id sid tid token lemma pos ejpam-5960 1 1 european european PROPN ejpam-5960 1 2 journal journal PROPN ejpam-5960 1 3 of of ADP ejpam-5960 1 4 pure pure ADJ ejpam-5960 1 5 and and CCONJ ejpam-5960 1 6 applied applied ADJ ejpam-5960 1 7 mathematics mathematic NOUN ejpam-5960 1 8 2025 2025 NUM ejpam-5960 1 9 , , PUNCT ejpam-5960 1 10 vol vol NOUN ejpam-5960 1 11 . . PROPN ejpam-5960 1 12 18 18 NUM ejpam-5960 1 13 , , PUNCT ejpam-5960 1 14 issue issue NOUN ejpam-5960 1 15 4 4 NUM ejpam-5960 1 16 , , PUNCT ejpam-5960 1 17 article article NOUN ejpam-5960 1 18 number number NOUN ejpam-5960 1 19 5960 5960 NUM ejpam-5960 1 20 issn issn PROPN ejpam-5960 1 21 1307 1307 NUM ejpam-5960 1 22 - - SYM ejpam-5960 1 23 5543 5543 NUM ejpam-5960 1 24 – – PUNCT ejpam-5960 1 25 ejpam.com ejpam.com X ejpam-5960 1 26 published publish VERB ejpam-5960 1 27 by by ADP ejpam-5960 1 28 new new PROPN ejpam-5960 1 29 york york PROPN ejpam-5960 1 30 business business PROPN ejpam-5960 1 31 global global NOUN ejpam-5960 1 32 on on ADP ejpam-5960 1 33 nearly nearly ADV ejpam-5960 1 34 𝛼–boundedness 𝛼–boundedness PROPN ejpam-5960 1 35 in in ADP ejpam-5960 1 36 𝐿–topological 𝐿–topological ADJ ejpam-5960 1 37 spaces space NOUN ejpam-5960 1 38 najah najah ADV ejpam-5960 1 39 a. a. PROPN ejpam-5960 1 40 alsaedi alsaedi PROPN ejpam-5960 1 41 department department PROPN ejpam-5960 1 42 of of ADP ejpam-5960 1 43 mathematics mathematic NOUN ejpam-5960 1 44 , , PUNCT ejpam-5960 1 45 faculty faculty NOUN ejpam-5960 1 46 of of ADP ejpam-5960 1 47 applied apply VERB ejpam-5960 1 48 science science NOUN ejpam-5960 1 49 , , PUNCT ejpam-5960 1 50 umm umm INTJ ejpam-5960 1 51 al al PROPN ejpam-5960 1 52 - - PUNCT ejpam-5960 1 53 qura qura PROPN ejpam-5960 1 54 university university PROPN ejpam-5960 1 55 , , PUNCT ejpam-5960 1 56 makkah makkah PROPN ejpam-5960 1 57 al al PROPN ejpam-5960 1 58 mukarramah mukarramah PROPN ejpam-5960 1 59 , , PUNCT ejpam-5960 1 60 saudi saudi PROPN ejpam-5960 1 61 arabia arabia PROPN ejpam-5960 1 62 abstract abstract NOUN ejpam-5960 1 63 . . PUNCT ejpam-5960 2 1 in in ADP ejpam-5960 2 2 this this DET ejpam-5960 2 3 paper paper NOUN ejpam-5960 2 4 , , PUNCT ejpam-5960 2 5 we we PRON ejpam-5960 2 6 introduce introduce VERB ejpam-5960 2 7 and and CCONJ ejpam-5960 2 8 study study VERB ejpam-5960 2 9 the the DET ejpam-5960 2 10 concept concept NOUN ejpam-5960 2 11 of of ADP ejpam-5960 2 12 nearly nearly ADV ejpam-5960 2 13 𝛼–boundedness 𝛼–boundedness PROPN ejpam-5960 2 14 on on ADP ejpam-5960 2 15 arbitrary arbitrary ADJ ejpam-5960 2 16 𝐿–subsets 𝐿–subset NOUN ejpam-5960 2 17 in in ADP ejpam-5960 2 18 𝐿–topological 𝐿–topological ADJ ejpam-5960 2 19 spaces space NOUN ejpam-5960 2 20 , , PUNCT ejpam-5960 2 21 which which PRON ejpam-5960 2 22 depends depend VERB ejpam-5960 2 23 on on ADP ejpam-5960 2 24 the the DET ejpam-5960 2 25 notion notion NOUN ejpam-5960 2 26 of of ADP ejpam-5960 2 27 𝛼–regular 𝛼–regular ADJ ejpam-5960 2 28 closed close VERB ejpam-5960 2 29 remoted remote VERB ejpam-5960 2 30 neighborhood neighborhood NOUN ejpam-5960 2 31 system system NOUN ejpam-5960 2 32 . . PUNCT ejpam-5960 3 1 several several ADJ ejpam-5960 3 2 characterizations characterization NOUN ejpam-5960 3 3 of of ADP ejpam-5960 3 4 nearly nearly ADV ejpam-5960 3 5 𝛼–boundedness 𝛼–boundedness PROPN ejpam-5960 3 6 in in ADP ejpam-5960 3 7 terms term NOUN ejpam-5960 3 8 of of ADP ejpam-5960 3 9 convergence convergence NOUN ejpam-5960 3 10 theory theory NOUN ejpam-5960 3 11 of of ADP ejpam-5960 3 12 𝛼–filters 𝛼–filter NOUN ejpam-5960 3 13 , , PUNCT ejpam-5960 3 14 𝛼–molecular 𝛼–molecular DET ejpam-5960 3 15 nets net NOUN ejpam-5960 3 16 and and CCONJ ejpam-5960 3 17 𝛼–ideals 𝛼–ideal NOUN ejpam-5960 3 18 are be AUX ejpam-5960 3 19 obtained obtain VERB ejpam-5960 3 20 . . PUNCT ejpam-5960 4 1 we we PRON ejpam-5960 4 2 prove prove VERB ejpam-5960 4 3 that that SCONJ ejpam-5960 4 4 the the DET ejpam-5960 4 5 concept concept NOUN ejpam-5960 4 6 is be AUX ejpam-5960 4 7 a a DET ejpam-5960 4 8 good good ADJ ejpam-5960 4 9 extension extension NOUN ejpam-5960 4 10 , , PUNCT ejpam-5960 4 11 productive productive ADJ ejpam-5960 4 12 , , PUNCT ejpam-5960 4 13 and and CCONJ ejpam-5960 4 14 topologically topologically ADV ejpam-5960 4 15 invariant invariant ADJ ejpam-5960 4 16 . . PUNCT ejpam-5960 5 1 2020 2020 NUM ejpam-5960 5 2 mathematics mathematics PROPN ejpam-5960 5 3 subject subject NOUN ejpam-5960 5 4 classifications classification NOUN ejpam-5960 5 5 : : PUNCT ejpam-5960 5 6 54a40 54a40 NUM ejpam-5960 5 7 key key ADJ ejpam-5960 5 8 words word NOUN ejpam-5960 5 9 and and CCONJ ejpam-5960 5 10 phrases phrase NOUN ejpam-5960 5 11 : : PUNCT ejpam-5960 5 12 nearly nearly ADV ejpam-5960 5 13 𝛼–boundedness 𝛼–boundedness PROPN ejpam-5960 5 14 , , PUNCT ejpam-5960 5 15 𝛼–regular 𝛼–regular PROPN ejpam-5960 5 16 closed close VERB ejpam-5960 5 17 remoted remoted ADJ ejpam-5960 5 18 neighborhood neighborhood NOUN ejpam-5960 5 19 , , PUNCT ejpam-5960 5 20 𝐿–topological 𝐿–topological ADJ ejpam-5960 5 21 space space NOUN ejpam-5960 5 22 , , PUNCT ejpam-5960 5 23 𝛼–filter 𝛼–filter NOUN ejpam-5960 5 24 , , PUNCT ejpam-5960 5 25 𝛼–molecular 𝛼–molecular DET ejpam-5960 5 26 nets net NOUN ejpam-5960 5 27 , , PUNCT ejpam-5960 5 28 𝛼–ideals 𝛼–ideal NOUN ejpam-5960 5 29 , , PUNCT ejpam-5960 5 30 nearly nearly ADV ejpam-5960 5 31 𝑄𝛼–compact 𝑄𝛼–compact PROPN ejpam-5960 5 32 1 1 NUM ejpam-5960 5 33 . . PUNCT ejpam-5960 6 1 introduction introduction NOUN ejpam-5960 6 2 boundedness boundedness NOUN ejpam-5960 6 3 , , PUNCT ejpam-5960 6 4 as as ADP ejpam-5960 6 5 a a DET ejpam-5960 6 6 natural natural ADJ ejpam-5960 6 7 generalization generalization NOUN ejpam-5960 6 8 of of ADP ejpam-5960 6 9 relative relative ADJ ejpam-5960 6 10 compactness compactness NOUN ejpam-5960 6 11 , , PUNCT ejpam-5960 6 12 was be AUX ejpam-5960 6 13 considered consider VERB ejpam-5960 6 14 by by ADP ejpam-5960 6 15 several several ADJ ejpam-5960 6 16 authors author NOUN ejpam-5960 6 17 ( ( PUNCT ejpam-5960 6 18 see see VERB ejpam-5960 6 19 [ [ X ejpam-5960 6 20 1 1 X ejpam-5960 6 21 ] ] PUNCT ejpam-5960 6 22 and and CCONJ ejpam-5960 6 23 [ [ X ejpam-5960 6 24 2 2 NUM ejpam-5960 6 25 ] ] PUNCT ejpam-5960 6 26 ) ) PUNCT ejpam-5960 6 27 . . PUNCT ejpam-5960 7 1 in in ADP ejpam-5960 7 2 1949 1949 NUM ejpam-5960 7 3 , , PUNCT ejpam-5960 7 4 hu hu PROPN ejpam-5960 8 1 [ [ X ejpam-5960 8 2 3 3 NUM ejpam-5960 8 3 ] ] PUNCT ejpam-5960 8 4 introduced introduce VERB ejpam-5960 8 5 the the DET ejpam-5960 8 6 notion notion NOUN ejpam-5960 8 7 of of ADP ejpam-5960 8 8 boundedness boundedness NOUN ejpam-5960 8 9 in in ADP ejpam-5960 8 10 general general ADJ ejpam-5960 8 11 topological topological ADJ ejpam-5960 8 12 spaces space NOUN ejpam-5960 8 13 and and CCONJ ejpam-5960 8 14 studied study VERB ejpam-5960 8 15 the the DET ejpam-5960 8 16 closure closure NOUN ejpam-5960 8 17 , , PUNCT ejpam-5960 8 18 interior interior NOUN ejpam-5960 8 19 , , PUNCT ejpam-5960 8 20 base base NOUN ejpam-5960 8 21 , , PUNCT ejpam-5960 8 22 and and CCONJ ejpam-5960 8 23 relativization relativization NOUN ejpam-5960 8 24 of