-2510 180 11 residual residual ADJ iajs-2510 180 12 quotient quotient NOUN iajs-2510 180 13 module module NOUN iajs-2510 180 14 and and CCONJ iajs-2510 180 15 p. p. NOUN iajs-2510 180 16 primary primary ADJ iajs-2510 180 17 submodule submodule NOUN iajs-2510 180 18 . . PUNCT iajs-2510 181 1 fuzzy fuzzy ADJ iajs-2510 181 2 sets set NOUN iajs-2510 181 3 and and CCONJ iajs-2510 181 4 systems system NOUN iajs-2510 181 5 . . PUNCT iajs-2510 181 6 1992 1992 NUM iajs-2510 181 7 , , PUNCT iajs-2510 181 8 51,333 51,333 NUM iajs-2510 181 9 - - SYM iajs-2510 181 10 344 344 NUM iajs-2510 181 11 . . NOUN iajs-2510 182 1 6 6 NUM iajs-2510 182 2 . . X iajs-2510 182 3 martinez martinez PROPN iajs-2510 182 4 , , PUNCT iajs-2510 182 5 l. l. PROPN iajs-2510 182 6 fuzzy fuzzy ADJ iajs-2510 182 7 module module NOUN iajs-2510 182 8 over over ADP iajs-2510 182 9 fuzzy fuzzy ADJ iajs-2510 182 10 rings ring NOUN iajs-2510 182 11 in in ADP iajs-2510 182 12 connection connection NOUN iajs-2510 182 13 with with ADP iajs-2510 182 14 fuzzy fuzzy ADJ iajs-2510 182 15 ideals ideal NOUN iajs-2510 182 16 of of ADP iajs-2510 182 17 rings ring NOUN iajs-2510 182 18 . . PUNCT iajs-2510 183 1 j. j. PROPN iajs-2510 183 2 fuzzy fuzzy PROPN iajs-2510 183 3 math math PROPN iajs-2510 183 4 . . PUNCT iajs-2510 184 1 1996 1996 NUM iajs-2510 184 2 , , PUNCT iajs-2510 184 3 4,843 4,843 NUM iajs-2510 184 4 - - SYM iajs-2510 184 5 857 857 NUM iajs-2510 184 6 . . PUNCT iajs-2510 185 1 7 7 X iajs-2510 185 2 . . X iajs-2510 185 3 yue yue PROPN iajs-2510 185 4 z. z. PROPN iajs-2510 185 5 prime prime PROPN iajs-2510 185 6 l l ADJ iajs-2510 185 7 - - ADJ iajs-2510 185 8 fuzzy fuzzy ADJ iajs-2510 185 9 ideals ideal NOUN iajs-2510 185 10 and and CCONJ iajs-2510 185 11 primary primary ADJ iajs-2510 185 12 l l ADJ iajs-2510 185 13 - - ADJ iajs-2510 185 14 fuzzy fuzzy ADJ iajs-2510 185 15 ideals ideal NOUN iajs-2510 185 16 . . PUNCT iajs-2510 186 1 fuzzy fuzzy ADJ iajs-2510 186 2 sets set NOUN iajs-2510 186 3 and and CCONJ iajs-2510 186 4 systems system NOUN iajs-2510 186 5 . . PUNCT iajs-2510 187 1 1988 1988 NUM iajs-2510 187 2 , , PUNCT iajs-2510 187 3 27 27 NUM iajs-2510 187 4 , , PUNCT iajs-2510 187 5 345 345 NUM iajs-2510 187 6 - - SYM iajs-2510 187 7 350 350 NUM iajs-2510 187 8 . . NOUN iajs-2510 187 9 8 8 NUM iajs-2510 187 10 . . PUNCT iajs-2510 188 1 kumar kumar PROPN iajs-2510 188 2 r. r. PROPN iajs-2510 188 3 fuzzy fuzzy PROPN iajs-2510 188 4 semi semi ADJ iajs-2510 188 5 - - ADJ iajs-2510 188 6 primary primary ADJ iajs-2510 188 7 ideals ideal NOUN iajs-2510 188 8 of of ADP iajs-2510 188 9 rings ring NOUN iajs-2510 188 10 . . PUNCT iajs-2510 189 1 fuzzy fuzzy ADJ iajs-2510 189 2 sets set NOUN iajs-2510 189 3 and and CCONJ iajs-2510 189 4 systems system NOUN iajs-2510 189 5 . . PUNCT iajs-2510 190 1 1991 1991 NUM iajs-2510 190 2 , , PUNCT iajs-2510 190 3 42 42 NUM iajs-2510 190 4 , , PUNCT iajs-2510 190 5 263272 263272 NUM iajs-2510 190 6 . . PUNCT iajs-2510 191 1 9 9 X iajs-2510 191 2 . . X iajs-2510 