of ADP ejpam-5960 8 25 boundedness boundedness NOUN ejpam-5960 8 26 . . PUNCT ejpam-5960 9 1 in in ADP ejpam-5960 9 2 - - PUNCT ejpam-5960 9 3 depth depth NOUN ejpam-5960 9 4 analysis analysis NOUN ejpam-5960 9 5 of of ADP ejpam-5960 9 6 boundedness boundedness NOUN ejpam-5960 9 7 and and CCONJ ejpam-5960 9 8 its its PRON ejpam-5960 9 9 various various ADJ ejpam-5960 9 10 weaker weak ADJ ejpam-5960 9 11 forms form NOUN ejpam-5960 9 12 was be AUX ejpam-5960 9 13 done do VERB ejpam-5960 9 14 by by ADP ejpam-5960 9 15 lamprinos lamprinos NOUN ejpam-5960 9 16 in in ADP ejpam-5960 9 17 [ [ X ejpam-5960 9 18 1 1 NUM ejpam-5960 9 19 ] ] PUNCT ejpam-5960 9 20 and and CCONJ ejpam-5960 9 21 [ [ X ejpam-5960 9 22 4 4 NUM ejpam-5960 9 23 ] ] PUNCT ejpam-5960 9 24 . . PUNCT ejpam-5960 10 1 a a DET ejpam-5960 10 2 subset subset ADJ ejpam-5960 10 3 𝐴 𝐴 PROPN ejpam-5960 10 4 of of ADP ejpam-5960 10 5 a a DET ejpam-5960 10 6 space space NOUN ejpam-5960 10 7 𝑋 𝑋 NOUN ejpam-5960 10 8 is be AUX ejpam-5960 10 9 said say VERB ejpam-5960 10 10 to to PART ejpam-5960 10 11 be be AUX ejpam-5960 10 12 bounded bound VERB ejpam-5960 10 13 if if SCONJ ejpam-5960 10 14 every every DET ejpam-5960 10 15 open open ADJ ejpam-5960 10 16 cover cover NOUN ejpam-5960 10 17 of of ADP ejpam-5960 10 18 𝑋 𝑋 PROPN ejpam-5960 10 19 has have VERB ejpam-5960 10 20 a a DET ejpam-5960 10 21 finite finite NOUN ejpam-5960 10 22 subfamily subfamily ADV ejpam-5960 10 23 that that PRON ejpam-5960 10 24 covers cover VERB ejpam-5960 10 25 𝐴. 𝐴. PROPN ejpam-5960 10 26 the the DET ejpam-5960 10 27 concept concept NOUN ejpam-5960 10 28 of of ADP ejpam-5960 10 29 a a DET ejpam-5960 10 30 bounded bounded ADJ ejpam-5960 10 31 set set NOUN ejpam-5960 10 32 is be AUX ejpam-5960 10 33 useful useful ADJ ejpam-5960 10 34 in in ADP ejpam-5960 10 35 investigating investigate VERB ejpam-5960 10 36 non non ADJ ejpam-5960 10 37 - - ADJ ejpam-5960 10 38 regular regular ADJ ejpam-5960 10 39 topological topological ADJ ejpam-5960 10 40 spaces space NOUN ejpam-5960 10 41 , , PUNCT ejpam-5960 10 42 since since SCONJ ejpam-5960 10 43 bounded bounded ADJ ejpam-5960 10 44 sets set NOUN ejpam-5960 10 45 in in ADP ejpam-5960 10 46 regular regular ADJ ejpam-5960 10 47 spaces space NOUN ejpam-5960 10 48 are be AUX ejpam-5960 10 49 compact compact ADJ ejpam-5960 10 50 . . PUNCT ejpam-5960 11 1 in in ADP ejpam-5960 11 2 1968 1968 NUM ejpam-5960 11 3 , , PUNCT ejpam-5960 11 4 chang chang PROPN ejpam-5960 11 5 [ [ X ejpam-5960 11 6 5 5 NUM ejpam-5960 11 7 ] ] PUNCT ejpam-5960 11 8 presented present VERB ejpam-5960 11 9 the the DET ejpam-5960 11 10 concept concept NOUN ejpam-5960 11 11 of of ADP ejpam-5960 11 12 fuzzy fuzzy ADJ ejpam-5960 11 13 compact compact ADJ ejpam-5960 11 14 . . PUNCT ejpam-5960 12 1 since since SCONJ ejpam-5960 12 2 then then ADV ejpam-5960 12 3 , , PUNCT ejpam-5960 12 4 it it PRON ejpam-5960 12 5 has have AUX ejpam-5960 12 6 been be AUX ejpam-5960 12 7 a a DET ejpam-5960 12 8 very very ADV ejpam-5960 12 9 important important ADJ ejpam-5960 12 10 topic topic NOUN ejpam-5960 12 11 to to PART ejpam-5960 12 12 define define VERB ejpam-5960 12 13 proper proper ADJ ejpam-5960 12 14 fuzzy fuzzy ADJ ejpam-5960 12 15 compactness compactness NOUN ejpam-5960 12 16 . . PUNCT ejpam-5960 13 1 many many ADJ ejpam-5960 13 2 authors author NOUN ejpam-5960 13 3 have have AUX ejpam-5960 13 4 written write VERB ejpam-5960 13 5 on on ADP ejpam-5960 13 6 this this DET ejpam-5960 13 7 problem problem NOUN ejpam-5960 13 8 and and CCONJ ejpam-5960 13 9 various various ADJ ejpam-5960 13 10 kinds kind NOUN ejpam-5960 13 11 of of ADP ejpam-5960 13 12 fuzzy fuzzy ADJ ejpam-5960 13 13 compactness compactness NOUN ejpam-5960 13 14 have have AUX ejpam-5960 13 15 been be AUX ejpam-5960 13 16 presented present VERB ejpam-5960 13 17 [ [ X ejpam-5960 13 18 6 6 NUM ejpam-5960 13 19 , , PUNCT ejpam-5960 13 20 7 7 NUM ejpam-5960 13 21 ] ] PUNCT ejpam-5960 13 22 . . PUNCT ejpam-5960 14 1 in in ADP ejpam-5960 14 2 1984 1984 NUM ejpam-5960 14 3 , , PUNCT ejpam-5960 14 4 li li PROPN ejpam-5960 15 1 [ [ X ejpam-5960 15 2 8 8 NUM ejpam-5960 15 3 ] ] PUNCT ejpam-5960 15 4 introduced introduce VERB ejpam-5960 15 5 the the DET ejpam-5960 15 6 fuzzy fuzzy ADJ ejpam-5960 15 7 𝑄𝛼–compactness 𝑄𝛼–compactness PROPN ejpam-5960 15 8 based base VERB ejpam-5960 15 9 upon upon SCONJ ejpam-5960 15 10 the the DET ejpam-5960 15 11 concept concept NOUN ejpam-5960 15 12 of of ADP ejpam-5960 15 13 𝑄–neighborhoods 𝑄–neighborhoods PROPN ejpam-5960 15 14 . . PUNCT ejpam-5960 16 1 in in ADP ejpam-5960 16 2 1992 1992 NUM ejpam-5960 16 3 , , PUNCT ejpam-5960 16 4 wang wang PROPN ejpam-5960 16 5 [ [ X ejpam-5960 16 6 9 9 NUM ejpam-5960 16 7 ] ] PUNCT ejpam-5960 16 8 generalized generalize VERB ejpam-5960 16 9 the the DET ejpam-5960 16 10 𝑄𝛼–compactness 𝑄𝛼–compactness PROPN ejpam-5960 16 11 to to ADP ejpam-5960 16 12 the the DET ejpam-5960 16 13 𝐿–fuzzy 𝐿–fuzzy ADJ ejpam-5960 16 14 topological topological ADJ ejpam-5960 16 15 spaces space NOUN ejpam-5960 16 16 . . PUNCT ejpam-5960 17 1 in in ADP ejpam-5960 17 2 1997 1997 NUM ejpam-5960 17 3 , , PUNCT ejpam-5960 17 4 georgiou georgiou PROPN ejpam-5960 17 5 and and CCONJ ejpam-5960 17 6 papadopoulos papadopoulos PROPN ejpam-5960 17 7 [ [ X ejpam-5960 17 8 10 10 NUM ejpam-5960 17 9 ] ] PUNCT ejpam-5960 17 10 gave give VERB ejpam-5960 17 11 a a DET ejpam-5960 17 12 characterization characterization NOUN ejpam-5960 17 13 of of ADP ejpam-5960 17 14 fuzzy fuzzy ADJ ejpam-5960 17 15 nearly nearly ADV ejpam-5960 17 16 compactness compactness NOUN ejpam-5960 17 17 by by ADP ejpam-5960 17 18 using use VERB ejpam-5960 17 19 the the DET ejpam-5960 17 20 notion notion NOUN ejpam-5960 17 21 of of ADP ejpam-5960 17 22 fuzzy fuzzy ADJ ejpam-5960 17 23 weakly weakly ADJ ejpam-5960 17 24 πœƒβ€“upper πœƒβ€“upper NOUN ejpam-5960 17 25 limit limit NOUN ejpam-5960 17 26 of of ADP ejpam-5960 17 27 fuzzy fuzzy ADJ ejpam-5960 17 28 nets net NOUN ejpam-5960 17 29 . . PUNCT ejpam-5960 18 1 also also ADV ejpam-5960 18 2 , , PUNCT ejpam-5960 18 3 he he PRON ejpam-5960 18 4 studied study VERB ejpam-5960 18 5 new new ADJ ejpam-5960 18 6 fuzzy fuzzy ADJ ejpam-5960 18 7 compactness compactness NOUN ejpam-5960 18 8 and and CCONJ ejpam-5960 18 9 fuzzy fuzzy ADJ ejpam-5960 18 10 boundedness boundedness NOUN ejpam-5960 18 11 in in ADP ejpam-5960 18 12 fuzzy fuzzy ADJ ejpam-5960 18 13 topological topological ADJ ejpam-5960 18 14 spaces space NOUN ejpam-5960 18 15 . . PUNCT ejpam-5960 19 1 recently recently ADV ejpam-5960 19 2 , , PUNCT ejpam-5960 19 3 georgiou georgiou PROPN ejpam-5960 19 4 and and CCONJ ejpam-5960 19 5 papadopoulos papadopoulos PROPN ejpam-5960 19 6 in in ADP ejpam-5960 19 7 [ [ X ejpam-5960 19 8 11 11 NUM ejpam-5960 19 9 , , PUNCT ejpam-5960 19 10 12 12 NUM ejpam-5960 19 11 ] ] PUNCT ejpam-5960 19 12 extended extend VERB ejpam-5960 19 13 the the DET ejpam-5960 19 14 concept concept NOUN ejpam-5960 19 15 of of ADP ejpam-5960 19 16 a a DET ejpam-5960 19 17 bounded bounded ADJ ejpam-5960 19 18 set set NOUN ejpam-5960 19 19 to to ADP ejpam-5960 19 20 fuzzy fuzzy ADJ ejpam-5960 19 21 topology topology NOUN ejpam-5960 19 22 ; ; PUNCT ejpam-5960 19 23 and and CCONJ ejpam-5960 19 24 introduced introduce VERB ejpam-5960 19 25 the the DET ejpam-5960 19 26 notion notion NOUN ejpam-5960 19 27 of of ADP ejpam-5960 19 28 fuzzy fuzzy ADJ ejpam-5960 19 29 boundedness boundedness NOUN ejpam-5960 19 30 using use VERB ejpam-5960 19 31 the the DET ejpam-5960 19 32 fuzzy fuzzy ADJ ejpam-5960 19 33 compactness compactness NOUN ejpam-5960 19 34 given give VERB ejpam-5960 19 35 by by ADP ejpam-5960 19 36 chang chang PROPN ejpam-5960 20 1 [ [ X ejpam-5960 20 2 5 5 NUM ejpam-5960 20 3 ] ] PUNCT ejpam-5960 20 4 , , PUNCT ejpam-5960 20 5 which which PRON ejpam-5960 20 6 is be AUX ejpam-5960 20 7 not not PART ejpam-5960 20 8 a a DET ejpam-5960 20 9 good good ADJ ejpam-5960 20 10 extension extension NOUN ejpam-5960 20 11 of of ADP ejpam-5960 20 12 ordinary ordinary ADJ ejpam-5960 20 13 compactness compactness NOUN ejpam-5960 20 14 ; ; PUNCT ejpam-5960 20 15 the the DET ejpam-5960 20 16 tychonoff tychonoff NOUN ejpam-5960 20 17 product product NOUN ejpam-5960 20 18 theorem theorem NOUN ejpam-5960 20 19 does do AUX ejpam-5960 20 20 not not PART ejpam-5960 20 21 hold hold VERB ejpam-5960 20 22 , , PUNCT ejpam-5960 20 23 and and CCONJ ejpam-5960 20 24 it it PRON ejpam-5960 20 25 contradicts contradict VERB ejpam-5960 20 26 some some DET ejpam-5960 20 27 kinds kind NOUN ejpam-5960 20 28 of of ADP ejpam-5960 20 29 separation separation NOUN ejpam-5960 20 30 doi doi NOUN ejpam-5960 20 31 : : PUNCT ejpam-5960 20 32 https://doi.org/10.29020/nybg.ejpam.v18i4.5960 https://doi.org/10.29020/nybg.ejpam.v18i4.5960 PROPN ejpam-5960 20 33 email email NOUN ejpam-5960 20 34 addresses address NOUN ejpam-5960 20 35 : : PUNCT ejpam-5960 20 36 dr-najah2008@hotmail.com dr-najah2008@hotmail.com X ejpam-5960 20 37 , , PUNCT ejpam-5960 20 38 nasadi@uqu.edu.sa nasadi@uqu.edu.sa PROPN ejpam-5960 20 39 ( ( PUNCT ejpam-5960 20 40 n. n. PROPN ejpam-5960 20 41 a. a. NOUN ejpam-5960 20 42 alsaedi alsaedi PROPN ejpam-5960 20 43 ) ) PUNCT ejpam-5960 20 44 https://www.ejpam.com https://www.ejpam.com NOUN ejpam-5960 21 1 1 1 NUM ejpam-5960 21 2 copyright copyright NOUN ejpam-5960 21 3 : : PUNCT ejpam-5960 21 4 Β© Β© PROPN ejpam-5960 21 5 2025 2025 NUM ejpam-5960 21 6 the the DET ejpam-5960 21 7 author(s author(s NOUN ejpam-5960 21 8 ) ) PUNCT ejpam-5960 21 9 . . PUNCT ejpam-5960 22 1 ( ( PUNCT ejpam-5960 22 2 cc cc NOUN ejpam-5960 22 3 by by ADP ejpam-5960 22 4 - - PUNCT ejpam-5960 22 5 nc nc PROPN ejpam-5960 22 6 4.0 4.0 NUM ejpam-5960 22 7 ) ) PUNCT ejpam-5960 22 8 n. n. NOUN ejpam-5960 22 9 a. a. NOUN ejpam-5960 22 10 alsaedi alsaedi PROPN ejpam-5960 22 11 / / SYM ejpam-5960 22 12 eur eur PROPN ejpam-5960 22 13 . . PUNCT ejpam-5960 23 1 j. j. PROPN ejpam-5960 23 2 pure pure PROPN ejpam-5960 23 3 appl appl PROPN ejpam-5960 23 4 . . PROPN ejpam-5960 23 5 math math PROPN ejpam-5960 23 6 , , PUNCT ejpam-5960 23 7 18 18 NUM ejpam-5960 23 8 ( ( PUNCT ejpam-5960 23 9 4 4 NUM ejpam-5960 23 10 ) ) PUNCT ejpam-5960 23 11 ( ( PUNCT ejpam-5960 23 12 2025 2025 NUM ejpam-5960 23 13 ) ) PUNCT ejpam-5960 23 14 , , PUNCT ejpam-5960 23 15 5960 5960 NUM ejpam-5960 23 16 2 2 NUM ejpam-5960 23 17 of of ADP ejpam-5960 23 18 22 22 NUM ejpam-5960 23 19 axioms axiom NOUN ejpam-5960 23 20 . . PUNCT ejpam-5960 24 1 hence hence ADV ejpam-5960 24 2 , , PUNCT ejpam-5960 24 3 the the DET ejpam-5960 24 4 notion notion NOUN ejpam-5960 24 5 of of ADP ejpam-5960 24 6 fuzzy fuzzy ADJ ejpam-5960 24 7 boundedness boundedness NOUN ejpam-5960 24 8 in in ADP ejpam-5960 24 9 [ [ X ejpam-5960 24 10 10 10 NUM ejpam-5960 24 11 ] ] PUNCT ejpam-5960 24 12 is be AUX ejpam-5960 24 13 not not PART ejpam-5960 24 14 a a DET ejpam-5960 24 15 good good ADJ ejpam-5960 24 16 extension extension NOUN ejpam-5960 24 17 of of ADP ejpam-5960 24 18 ordinary ordinary ADJ ejpam-5960 24 19 bounded bound VERB ejpam-5960 24 20 , , PUNCT ejpam-5960 24 21 and and CCONJ ejpam-5960 24 22 so so ADV ejpam-5960 24 23 it it PRON ejpam-5960 24 24 is be AUX ejpam-5960 24 25 unsatisfactory unsatisfactory ADJ ejpam-5960 24 26 . . PUNCT ejpam-5960 25 1 in in ADP ejpam-5960 25 2 2003 2003 NUM ejpam-5960 25 3 , , PUNCT ejpam-5960 25 4 nouh nouh NOUN ejpam-5960 26 1 [ [ X ejpam-5960 26 2 13 13 NUM ejpam-5960 26 3 ] ] PUNCT ejpam-5960 26 4 introduced introduce VERB ejpam-5960 26 5 the the DET ejpam-5960 26 6 concept concept NOUN ejpam-5960 26 7 of of ADP ejpam-5960 26 8 𝑁–boundedness 𝑁–boundedness NOUN ejpam-5960 26 9 on on ADP ejpam-5960 26 10 an an DET ejpam-5960 26 11 arbitrary arbitrary ADJ ejpam-5960 26 12 𝐿 𝐿 PROPN ejpam-5960 26 13 – – PUNCT ejpam-5960 26 14 subset subset NOUN ejpam-5960 26 15 in in ADP ejpam-5960 26 16 𝐿–topological 𝐿–topological ADJ ejpam-5960 26 17 spaces space NOUN ejpam-5960 26 18 , , PUNCT ejpam-5960 26 19 and and CCONJ ejpam-5960 26 20 he he PRON ejpam-5960 26 21 gave give VERB ejpam-5960 26 22 new new ADJ ejpam-5960 26 23 characterizations characterization NOUN ejpam-5960 26 24 and and CCONJ ejpam-5960 26 25 properties property NOUN ejpam-5960 26 26 of of ADP ejpam-5960 26 27 𝑁–boundedness 𝑁–boundedness ADJ ejpam-5960 26 28 in in ADP ejpam-5960 26 29 terms term NOUN ejpam-5960 26 30 of of ADP ejpam-5960 26 31 the the DET ejpam-5960 26 32 convergence convergence NOUN ejpam-5960 26 33 theory theory NOUN ejpam-5960 26 34 of of ADP ejpam-5960 26 35 𝛼–nets 𝛼–net NOUN ejpam-5960 26 36 , , PUNCT ejpam-5960 26 37 𝛼–filters 𝛼–filter NOUN ejpam-5960 26 38 , , PUNCT ejpam-5960 26 39 and and CCONJ ejpam-5960 26 40 𝛼–ideals 𝛼–ideal NOUN ejpam-5960 26 41 . . PUNCT ejpam-5960 27 1 he he PRON ejpam-5960 27 2 proved prove VERB ejpam-5960 27 3 that that SCONJ ejpam-5960 27 4 the the DET ejpam-5960 27 5 concept concept NOUN ejpam-5960 27 6 of of ADP ejpam-5960 27 7 𝑁–boundedness 𝑁–boundedness ADJ ejpam-5960 27 8 is be AUX ejpam-5960 27 9 a a DET ejpam-5960 27 10 good good ADJ ejpam-5960 27 11 extension extension NOUN ejpam-5960 27 12 , , PUNCT ejpam-5960 27 13 productive productive ADJ ejpam-5960 27 14 , , PUNCT ejpam-5960 27 15 and and CCONJ ejpam-5960 27 16 topologically topologically ADV ejpam-5960 27 17 invariant invariant ADJ ejpam-5960 27 18 . . PUNCT ejpam-5960 28 1 since since SCONJ ejpam-5960 28 2 there there PRON ejpam-5960 28 3 are be VERB ejpam-5960 28 4 not not PART ejpam-5960 28 5 enough enough ADJ ejpam-5960 28 6 studies study NOUN ejpam-5960 28 7 on on ADP ejpam-5960 28 8 the the DET ejpam-5960 28 9 concept concept NOUN ejpam-5960 28 10 of of ADP ejpam-5960 28 11 boundedness boundedness NOUN ejpam-5960 28 12 in in ADP ejpam-5960 28 13 𝐿–topological 𝐿–topological ADJ ejpam-5960 28 14 spaces space NOUN ejpam-5960 28 15 . . PUNCT ejpam-5960 29 1 so so ADV ejpam-5960 29 2 in in ADP ejpam-5960 29 3 2023 2023 NUM ejpam-5960 29 4 , , PUNCT ejpam-5960 29 5 alsaedi alsaedi VERB ejpam-5960 29 6 [ [ X ejpam-5960 29 7 14 14 NUM ejpam-5960 29 8 ] ] PUNCT ejpam-5960 29 9 introduced introduce VERB ejpam-5960 29 10 the the DET ejpam-5960 29 11 concept concept NOUN ejpam-5960 29 12 of of ADP ejpam-5960 29 13 nearly nearly ADV ejpam-5960 29 14 Ο‰ Ο‰ NOUN ejpam-5960 29 15 – – PUNCT ejpam-5960 29 16 