192 1 maysoun maysoun NOUN iajs-2510 192 2 , , PUNCT iajs-2510 192 3 a. a. PROPN iajs-2510 192 4 h. h. PROPN iajs-2510 193 1 f f X iajs-2510 193 2 - - PUNCT iajs-2510 193 3 regular regular ADJ iajs-2510 193 4 fuzzy fuzzy ADJ iajs-2510 193 5 modules module NOUN iajs-2510 193 6 . . PUNCT iajs-2510 194 1 m.sc m.sc NOUN iajs-2510 194 2 . . PUNCT iajs-2510 195 1 thesis thesis NOUN iajs-2510 195 2 , , PUNCT iajs-2510 195 3 university university NOUN iajs-2510 195 4 of of ADP iajs-2510 195 5 baghdad baghdad PROPN iajs-2510 195 6 , , PUNCT iajs-2510 195 7 2002 2002 NUM iajs-2510 195 8 . . PUNCT iajs-2510 196 1 10 10 NUM iajs-2510 196 2 . . PUNCT iajs-2510 197 1 kumar kumar PROPN iajs-2510 197 2 r. r. PROPN iajs-2510 197 3 , , PUNCT iajs-2510 197 4 s. s. PROPN iajs-2510 197 5 k. k. PROPN iajs-2510 197 6 ; ; PUNCT iajs-2510 197 7 bhambir bhambir PROPN iajs-2510 197 8 , , PUNCT iajs-2510 197 9 kumar kumar PROPN iajs-2510 197 10 p. p. PROPN iajs-2510 197 11 fuzzy fuzzy PROPN iajs-2510 197 12 submodule submodule NOUN iajs-2510 197 13 of of ADP iajs-2510 197 14 some some DET iajs-2510 197 15 analogous analogous ADJ iajs-2510 197 16 and and CCONJ iajs-2510 197 17 deviation deviation NOUN iajs-2510 197 18 . . PUNCT iajs-2510 198 1 fuzzy fuzzy ADJ iajs-2510 198 2 sets set NOUN iajs-2510 198 3 and and CCONJ iajs-2510 198 4 systems system NOUN iajs-2510 198 5 . . PUNCT iajs-2510 199 1 1995 1995 NUM iajs-2510 199 2 , , PUNCT iajs-2510 199 3 70,125 70,125 NUM iajs-2510 199 4 - - SYM iajs-2510 199 5 130 130 NUM iajs-2510 199 6 . . PUNCT iajs-2510 200 1 11 11 NUM iajs-2510 200 2 . . PUNCT iajs-2510 201 1 mukhejee mukhejee NOUN iajs-2510 201 2 , , PUNCT iajs-2510 201 3 t. t. PROPN iajs-2510 201 4 k. k. PROPN iajs-2510 201 5 ; ; PUNCT iajs-2510 201 6 sen sen PROPN iajs-2510 201 7 , , PUNCT iajs-2510 201 8 m. m. PROPN iajs-2510 201 9 k. k. PROPN iajs-2510 201 10 ; ; PUNCT iajs-2510 201 11 roy roy PROPN iajs-2510 201 12 d. d. PROPN iajs-2510 201 13 on on ADP iajs-2510 201 14 submodule submodule PROPN iajs-2510 201 15 and and CCONJ iajs-2510 201 16 their their PRON iajs-2510 201 17 radicals radical NOUN iajs-2510 201 18 . . PUNCT iajs-2510 202 1 j. j. PROPN iajs-2510 202 2 fuzzy fuzzy PROPN iajs-2510 202 3 math math PROPN iajs-2510 202 4 . . PUNCT iajs-2510 203 1 1996 1996 NUM iajs-2510 203 2 , , PUNCT iajs-2510 203 3 4,549 4,549 NUM iajs-2510 203 4 - - SYM iajs-2510 203 5 558 558 NUM iajs-2510 203 6 . . PUNCT iajs-2510 204 1 12 12 NUM iajs-2510 204 2 . . PUNCT iajs-2510 205 1 rabi rabi NOUN iajs-2510 205 2 , , PUNCT iajs-2510 205 3 h. h. PROPN iajs-2510 205 4 j. j. PROPN iajs-2510 205 5 prime prime PROPN iajs-2510 205 6 fuzzy fuzzy ADJ iajs-2510 205 7 submodules submodule NOUN iajs-2510 205 8 and and CCONJ iajs-2510 205 9 prime prime ADJ iajs-2510 205 10 fuzzy fuzzy ADJ iajs-2510 205 11 modules module NOUN iajs-2510 205 12 . . PUNCT iajs-2510 206 1 m. m. PROPN iajs-2510 206 2 sc sc PROPN iajs-2510 206 3 . . PUNCT iajs-2510 207 1 thesis thesis PROPN iajs-2510 207 2 , , PUNCT