boundedness boundedness NOUN ejpam-5960 29 17 on on ADP ejpam-5960 29 18 an an DET ejpam-5960 29 19 arbitrary arbitrary ADJ ejpam-5960 29 20 𝐿–subset 𝐿–subset NOUN ejpam-5960 29 21 in in ADP ejpam-5960 29 22 𝐿–topological 𝐿–topological ADJ ejpam-5960 29 23 spaces space NOUN ejpam-5960 29 24 by by ADP ejpam-5960 29 25 using use VERB ejpam-5960 29 26 the the DET ejpam-5960 29 27 notion notion NOUN ejpam-5960 29 28 of of ADP ejpam-5960 29 29 Ο‰ Ο‰ ADJ ejpam-5960 29 30 – – PUNCT ejpam-5960 29 31 upper upper ADJ ejpam-5960 29 32 limit limit NOUN ejpam-5960 29 33 of of ADP ejpam-5960 29 34 Ο‰ Ο‰ NOUN ejpam-5960 29 35 – – PUNCT ejpam-5960 29 36 nets net NOUN ejpam-5960 29 37 . . PUNCT ejpam-5960 30 1 in in ADP ejpam-5960 30 2 this this DET ejpam-5960 30 3 paper paper NOUN ejpam-5960 30 4 , , PUNCT ejpam-5960 30 5 we we PRON ejpam-5960 30 6 generalize generalize VERB ejpam-5960 30 7 the the DET ejpam-5960 30 8 nearly nearly ADV ejpam-5960 30 9 Ο‰ Ο‰ NOUN ejpam-5960 30 10 – – PUNCT ejpam-5960 30 11 boundedness boundedness NOUN ejpam-5960 30 12 to to ADP ejpam-5960 30 13 nearly nearly ADV ejpam-5960 30 14 𝛼–boundedness 𝛼–boundedness PROPN ejpam-5960 30 15 , , PUNCT ejpam-5960 30 16 where where SCONJ ejpam-5960 30 17 we we PRON ejpam-5960 30 18 will will AUX ejpam-5960 30 19 study study VERB ejpam-5960 30 20 this this DET ejpam-5960 30 21 concept concept NOUN ejpam-5960 30 22 on on ADP ejpam-5960 30 23 arbitrary arbitrary ADJ ejpam-5960 30 24 𝐿–subsets 𝐿–subset NOUN ejpam-5960 30 25 in in ADP ejpam-5960 30 26 𝐿–topological 𝐿–topological ADJ ejpam-5960 30 27 spaces space NOUN ejpam-5960 30 28 along along ADP ejpam-5960 30 29 the the DET ejpam-5960 30 30 line line NOUN ejpam-5960 30 31 of of ADP ejpam-5960 30 32 nearly nearly ADV ejpam-5960 30 33 𝑄𝛼–compactness 𝑄𝛼–compactness PROPN ejpam-5960 30 34 defined define VERB ejpam-5960 30 35 by by ADP ejpam-5960 30 36 wang wang PROPN ejpam-5960 31 1 [ [ X ejpam-5960 31 2 9 9 NUM ejpam-5960 31 3 ] ] PUNCT ejpam-5960 31 4 and and CCONJ ejpam-5960 31 5 𝛼–regular 𝛼–regular ADJ ejpam-5960 31 6 closed close VERB ejpam-5960 31 7 remoted remoted ADJ ejpam-5960 31 8 neighborhood neighborhood NOUN ejpam-5960 31 9 due due ADP ejpam-5960 31 10 to to ADP ejpam-5960 31 11 zhao zhao X ejpam-5960 31 12 [ [ X ejpam-5960 31 13 15 15 NUM ejpam-5960 31 14 ] ] PUNCT ejpam-5960 31 15 . . PUNCT ejpam-5960 32 1 then then ADV ejpam-5960 32 2 we we PRON ejpam-5960 32 3 give give VERB ejpam-5960 32 4 new new ADJ ejpam-5960 32 5 characterizations characterization NOUN ejpam-5960 32 6 and and CCONJ ejpam-5960 32 7 properties property NOUN ejpam-5960 32 8 of of ADP ejpam-5960 32 9 nearly nearly ADV ejpam-5960 32 10 𝛼–boundedness 𝛼–boundedness PROPN ejpam-5960 32 11 in in ADP ejpam-5960 32 12 terms term NOUN ejpam-5960 32 13 of of ADP ejpam-5960 32 14 the the DET ejpam-5960 32 15 convergence convergence NOUN ejpam-5960 32 16 theory theory NOUN ejpam-5960 32 17 of of ADP ejpam-5960 32 18 constant constant ADJ ejpam-5960 32 19 𝛼–filter 𝛼–filter NOUN ejpam-5960 32 20 , , PUNCT ejpam-5960 32 21 𝛼 𝛼 ADJ ejpam-5960 32 22 – – PUNCT ejpam-5960 32 23 molecular molecular ADJ ejpam-5960 32 24 nets net NOUN ejpam-5960 32 25 , , PUNCT ejpam-5960 32 26 and and CCONJ ejpam-5960 32 27 𝛼–ideals 𝛼–ideal NOUN ejpam-5960 32 28 . . PUNCT ejpam-5960 33 1 we we PRON ejpam-5960 33 2 prove prove VERB ejpam-5960 33 3 that that SCONJ ejpam-5960 33 4 the the DET ejpam-5960 33 5 notion notion NOUN ejpam-5960 33 6 is be AUX ejpam-5960 33 7 a a DET ejpam-5960 33 8 good good ADJ ejpam-5960 33 9 extension extension NOUN ejpam-5960 33 10 , , PUNCT ejpam-5960 33 11 productive productive ADJ ejpam-5960 33 12 , , PUNCT ejpam-5960 33 13 and and CCONJ ejpam-5960 33 14 topologically topologically ADV ejpam-5960 33 15 invariant invariant ADJ ejpam-5960 33 16 . . PUNCT ejpam-5960 34 1 2 2 X ejpam-5960 34 2 . . X ejpam-5960 34 3 preliminaries preliminary NOUN ejpam-5960 34 4 throughout throughout ADP ejpam-5960 34 5 this this DET ejpam-5960 34 6 paper paper NOUN ejpam-5960 34 7 𝐿 𝐿 PROPN ejpam-5960 34 8 = = PUNCT ejpam-5960 34 9 ⟨𝐿 ⟨𝐿 PROPN ejpam-5960 34 10 , , PUNCT ejpam-5960 34 11 ≀,∧,∨,βˆ— ≀,∧,∨,βˆ— PROPN ejpam-5960 34 12 ⟩ ⟩ NOUN ejpam-5960 34 13 denotes denote VERB ejpam-5960 34 14 a a DET ejpam-5960 34 15 completely completely ADV ejpam-5960 34 16 distributive distributive ADJ ejpam-5960 34 17 complete complete ADJ ejpam-5960 34 18 lattice lattice NOUN ejpam-5960 34 19 with with ADP ejpam-5960 34 20 a a DET ejpam-5960 34 21 smallest small ADJ ejpam-5960 34 22 element element NOUN ejpam-5960 34 23 0 0 NUM ejpam-5960 34 24 and and CCONJ ejpam-5960 34 25 a a DET ejpam-5960 34 26 largest large ADJ ejpam-5960 34 27 element element NOUN ejpam-5960 34 28 1 1 NUM ejpam-5960 34 29 ( ( PUNCT ejpam-5960 34 30 0 0 NUM ejpam-5960 34 31 β‰  β‰  PROPN ejpam-5960 34 32 1 1 NUM ejpam-5960 34 33 ) ) PUNCT ejpam-5960 34 34 and and CCONJ ejpam-5960 34 35 with with ADP ejpam-5960 34 36 an an DET ejpam-5960 34 37 order order NOUN ejpam-5960 34 38 – – PUNCT ejpam-5960 34 39 reversing reverse VERB ejpam-5960 34 40 involution involution NOUN ejpam-5960 34 41 on on ADP ejpam-5960 34 42 it it PRON ejpam-5960 34 43 . . PUNCT ejpam-5960 35 1 an an DET ejpam-5960 35 2 𝛼 𝛼 PROPN ejpam-5960 35 3 ∈ ∈ PROPN ejpam-5960 35 4 𝐿 𝐿 PROPN ejpam-5960 35 5 is be AUX ejpam-5960 35 6 called call VERB ejpam-5960 35 7 a a DET ejpam-5960 35 8 molecule molecule NOUN ejpam-5960 35 9 of of ADP ejpam-5960 35 10 𝐿 𝐿 PROPN ejpam-5960 35 11 if if SCONJ ejpam-5960 35 12 𝛼 𝛼 PRON ejpam-5960 35 13 β‰  β‰  PROPN ejpam-5960 35 14 0 0 NUM ejpam-5960 35 15 and and CCONJ ejpam-5960 35 16 0 0 NUM ejpam-5960 35 17 ≀ ≀ NOUN ejpam-5960 35 18 𝑣 𝑣 ADP ejpam-5960 35 19 ∨ ∨ NUM ejpam-5960 35 20 𝛾 𝛾 PROPN ejpam-5960 35 21 ≀ ≀ NOUN ejpam-5960 35 22 𝛼 𝛼 NOUN ejpam-5960 35 23 implies imply VERB ejpam-5960 35 24 𝑣 𝑣 ADP ejpam-5960 35 25 ≀ ≀ NOUN ejpam-5960 35 26 𝛾 𝛾 ADP ejpam-5960 35 27 or or CCONJ ejpam-5960 35 28 𝛾 𝛾 ADP ejpam-5960 35 29 ≀ ≀ NOUN ejpam-5960 35 30 𝑣 𝑣 ADP ejpam-5960 35 31 , , PUNCT ejpam-5960 35 32 for for ADP ejpam-5960 35 33 all all DET ejpam-5960 35 34 𝑣 𝑣 NOUN ejpam-5960 35 35 , , PUNCT ejpam-5960 35 36 𝛾 𝛾 AUX ejpam-5960 35 37 ∈ ∈ ADV ejpam-5960 35 38 𝐿. 𝐿. VERB ejpam-5960 35 39 the the DET ejpam-5960 35 40 set set NOUN ejpam-5960 35 41 of of ADP ejpam-5960 35 42 all all DET ejpam-5960 35 43 molecules molecule NOUN ejpam-5960 35 44 of of ADP ejpam-5960 35 45 𝐿 𝐿 PROPN ejpam-5960 35 46 is be AUX ejpam-5960 35 47 denoted denote VERB ejpam-5960 35 48 by by ADP ejpam-5960 35 49 𝑀 𝑀 PROPN ejpam-5960 35 50 ( ( PUNCT ejpam-5960 35 51 𝐿 𝐿 PROPN ejpam-5960 35 52 ) ) PUNCT ejpam-5960 35 53 . . PUNCT ejpam-5960 36 1 let let VERB ejpam-5960 36 2 𝑋 𝑋 NOUN ejpam-5960 36 3 be be AUX ejpam-5960 36 4 a a DET ejpam-5960 36 5 nonempty nonempty ADJ ejpam-5960 36 6 set set VERB ejpam-5960 36 7 . . PUNCT ejpam-5960 37 1 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 37 2 denotes denote VERB ejpam-5960 37 3 the the DET ejpam-5960 37 4 family family NOUN ejpam-5960 37 5 of of ADP ejpam-5960 37 6 all all DET ejpam-5960 37 7 mappings mapping NOUN ejpam-5960 37 8 from from ADP ejpam-5960 37 9 𝑋 𝑋 PROPN ejpam-5960 37 10 to to AUX ejpam-5960 37 11 𝐿. 𝐿. VERB ejpam-5960 37 12 the the DET ejpam-5960 37 13 elements element NOUN ejpam-5960 37 14 of of ADP ejpam-5960 37 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 37 16 are be AUX ejpam-5960 37 17 called call VERB ejpam-5960 37 18 𝐿–subsets 𝐿–subset NOUN ejpam-5960 37 19 on on ADP ejpam-5960 37 20 𝑋. 𝑋. PROPN ejpam-5960 37 21 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 37 22 can can AUX ejpam-5960 37 23 be be AUX ejpam-5960 37 24 made make VERB ejpam-5960 37 25 into into ADP ejpam-5960 37 26 a a DET ejpam-5960 37 27 lattice lattice NOUN ejpam-5960 37 28 by by ADP ejpam-5960 37 29 inducing induce VERB ejpam-5960 37 30 the the DET ejpam-5960 37 31 order order NOUN ejpam-5960 37 32 and and CCONJ ejpam-5960 37 33 involution involution NOUN ejpam-5960 37 34 from from ADP ejpam-5960 37 35 𝐿. 𝐿. PROPN ejpam-5960 37 36 we we PRON ejpam-5960 37 37 denote denote VERB ejpam-5960 37 38 the the DET ejpam-5960 37 39 smallest small ADJ ejpam-5960 37 40 element element NOUN ejpam-5960 37 41 and and CCONJ ejpam-5960 37 42 the the DET ejpam-5960 37 43 largest large ADJ ejpam-5960 37 44 element element NOUN ejpam-5960 37 45 of of ADP ejpam-5960 37 46 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 37 47 by by ADP ejpam-5960 37 48 0𝑋 0𝑋 NOUN ejpam-5960 37 49 and and CCONJ ejpam-5960 37 50 1𝑋 1𝑋 NOUN ejpam-5960 37 51 , , PUNCT ejpam-5960 37 52 respectively respectively ADV ejpam-5960 37 53 . . PUNCT ejpam-5960 38 1 if if SCONJ ejpam-5960 38 2 𝛼 𝛼 PROPN ejpam-5960 38 3 ∈ ∈ PROPN ejpam-5960 38 4 𝐿 𝐿 PROPN ejpam-5960 38 5 , , PUNCT ejpam-5960 38 6 then then ADV ejpam-5960 38 7 the the DET ejpam-5960 38 8 constant constant ADJ ejpam-5960 38 9 mapping mapping NOUN ejpam-5960 38 10 𝛼𝑋 𝛼𝑋 NOUN ejpam-5960 38 11 : : PUNCT ejpam-5960 38 12 𝑋 𝑋 PROPN ejpam-5960 38 13 β†’ β†’ SYM ejpam-5960 38 14 { { PUNCT ejpam-5960 38 15 𝛼 𝛼 X ejpam-5960 38 16 } } PUNCT ejpam-5960 38 17 is be AUX ejpam-5960 38 18 𝐿–subset 𝐿–subset ADJ ejpam-5960 38 19 [ [ PUNCT ejpam-5960 38 20 16 16 NUM ejpam-5960 38 21 ] ] PUNCT ejpam-5960 38 22 . . PUNCT ejpam-5960 39 1 an an DET ejpam-5960 39 2 𝐿–point 𝐿–point NOUN ejpam-5960 39 3 ( ( PUNCT ejpam-5960 39 4 or or CCONJ ejpam-5960 39 5 molecule molecule NOUN ejpam-5960 39 6 on on ADP ejpam-5960 39 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 39 8 ) ) PUNCT ejpam-5960 39 9 , , PUNCT ejpam-5960 39 10 denoted denote VERB ejpam-5960 39 11 by by ADP ejpam-5960 39 12 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 39 13 , , PUNCT ejpam-5960 39 14 𝛼 𝛼 PROPN ejpam-5960 39 15 ∈ ∈ PROPN ejpam-5960 39 16 𝑀 𝑀 PROPN ejpam-5960 39 17 ( ( PUNCT ejpam-5960 39 18 𝐿 𝐿 PROPN ejpam-5960 39 19 ) ) PUNCT ejpam-5960 39 20 is be AUX ejpam-5960 39 21 a a DET ejpam-5960 39 22 𝐿–subset 𝐿–subset NOUN ejpam-5960 39 23 which which PRON ejpam-5960 39 24 is be AUX ejpam-5960 39 25 defined define VERB ejpam-5960 39 26 by by ADP ejpam-5960 39 27 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 39 28 ( ( PUNCT ejpam-5960 39 29 𝑦 𝑦 NOUN ejpam-5960 39 30 ) ) PUNCT ejpam-5960 39 31 = = PRON ejpam-5960 39 32 { { PUNCT ejpam-5960 39 33 𝛼 𝛼 NOUN ejpam-5960 39 34 : : PUNCT ejpam-5960 39 35 π‘₯ π‘₯ X ejpam-5960 39 36 = = SYM ejpam-5960 39 37 𝑦 𝑦 NOUN ejpam-5960 39 38 0 0 NUM ejpam-5960 39 39 : : PUNCT ejpam-5960 39 40 π‘₯ π‘₯ PROPN ejpam-5960 39 41 β‰  β‰  PROPN ejpam-5960 39 42 𝑦 𝑦 NUM ejpam-5960 39 43 the the DET ejpam-5960 39 44 family family NOUN ejpam-5960 39 45 of of ADP ejpam-5960 39 46 all all DET ejpam-5960 39 47 molecules molecule NOUN ejpam-5960 39 48 of of ADP ejpam-5960 39 49 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 39 50 is be AUX ejpam-5960 39 51 denoted denote VERB ejpam-5960 39 52 by by ADP ejpam-5960 39 53 𝑀 𝑀 PROPN ejpam-5960 39 54 ( ( PUNCT ejpam-5960 39 55 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 39 56 ) ) PUNCT ejpam-5960 40 1 [ [ X ejpam-5960 40 2 17 17 NUM ejpam-5960 40 3 ] ] PUNCT ejpam-5960 40 4 . . PUNCT ejpam-5960 41 1 for for ADP ejpam-5960 41 2 πœ‡ πœ‡ ADP ejpam-5960 41 3 ∈ ∈ PROPN ejpam-5960 41 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 41 5 and and CCONJ ejpam-5960 41 6 𝛼 𝛼 ADP ejpam-5960 41 7 ∈ ∈ PROPN ejpam-5960 41 8 𝐿 𝐿 NOUN ejpam-5960 41 9 we we PRON ejpam-5960 41 10 defined define VERB ejpam-5960 41 11 the the DET ejpam-5960 41 12 set set ADJ ejpam-5960 41 13 πœ‡π‘€π›Ό πœ‡π‘€π›Ό NOUN ejpam-5960 41 14 = = SYM ejpam-5960 41 15 { { PUNCT ejpam-5960 41 16 π‘₯ π‘₯ PUNCT ejpam-5960 41 17 ∈ ∈ PROPN ejpam-5960 41 18 𝑋 𝑋 NOUN ejpam-5960 41 19 : : PUNCT ejpam-5960 41 20 πœ‡(π‘₯ πœ‡(π‘₯ PROPN ejpam-5960 41 21 ) ) PUNCT ejpam-5960 41 22 β‰₯ β‰₯ PRON ejpam-5960 41 23 𝛼 𝛼 NOUN ejpam-5960 41 24 } } PUNCT ejpam-5960 41 25 , , PUNCT ejpam-5960 41 26 which which PRON ejpam-5960 41 27 it it PRON ejpam-5960 41 28 is be AUX ejpam-5960 41 29 called call VERB ejpam-5960 41 30 weak weak ADJ ejpam-5960 41 31 𝛼 𝛼 NOUN ejpam-5960 41 32 – – PUNCT ejpam-5960 41 33 cut cut NOUN ejpam-5960 41 34 of of ADP ejpam-5960 41 35 πœ‡. πœ‡. NOUN ejpam-5960 41 36 the the DET ejpam-5960 41 37 set set ADJ ejpam-5960 41 38 πœ‡π‘ π›Ό πœ‡π‘ π›Ό NOUN ejpam-5960 41 39 = = SYM ejpam-5960 41 40 { { PUNCT ejpam-5960 41 41 π‘₯ π‘₯ NOUN ejpam-5960 41 42 ∈ ∈ PROPN ejpam-5960 41 43 𝑋 𝑋 NOUN ejpam-5960 41 44 : : PUNCT ejpam-5960 41 45 πœ‡(π‘₯ πœ‡(π‘₯ PROPN ejpam-5960 41 46 ) ) PUNCT ejpam-5960 41 47 β‰° β‰° X ejpam-5960 41 48 𝛼 𝛼 PART ejpam-5960 41 49 } } PUNCT ejpam-5960 41 50 , , PUNCT ejpam-5960 41 51 it it PRON ejpam-5960 41 52 is be AUX ejpam-5960 41 53 called call VERB ejpam-5960 41 54 strong strong ADJ ejpam-5960 41 55 𝛼–cut 𝛼–cut NUM ejpam-5960 41 56 of of ADP ejpam-5960 41 57 πœ‡ πœ‡ ADP ejpam-5960 41 58 and and CCONJ ejpam-5960 41 59 supp(πœ‡ supp(πœ‡ NOUN ejpam-5960 41 60 ) ) PUNCT ejpam-5960 41 61 = = PRON ejpam-5960 41 62 { { PUNCT ejpam-5960 41 63 π‘₯ π‘₯ PUNCT ejpam-5960 41 64 ∈ ∈ PROPN ejpam-5960 41 65 𝑋 𝑋 NOUN ejpam-5960 41 66 : : PUNCT ejpam-5960 41 67 πœ‡(π‘₯ πœ‡(π‘₯ PROPN ejpam-5960 41 68 ) ) PUNCT ejpam-5960 41 69 > > X ejpam-5960 41 70 0 0 X ejpam-5960 41 71 } } PUNCT ejpam-5960 41 72 is be AUX ejpam-5960 41 73 called call VERB ejpam-5960 41 74 support support NOUN ejpam-5960 41 75 of of ADP ejpam-5960 41 76 πœ‡ πœ‡ X ejpam-5960 41 77 [ [ X ejpam-5960 41 78 18 18 NUM ejpam-5960 41 79 ] ] PUNCT ejpam-5960 41 80 . . PUNCT ejpam-5960 42 1 for for ADP ejpam-5960 42 2 any any DET ejpam-5960 42 3 πœ† πœ† PROPN ejpam-5960 42 4 ∈ ∈ PROPN ejpam-5960 42 5 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 42 6 and and CCONJ ejpam-5960 42 7 𝛼 𝛼 NOUN ejpam-5960 42 8 ∈ ∈ PROPN ejpam-5960 42 9 𝑀 𝑀 PROPN ejpam-5960 42 10 ( ( PUNCT ejpam-5960 42 11 𝐿 𝐿 PROPN ejpam-5960 42 12 ) ) PUNCT ejpam-5960 42 13 with with ADP ejpam-5960 42 14 𝛼′ 𝛼′ PROPN ejpam-5960 42 15 β‰₯ β‰₯ NUM ejpam-5960 42 16 𝛼 𝛼 NOUN ejpam-5960 42 17 , , PUNCT ejpam-5960 42 18 we we PRON ejpam-5960 42 19 have have VERB ejpam-5960 42 20 ( ( PUNCT ejpam-5960 42 21 πœ†π‘€π›Ό)β€² πœ†π‘€π›Ό)β€² NUM ejpam-5960 42 22 βŠ† βŠ† NUM ejpam-5960 42 23 ( ( PUNCT ejpam-5960 42 24 πœ†β€²)𝑀𝛼. πœ†β€²)𝑀𝛼. X ejpam-5960 42 25 for for ADP ejpam-5960 42 26 ψ ψ PRON