iajs-2510 207 3 university university NOUN iajs-2510 207 4 of of ADP iajs-2510 207 5 baghdad baghdad PROPN iajs-2510 207 6 , , PUNCT iajs-2510 207 7 2001 2001 NUM iajs-2510 207 8 . . PUNCT iajs-2510 208 1 13 13 NUM iajs-2510 208 2 . . X iajs-2510 208 3 athab athab PROPN iajs-2510 208 4 , , PUNCT iajs-2510 208 5 e. e. PROPN iajs-2510 208 6 a. a. PROPN iajs-2510 208 7 prime prime PROPN iajs-2510 208 8 and and CCONJ iajs-2510 208 9 semi semi ADJ iajs-2510 208 10 - - ADJ iajs-2510 208 11 prime prime ADJ iajs-2510 208 12 submodules submodule NOUN iajs-2510 208 13 . . PUNCT iajs-2510 209 1 m. m. PROPN iajs-2510 209 2 sc sc PROPN iajs-2510 209 3 . . PUNCT iajs-2510 210 1 thesis thesis PROPN iajs-2510 210 2 , , PUNCT iajs-2510 210 3 university university NOUN iajs-2510 210 4 of of ADP iajs-2510 210 5 baghdad baghdad PROPN iajs-2510 210 6 , , PUNCT iajs-2510 210 7 1996 1996 NUM iajs-2510 210 8 . . PUNCT iajs-2510 211 1 14 14 NUM iajs-2510 211 2 . . PUNCT iajs-2510 211 3 hadi hadi PROPN iajs-2510 211 4 , , PUNCT iajs-2510 211 5 i. i. PROPN iajs-2510 211 6 m m PROPN iajs-2510 211 7 .a .a PROPN iajs-2510 211 8 . . PUNCT iajs-2510 212 1 semi semi ADJ iajs-2510 212 2 - - ADJ iajs-2510 212 3 prime prime ADJ iajs-2510 212 4 fuzzy fuzzy ADJ iajs-2510 212 5 submodules submodule NOUN iajs-2510 212 6 of of ADP iajs-2510 212 7 fuzzy fuzzy ADJ iajs-2510 212 8 modules module NOUN iajs-2510 212 9 . . PUNCT iajs-2510 213 1 ibn ibn PROPN iajs-2510 213 2 - - PUNCT iajs-2510 213 3 haitham haitham PROPN iajs-2510 213 4 j. j. PROPN iajs-2510 213 5 for for ADP iajs-2510 213 6 pure pure ADJ iajs-2510 213 7 and and CCONJ iajs-2510 213 8 appl appl NOUN iajs-2510 213 9 . . PUNCT iajs-2510 214 1 sci sci PROPN iajs-2510 214 2 . . PROPN iajs-2510 214 3 , , PUNCT iajs-2510 214 4 2004 2004 NUM iajs-2510 214 5 , , PUNCT iajs-2510 214 6 17,3 17,3 NUM iajs-2510 214 7 , , PUNCT iajs-2510 214 8 112 112 NUM iajs-2510 214 9 - - SYM iajs-2510 214 10 123 123 NUM iajs-2510 214 11 . . PUNCT iajs-2510 215 1 15 15 NUM iajs-2510 215 2 . . PUNCT iajs-2510 216 1 hassan hassan PROPN iajs-2510 216 2 , , PUNCT iajs-2510 216 3 k. k. PROPN iajs-2510 216 4 m. m. PROPN iajs-2510 216 5 ; ; PUNCT iajs-2510 216 6 hatam hatam NOUN iajs-2510 216 7 , , PUNCT iajs-2510 216 8 y. y. PROPN iajs-2510 216 9 k. k. PROPN iajs-2510 217 1 essential essential ADJ iajs-2510 217 2 fuzzy fuzzy ADJ iajs-2510 217 3 submodules submodule NOUN iajs-2510 217 4 and and CCONJ iajs-2510 217 5 closed close VERB iajs-2510 217 6 fuzzy fuzzy ADJ iajs-2510 217 7 submodules submodule NOUN iajs-2510 217 8 . . PUNCT iajs-2510 218 1 iraq iraq PROPN iajs-2510 218 2 . . PUNCT iajs-2510 219 1 j. j. PROPN iajs-2510 219 2 of of ADP iajs-2510 219 3 science science PROPN iajs-2510 219 4 , , PUNCT iajs-2510 219 5 2020 2020 NUM iajs-2510 219 6 , , PUNCT iajs-2510 219 7 61 61 NUM iajs-2510 219 8 , , PUNCT iajs-2510 219 9 4 4 NUM iajs-2510 219 10 , , PUNCT iajs-2510 219 11 890 890 NUM iajs-2510 219 12 - - SYM iajs-2510 219 13 897 897 NUM iajs-2510 219 14 . . PUNCT