ejpam-5960 42 27 βŠ† βŠ† NUM ejpam-5960 42 28 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 42 29 , , PUNCT ejpam-5960 42 30 we we PRON ejpam-5960 42 31 define define VERB ejpam-5960 42 32 2(ψ 2(ψ NUM ejpam-5960 42 33 ) ) PUNCT ejpam-5960 42 34 by by ADP ejpam-5960 42 35 the the DET ejpam-5960 42 36 set set NOUN ejpam-5960 42 37 { { PUNCT ejpam-5960 42 38 πœ‘ πœ‘ NOUN ejpam-5960 42 39 βŠ† βŠ† NUM ejpam-5960 42 40 ψ ψ NOUN ejpam-5960 42 41 : : PUNCT ejpam-5960 42 42 πœ‘ πœ‘ PROPN ejpam-5960 42 43 is be AUX ejpam-5960 42 44 finite finite ADJ ejpam-5960 42 45 subfamily subfamily ADV ejpam-5960 42 46 of of ADP ejpam-5960 42 47 ψ ψ NOUN ejpam-5960 42 48 } } PUNCT ejpam-5960 42 49 . . PUNCT ejpam-5960 43 1 an an DET ejpam-5960 43 2 𝐿–topology 𝐿–topology NOUN ejpam-5960 43 3 on on ADP ejpam-5960 43 4 𝑋 𝑋 PROPN ejpam-5960 43 5 is be AUX ejpam-5960 43 6 a a DET ejpam-5960 43 7 subfamily subfamily ADV ejpam-5960 43 8 𝜏 𝜏 NOUN ejpam-5960 43 9 of of ADP ejpam-5960 43 10 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 43 11 closed close VERB ejpam-5960 43 12 under under ADP ejpam-5960 43 13 arbitrary arbitrary ADJ ejpam-5960 43 14 unions union NOUN ejpam-5960 43 15 and and CCONJ ejpam-5960 43 16 finite finite ADJ ejpam-5960 43 17 intersections intersection NOUN ejpam-5960 43 18 . . PUNCT ejpam-5960 44 1 the the DET ejpam-5960 44 2 pair pair NOUN ejpam-5960 44 3 ( ( PUNCT ejpam-5960 44 4 𝑋 𝑋 PROPN ejpam-5960 44 5 , , PUNCT ejpam-5960 44 6 𝜏 𝜏 NOUN ejpam-5960 44 7 ) ) PUNCT ejpam-5960 44 8 is be AUX ejpam-5960 44 9 called call VERB ejpam-5960 44 10 an an DET ejpam-5960 44 11 𝐿–topological 𝐿–topological ADJ ejpam-5960 44 12 space space NOUN ejpam-5960 44 13 ( ( PUNCT ejpam-5960 44 14 or or CCONJ ejpam-5960 44 15 𝐿–ts 𝐿–ts NOUN ejpam-5960 44 16 , , PUNCT ejpam-5960 44 17 for for ADP ejpam-5960 44 18 short short ADJ ejpam-5960 44 19 ) ) PUNCT ejpam-5960 45 1 [ [ X ejpam-5960 45 2 19 19 NUM ejpam-5960 45 3 ] ] PUNCT ejpam-5960 45 4 . . PUNCT ejpam-5960 46 1 if if SCONJ ejpam-5960 46 2 ( ( PUNCT ejpam-5960 46 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 46 4 , , PUNCT ejpam-5960 46 5 𝜏 𝜏 NOUN ejpam-5960 46 6 ) ) PUNCT ejpam-5960 46 7 is be AUX ejpam-5960 46 8 an an DET ejpam-5960 46 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 46 10 , , PUNCT ejpam-5960 46 11 then then ADV ejpam-5960 46 12 for for ADP ejpam-5960 46 13 πœ‚ πœ‚ PROPN ejpam-5960 46 14 ∈ ∈ PROPN ejpam-5960 46 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 46 16 , , PUNCT ejpam-5960 46 17 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 46 18 ( ( PUNCT ejpam-5960 46 19 πœ‚ πœ‚ NOUN ejpam-5960 46 20 ) ) PUNCT ejpam-5960 46 21 , , PUNCT ejpam-5960 46 22 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 46 23 ( ( PUNCT ejpam-5960 46 24 πœ‚ πœ‚ NOUN ejpam-5960 46 25 ) ) PUNCT ejpam-5960 46 26 and and CCONJ ejpam-5960 46 27 πœ‚β€² πœ‚β€² NOUN ejpam-5960 46 28 will will AUX ejpam-5960 46 29 denote denote VERB ejpam-5960 46 30 the the DET ejpam-5960 46 31 closure closure NOUN ejpam-5960 46 32 , , PUNCT ejpam-5960 46 33 n. n. NOUN ejpam-5960 46 34 a. a. NOUN ejpam-5960 46 35 alsaedi alsaedi PROPN ejpam-5960 46 36 / / SYM ejpam-5960 46 37 eur eur PROPN ejpam-5960 46 38 . . PUNCT ejpam-5960 47 1 j. j. PROPN ejpam-5960 47 2 pure pure PROPN ejpam-5960 47 3 appl appl PROPN ejpam-5960 47 4 . . PROPN ejpam-5960 47 5 math math PROPN ejpam-5960 47 6 , , PUNCT ejpam-5960 47 7 18 18 NUM ejpam-5960 47 8 ( ( PUNCT ejpam-5960 47 9 4 4 NUM ejpam-5960 47 10 ) ) PUNCT ejpam-5960 47 11 ( ( PUNCT ejpam-5960 47 12 2025 2025 NUM ejpam-5960 47 13 ) ) PUNCT ejpam-5960 47 14 , , PUNCT ejpam-5960 47 15 5960 5960 NUM ejpam-5960 47 16 3 3 NUM ejpam-5960 47 17 of of ADP ejpam-5960 47 18 22 22 NUM ejpam-5960 47 19 interior interior NOUN ejpam-5960 47 20 , , PUNCT ejpam-5960 47 21 and and CCONJ ejpam-5960 47 22 complement complement NOUN ejpam-5960 47 23 of of ADP ejpam-5960 47 24 πœ‚. πœ‚. NOUN ejpam-5960 47 25 a a DET ejpam-5960 47 26 mapping mapping NOUN ejpam-5960 47 27 𝑓 𝑓 X ejpam-5960 47 28 : : PUNCT ejpam-5960 47 29 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 47 30 β†’ β†’ SYM ejpam-5960 47 31 πΏπ‘Œ πΏπ‘Œ PROPN ejpam-5960 47 32 is be AUX ejpam-5960 47 33 an an DET ejpam-5960 47 34 𝐿–valued 𝐿–value VERB ejpam-5960 47 35 zadeh zadeh PROPN ejpam-5960 47 36 function function NOUN ejpam-5960 47 37 induced induce VERB ejpam-5960 47 38 by by ADP ejpam-5960 47 39 a a DET ejpam-5960 47 40 mapping mapping NOUN ejpam-5960 47 41 𝑓 𝑓 X ejpam-5960 47 42 : : PUNCT ejpam-5960 47 43 𝑋 𝑋 PROPN ejpam-5960 47 44 β†’ β†’ SYM ejpam-5960 47 45 π‘Œ π‘Œ PROPN ejpam-5960 47 46 , , PUNCT ejpam-5960 47 47 iff iff PROPN ejpam-5960 47 48 𝑓 𝑓 PROPN ejpam-5960 47 49 ( ( PUNCT ejpam-5960 47 50 πœ‡ πœ‡ NOUN ejpam-5960 47 51 ) ) PUNCT ejpam-5960 47 52 ( ( PUNCT ejpam-5960 47 53 𝑦 𝑦 NOUN ejpam-5960 47 54 ) ) PUNCT ejpam-5960 47 55 = = SYM ejpam-5960 47 56 ∨{πœ‡(π‘₯ ∨{πœ‡(π‘₯ NOUN ejpam-5960 47 57 ) ) PUNCT ejpam-5960 47 58 : : PUNCT ejpam-5960 48 1 𝑓 𝑓 X ejpam-5960 48 2 ( ( PUNCT ejpam-5960 48 3 π‘₯ π‘₯ NOUN ejpam-5960 48 4 ) ) PUNCT ejpam-5960 48 5 = = SYM ejpam-5960 48 6 𝑦 𝑦 X ejpam-5960 48 7 } } PUNCT ejpam-5960 48 8 for for ADP ejpam-5960 48 9 every every DET ejpam-5960 48 10 πœ‡ πœ‡ ADP ejpam-5960 48 11 ∈ ∈ ADJ ejpam-5960 48 12 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 48 13 and and CCONJ ejpam-5960 48 14 every every DET ejpam-5960 48 15 𝑦 𝑦 NOUN ejpam-5960 48 16 ∈ ∈ NOUN ejpam-5960 48 17 π‘Œ π‘Œ PROPN ejpam-5960 49 1 [ [ X ejpam-5960 49 2 17 17 NUM ejpam-5960 49 3 ] ] PUNCT ejpam-5960 49 4 . . PUNCT ejpam-5960 50 1 an an DET ejpam-5960 50 2 𝐿–ts 𝐿–ts NOUN ejpam-5960 50 3 ( ( PUNCT ejpam-5960 50 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 50 5 , , PUNCT ejpam-5960 50 6 𝜏 𝜏 NOUN ejpam-5960 50 7 ) ) PUNCT ejpam-5960 50 8 is be AUX ejpam-5960 50 9 called call VERB ejpam-5960 50 10 fully fully ADV ejpam-5960 50 11 stratified stratified ADJ ejpam-5960 50 12 if if SCONJ ejpam-5960 50 13 for for ADP ejpam-5960 50 14 each each DET ejpam-5960 50 15 𝛼 𝛼 PROPN ejpam-5960 50 16 ∈ ∈ PROPN ejpam-5960 50 17 𝐿 𝐿 PROPN ejpam-5960 50 18 , , PUNCT ejpam-5960 50 19 𝛼 𝛼 PRON ejpam-5960 50 20 ∈𝜏[18 ∈𝜏[18 NOUN ejpam-5960 50 21 ] ] PUNCT ejpam-5960 50 22 . . PUNCT ejpam-5960 51 1 if if SCONJ ejpam-5960 51 2 ( ( PUNCT ejpam-5960 51 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 51 4 , , PUNCT ejpam-5960 51 5 𝜏 𝜏 NOUN ejpam-5960 51 6 ) ) PUNCT ejpam-5960 51 7 is be AUX ejpam-5960 51 8 an an DET ejpam-5960 51 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 51 10 , , PUNCT ejpam-5960 51 11 then then ADV ejpam-5960 51 12 the the DET ejpam-5960 51 13 family family NOUN ejpam-5960 51 14 of of ADP ejpam-5960 51 15 all all DET ejpam-5960 51 16 crisp crisp ADJ ejpam-5960 51 17 open open ADJ ejpam-5960 51 18 sets set NOUN ejpam-5960 51 19 in in ADP ejpam-5960 51 20 𝜏 𝜏 NOUN ejpam-5960 51 21 is be AUX ejpam-5960 51 22 denoted denote VERB ejpam-5960 51 23 by by ADP ejpam-5960 51 24 [ [ X ejpam-5960 51 25 𝜏 𝜏 X ejpam-5960 51 26 ] ] X ejpam-5960 51 27 i.e. i.e. X ejpam-5960 51 28 , , PUNCT ejpam-5960 51 29 ( ( PUNCT ejpam-5960 51 30 𝑋 𝑋 NOUN ejpam-5960 51 31 , , PUNCT ejpam-5960 51 32 [ [ X ejpam-5960 51 33 𝜏 𝜏 X ejpam-5960 51 34 ] ] PUNCT ejpam-5960 51 35 ) ) PUNCT ejpam-5960 51 36 is be AUX ejpam-5960 51 37 a a DET ejpam-5960 51 38 crisp crisp ADJ ejpam-5960 51 39 topological topological ADJ ejpam-5960 51 40 space space NOUN ejpam-5960 52 1 [ [ X ejpam-5960 52 2 20 20 NUM ejpam-5960 52 3 ] ] PUNCT ejpam-5960 52 4 . . PUNCT ejpam-5960 53 1 definition definition NOUN ejpam-5960 53 2 2.1 2.1 NUM ejpam-5960 54 1 [ [ X ejpam-5960 54 2 21 21 NUM ejpam-5960 54 3 ] ] PUNCT ejpam-5960 54 4 . . PUNCT ejpam-5960 55 1 if if SCONJ ejpam-5960 55 2 ( ( PUNCT ejpam-5960 55 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 55 4 , , PUNCT ejpam-5960 55 5 𝜏 𝜏 NOUN ejpam-5960 55 6 ) ) PUNCT ejpam-5960 55 7 is be AUX ejpam-5960 55 8 𝐿–ts 𝐿–ts NOUN ejpam-5960 55 9 , , PUNCT ejpam-5960 55 10 then then ADV ejpam-5960 55 11 πœ‡ πœ‡ SCONJ ejpam-5960 55 12 ∈ ∈ PROPN ejpam-5960 55 13 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 55 14 is be AUX ejpam-5960 55 15 called call VERB ejpam-5960 55 16 a a DET ejpam-5960 55 17 regular regular ADJ ejpam-5960 55 18 open open ADJ ejpam-5960 55 19 set set NOUN ejpam-5960 55 20 iff iff PROPN ejpam-5960 55 21 πœ‡ πœ‡ ADP ejpam-5960 55 22 = = PROPN ejpam-5960 55 23 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 55 24 ( ( PUNCT ejpam-5960 55 25 𝑐𝑙 𝑐𝑙 X ejpam-5960 55 26 ( ( PUNCT ejpam-5960 55 27 πœ‡ πœ‡ NOUN ejpam-5960 55 28 ) ) PUNCT ejpam-5960 55 29 ) ) PUNCT ejpam-5960 55 30 . . PUNCT ejpam-5960 56 1 the the DET ejpam-5960 56 2 family family NOUN ejpam-5960 56 3 of of ADP ejpam-5960 56 4 all all DET ejpam-5960 56 5 regular regular ADJ ejpam-5960 56 6 open open ADJ ejpam-5960 56 7 sets set NOUN ejpam-5960 56 8 is be AUX ejpam-5960 56 9 denoted denote VERB ejpam-5960 56 10 by by ADP ejpam-5960 56 11 𝑅𝑂 𝑅𝑂 PROPN ejpam-5960 56 12 ( ( PUNCT ejpam-5960 56 13 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 56 14 , , PUNCT ejpam-5960 56 15 𝜏 𝜏 NOUN ejpam-5960 56 16 ) ) PUNCT ejpam-5960 56 17 . . PUNCT ejpam-5960 57 1 the the DET ejpam-5960 57 2 complement complement NOUN ejpam-5960 57 3 of of ADP ejpam-5960 57 4 a a DET ejpam-5960 57 5 regular regular ADJ ejpam-5960 57 6 open open ADJ ejpam-5960 57 7 set set NOUN ejpam-5960 57 8 is be AUX ejpam-5960 57 9 called call VERB ejpam-5960 57 10 a a DET ejpam-5960 57 11 regular regular ADJ ejpam-5960 57 12 closed close VERB ejpam-5960 57 13 set set NOUN ejpam-5960 57 14 and and CCONJ ejpam-5960 57 15 satisfies satisfie NOUN ejpam-5960 57 16 πœ‡π‘ πœ‡π‘ NOUN ejpam-5960 57 17 = = SYM ejpam-5960 57 18 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 57 19 ( ( PUNCT ejpam-5960 57 20 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 57 21 ( ( PUNCT ejpam-5960 57 22 πœ‡ πœ‡ NOUN ejpam-5960 57 23 ) ) PUNCT ejpam-5960 57 24 ) ) PUNCT ejpam-5960 57 25 . . PUNCT ejpam-5960 58 1 the the DET ejpam-5960 58 2 family family NOUN ejpam-5960 58 3 of of ADP ejpam-5960 58 4 all all DET ejpam-5960 58 5 regular regular ADJ ejpam-5960 58 6 closed closed ADJ ejpam-5960 58 7 sets set NOUN ejpam-5960 58 8 is be AUX ejpam-5960 58 9 denoted denote VERB ejpam-5960 58 10 by by ADP ejpam-5960 58 11 𝑅𝐢(l𝑋 𝑅𝐢(l𝑋 PROPN ejpam-5960 58 12 , , PUNCT ejpam-5960 58 13 𝜏 𝜏 NOUN ejpam-5960 58 14 ) ) PUNCT ejpam-5960 58 15 . . PUNCT ejpam-5960 59 1 definition definition NOUN ejpam-5960 59 2 2.2 2.2 NUM ejpam-5960 60 1 [ [ X ejpam-5960 60 2 21 21 NUM ejpam-5960 60 3 ] ] PUNCT ejpam-5960 60 4 . . PUNCT ejpam-5960 61 1 the the DET ejpam-5960 61 2 𝐿–valued 𝐿–valued PROPN ejpam-5960 61 3 zadeh zadeh PROPN ejpam-5960 61 4 mapping mapping NOUN ejpam-5960 61 5 𝑓𝐿 𝑓𝐿 NOUN ejpam-5960 61 6 : : PUNCT ejpam-5960 61 7 ( ( PUNCT ejpam-5960 61 8 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 61 9 , , PUNCT ejpam-5960 61 10 𝜏 𝜏 NOUN ejpam-5960 61 11 ) ) PUNCT ejpam-5960 61 12 β†’ β†’ SYM ejpam-5960 61 13 ( ( PUNCT ejpam-5960 61 14 πΏπ‘Œ πΏπ‘Œ PROPN ejpam-5960 61 15 , , PUNCT ejpam-5960 61 16 Ξ΄ Ξ΄ PROPN ejpam-5960 61 17 ) ) PUNCT ejpam-5960 61 18 is be AUX ejpam-5960 61 19 called call VERB ejpam-5960 61 20 almost almost ADV ejpam-5960 61 21 𝐿–continuous 𝐿–continuous ADJ ejpam-5960 61 22 iff iff PROPN ejpam-5960 61 23 𝑓 𝑓 DET ejpam-5960 61 24 βˆ’1 βˆ’1 NOUN ejpam-5960 61 25 𝐿 𝐿 PROPN ejpam-5960 61 26 ( ( PUNCT ejpam-5960 61 27 πœ‚ πœ‚ NOUN ejpam-5960 61 28 ) ) PUNCT ejpam-5960 61 29 ∈ ∈ NOUN ejpam-5960 61 30 πœβ€² πœβ€² NOUN ejpam-5960 61 31 for for ADP ejpam-5960 61 32 each each DET ejpam-5960 61 33 πœ‚ πœ‚ PROPN ejpam-5960 61 34 ∈ ∈ PROPN ejpam-5960 61 35 𝑅𝐢 𝑅𝐢 PROPN ejpam-5960 61 36 ( ( PUNCT ejpam-5960 61 37 πΏπ‘Œ πΏπ‘Œ PROPN ejpam-5960 61 38 , , PUNCT ejpam-5960 61 39 Ξ΄ Ξ΄ PROPN ejpam-5960 61 40 ) ) PUNCT ejpam-5960 61 41 . . PUNCT ejpam-5960 62 1 definition definition NOUN ejpam-5960 62 2 2.3 2.3 NUM ejpam-5960 63 1 [ [ X ejpam-5960 63 2 9 9 NUM ejpam-5960 63 3 ] ] PUNCT ejpam-5960 63 4 . . PUNCT ejpam-5960 64 1 let let VERB ejpam-5960 64 2 ( ( PUNCT ejpam-5960 64 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 64 4 , , PUNCT ejpam-5960 64 5 𝜏 𝜏 NOUN ejpam-5960 64 6 ) ) PUNCT ejpam-5960 64 7 be be VERB ejpam-5960 64 8 an an DET ejpam-5960 64 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 64 10 and and CCONJ ejpam-5960 64 11 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 64 12 ∈ ∈ PROPN ejpam-5960 64 13 𝑀 𝑀 PROPN ejpam-5960 64 14 ( ( PUNCT ejpam-5960 64 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 64 16 ) ) PUNCT ejpam-5960 64 17 . . PUNCT ejpam-5960 65 1 then then ADV ejpam-5960 65 2 πœ† πœ† PROPN ejpam-5960 65 3 ∈ ∈ PROPN ejpam-5960 65 4 πœβ€² πœβ€² NOUN ejpam-5960 65 5 is be AUX ejpam-5960 65 6 called call VERB ejpam-5960 65 7 an an DET ejpam-5960 65 8 remoted remoted ADJ ejpam-5960 65 9 neighborhood neighborhood NOUN ejpam-5960 65 10 ( ( PUNCT ejpam-5960 65 11 r r NOUN ejpam-5960 65 12 – – PUNCT ejpam-5960 65 13 nbd nbd PROPN ejpam-5960 65 14 , , PUNCT ejpam-5960 65 15 for for ADP ejpam-5960 65 16 short short ADJ ejpam-5960 65 17 ) ) PUNCT ejpam-5960 65 18 of of ADP ejpam-5960 65 19 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 65 20 if if SCONJ ejpam-5960 65 21 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 65 22 βˆ‰ βˆ‰ PROPN ejpam-5960 65 23 πœ†. πœ†. VERB ejpam-5960 65 24 the the DET ejpam-5960 65 25 set set NOUN ejpam-5960 65 26 of of ADP ejpam-5960 65 27 all all DET ejpam-5960 65 28 r r NOUN ejpam-5960 65 29 – – PUNCT ejpam-5960 65 30 nbds nbds NOUN ejpam-5960 65 31 of of ADP ejpam-5960 65 32 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 65 33 is be AUX ejpam-5960 65 34 called call VERB ejpam-5960 65 35 remoted remoted ADJ ejpam-5960 65 36 neighborhood neighborhood NOUN ejpam-5960 65 37 system system NOUN ejpam-5960 65 38 and and CCONJ ejpam-5960 65 39 is be AUX ejpam-5960 65 40 denoted denote VERB ejpam-5960 65 41 by by ADP ejpam-5960 65 42 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 65 43 . . PUNCT ejpam-5960 66 1 definition definition NOUN ejpam-5960 66 2 2.4 2.4 NUM ejpam-5960 66 3 [ [ X ejpam-5960 66 4 15 15 NUM ejpam-5960 66 5 ] ] PUNCT ejpam-5960 66 6 : : PUNCT ejpam-5960 66 7 let let VERB ejpam-5960 66 8 ( ( PUNCT ejpam-5960 66 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 66 10 , , PUNCT ejpam-5960 66 11 𝜏 𝜏 NOUN ejpam-5960 66 12 ) ) PUNCT ejpam-5960 66 13 be be VERB ejpam-5960 66 14 an an DET ejpam-5960 66 15 𝐿-ts 𝐿-ts NOUN ejpam-5960 66 16 , , PUNCT ejpam-5960 66 17 πœ‡ πœ‡ ADP ejpam-5960 66 18 ∈ ∈ ADP ejpam-5960 66 19 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 66 20 and and CCONJ ejpam-5960 66 21 𝛼 𝛼 NOUN ejpam-5960 66 22 ∈ ∈ PROPN ejpam-5960 66 23 𝑀 𝑀 PROPN ejpam-5960 66 24 ( ( PUNCT ejpam-5960 66 25 𝐿 𝐿 PROPN ejpam-5960 66 26 ) ) PUNCT ejpam-5960 66 27 . . PUNCT ejpam-5960 67 1 ψ ψ X ejpam-5960 67 2 βŠ‚ βŠ‚ X ejpam-5960 67 3 πœβ€² πœβ€² PROPN ejpam-5960 67 4 is be AUX ejpam-5960 67 5 called call VERB ejpam-5960 67 6 an an DET ejpam-5960 67 7 : : PUNCT ejpam-5960 67 8 ( ( PUNCT ejpam-5960 67 9 i i NOUN ejpam-5960 67 10 ) ) PUNCT ejpam-5960 67 11 𝛼-remoted 𝛼-remote VERB ejpam-5960 67 12 neighborhood neighborhood NOUN ejpam-5960 67 13 family family NOUN ejpam-5960 67 14 of of ADP ejpam-5960 67 15 πœ‡ πœ‡ ADP ejpam-5960 67 16 , , PUNCT ejpam-5960 67 17 briefly briefly NOUN ejpam-5960 67 18 𝛼-rf 𝛼-rf PROPN ejpam-5960 67 19 of of ADP ejpam-5960 67 20 πœ‡ πœ‡ ADP ejpam-5960 67 21 , , PUNCT ejpam-5960 67 22 if if SCONJ ejpam-5960 67 23 for for ADP ejpam-5960 67 24 each each DET ejpam-5960 67 25 𝐿–point 𝐿–point NOUN ejpam-5960 67 26 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 67 27 ∈ ∈ PROPN ejpam-5960 67 28 πœ‡ πœ‡ SCONJ ejpam-5960 67 29 there there PRON ejpam-5960 67 30 is be VERB ejpam-5960 67 31 πœ† πœ† DET ejpam-5960 67 32 ∈ ∈ PROPN ejpam-5960 67 33 ψ ψ NOUN ejpam-5960 67 34 such such ADJ ejpam-5960 67 35 that that SCONJ ejpam-5960 67 36 πœ† πœ† DET ejpam-5960 67 37 ∈ ∈ PROPN ejpam-5960 67 38 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 67 39 . . PUNCT ejpam-5960 68 1 ( ( PUNCT ejpam-5960 68 2 ii ii NOUN ejpam-5960 68 3 ) ) PUNCT ejpam-5960 68 4 οΏ½ οΏ½ PROPN ejpam-5960 68 5 Μ„ Μ„ NOUN ejpam-5960 68 6 οΏ½ οΏ½ PROPN ejpam-5960 68 7 -remoted -remote VERB ejpam-5960 68 8 neighborhood neighborhood NOUN ejpam-5960 68 9 family family NOUN ejpam-5960 68 10 of of ADP ejpam-5960 68 11 πœ‡ πœ‡ ADP ejpam-5960 68 12 , , PUNCT ejpam-5960 68 13 briefly briefly NOUN ejpam-5960 68 14 οΏ½ οΏ½ PROPN ejpam-5960 68 15 Μ„ Μ„ PROPN ejpam-5960 68 16 οΏ½ οΏ½ PROPN ejpam-5960 68 17 -rf -rf NOUN ejpam-5960 68 18 of of ADP ejpam-5960 68 19 πœ‡ πœ‡ ADP ejpam-5960 68 20 , , PUNCT ejpam-5960 68 21 if if SCONJ ejpam-5960 68 22 there there PRON ejpam-5960 68 23 exists exist VERB ejpam-5960 68 24 𝛾 𝛾 PROPN ejpam-5960 68 25 ∈ ∈ PROPN ejpam-5960 68 26 π›½βˆ—(𝛼 π›½βˆ—(𝛼 PROPN ejpam-5960 68 27 ) ) PUNCT ejpam-5960 68 28 such such ADJ ejpam-5960 68 29 that that SCONJ ejpam-5960 68 30 ψ ψ NOUN ejpam-5960 68 31 is be AUX ejpam-5960 68 32 a a DET ejpam-5960 68 33 𝛾-rf 𝛾-rf PROPN ejpam-5960 68 34 of of ADP ejpam-5960 68 35 πœ‡ πœ‡ ADP ejpam-5960 68 36 , , PUNCT ejpam-5960 68 37 where where SCONJ ejpam-5960 68 38 π›½βˆ—(𝛼 π›½βˆ—(𝛼 NOUN ejpam-5960 68 39 ) ) PUNCT ejpam-5960 68 40 = = SYM ejpam-5960 68 41 𝛽(𝛼 𝛽(𝛼 PROPN ejpam-5960 68 42 ) ) PUNCT ejpam-5960 68 43 ∩ ∩ PROPN ejpam-5960 68 44 𝑀 𝑀 PROPN ejpam-5960 68 45 ( ( PUNCT ejpam-5960 68 46 𝐿 𝐿 PROPN ejpam-5960 68 47 ) ) PUNCT ejpam-5960 68 48 , , PUNCT ejpam-5960 68 49 and and CCONJ ejpam-5960 68 50 𝛽(𝛼 𝛽(𝛼 PROPN ejpam-5960 68 51 ) ) PUNCT ejpam-5960 68 52 denotes denote VERB ejpam-5960 68 53 the the DET ejpam-5960 68 54 union union NOUN ejpam-5960 68 55 of of ADP ejpam-5960 68 56 all all DET ejpam-5960 68 57 the the DET ejpam-5960 68 58 minimal minimal ADJ ejpam-5960 68 59 sets set NOUN ejpam-5960 68 60 relative relative ADJ ejpam-5960 68 61 to to ADP ejpam-5960 68 62 𝛼. 𝛼. NOUN ejpam-5960 68 63 definition definition NOUN ejpam-5960 68 64 2.5 2.5 NUM ejpam-5960 69 1 [ [ X ejpam-5960 69 2 22 22 NUM ejpam-5960 69 3 ] ] PUNCT ejpam-5960 69 4 : : PUNCT ejpam-5960 69 5 let let VERB ejpam-5960 69 6 ( ( PUNCT ejpam-5960 69 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 69 8 , , PUNCT ejpam-5960 69 9 𝜏 𝜏 NOUN ejpam-5960 69 10 ) ) PUNCT ejpam-5960 69 11 be be VERB ejpam-5960 69 12 an an DET ejpam-5960 69 13 𝐿-ts 𝐿-ts NOUN ejpam-5960 69 14 , , PUNCT ejpam-5960 69 15 πœ‡ πœ‡ ADP ejpam-5960 69 16 ∈ ∈ ADP ejpam-5960 69 17 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 69 18 and and CCONJ ejpam-5960 69 19 𝛼 𝛼 NOUN ejpam-5960 69 20 ∈ ∈ PROPN ejpam-5960 69 21 𝑀 𝑀 PROPN ejpam-5960 69 22 ( ( PUNCT ejpam-5960 69 23 𝐿 𝐿 PROPN ejpam-5960 69 24 ) ) PUNCT ejpam-5960 69 25 . . PUNCT ejpam-5960 70 1 then then ADV ejpam-5960 70 2 ψ ψ X ejpam-5960 70 3 βŠ‚ βŠ‚ PROPN ejpam-5960 70 4 𝑅𝐢 𝑅𝐢 PROPN ejpam-5960 70 5 ( ( PUNCT ejpam-5960 70 6 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 70 7 , , PUNCT ejpam-5960 70 8 𝜏 𝜏 NOUN ejpam-5960 70 9 ) ) PUNCT ejpam-5960 70 10 is be AUX ejpam-5960 70 11 called call VERB ejpam-5960 70 12 an an DET ejpam-5960 70 13 𝛼-regular 𝛼-regular PROPN ejpam-5960 70 14 closed close VERB ejpam-5960 70 15 remoted remote VERB ejpam-5960 70 16 neighborhood neighborhood NOUN ejpam-5960 70 17 family family NOUN ejpam-5960 70 18 of of ADP ejpam-5960 70 19 πœ‡ πœ‡ ADP ejpam-5960 70 20 , , PUNCT ejpam-5960 70 21 briefly briefly NOUN ejpam-5960 70 22 𝛼–rcrf 𝛼–rcrf PROPN ejpam-5960 70 23 of of ADP ejpam-5960 70 24 πœ‡ πœ‡ ADP ejpam-5960 70 25 , , PUNCT ejpam-5960 70 26 if if SCONJ ejpam-5960 70 27 for for ADP ejpam-5960 70 28 each each DET ejpam-5960 70 29 𝐿-point 𝐿-point PROPN ejpam-5960 70 30 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 70 31 ∈ ∈ PROPN ejpam-5960 70 32 πœ‡ πœ‡ SCONJ ejpam-5960 70 33 there there PRON ejpam-5960 70 34 is be VERB ejpam-5960 70 35 πœ† πœ† DET ejpam-5960 70 36 ∈ ∈ PROPN ejpam-5960 70 37 ψ ψ NOUN ejpam-5960 70 38 such such ADJ ejpam-5960 70 39 that that SCONJ ejpam-5960 70 40 πœ† πœ† DET ejpam-5960 70 41 ∈ ∈ PROPN ejpam-5960 70 42 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 70 43 . . PUNCT ejpam-5960 71 1 definition definition NOUN ejpam-5960 71 2 2.6 2.6 NUM ejpam-5960 72 1 [ [ X ejpam-5960 72 2 21 21 NUM ejpam-5960 72 3 ] ] PUNCT ejpam-5960 72 4 . . PUNCT ejpam-5960 73 1 let let VERB ejpam-5960 73 2 ( ( PUNCT ejpam-5960 73 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 73 4 , , PUNCT ejpam-5960 73 5 𝜏 𝜏 NOUN ejpam-5960 73 6 ) ) PUNCT ejpam-5960 73 7 be be VERB ejpam-5960 73 8 an an DET ejpam-5960 73 9 𝐿-ts 𝐿-ts NOUN ejpam-5960 73 10 , , PUNCT ejpam-5960 73 11 πœ‡ πœ‡ ADP ejpam-5960 73 12 ∈ ∈ PROPN ejpam-5960 73 13 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 73 14 and and CCONJ ejpam-5960 73 15 𝛼′ 𝛼′ NUM ejpam-5960 73 16 ∈ ∈ PROPN ejpam-5960 73 17 𝑀 𝑀 PROPN ejpam-5960 73 18 ( ( PUNCT ejpam-5960 73 19 𝐿 𝐿 PROPN ejpam-5960 73 20 ) ) PUNCT ejpam-5960 73 21 . . PUNCT ejpam-5960 74 1 then then ADV ejpam-5960 74 2 the the DET ejpam-5960 74 3 family family NOUN ejpam-5960 74 4 ψ ψ ADP ejpam-5960 74 5 βŠ† βŠ† NUM ejpam-5960 74 6 𝜏 𝜏 NOUN ejpam-5960 74 7 is be AUX ejpam-5960 74 8 called call VERB ejpam-5960 74 9 an an DET ejpam-5960 74 10 : : PUNCT ejpam-5960 74 11 ( ( PUNCT ejpam-5960 74 12 i i NOUN ejpam-5960 74 13 ) ) PUNCT ejpam-5960 74 14 𝛼–cover 𝛼–cover NOUN ejpam-5960 74 15 of of ADP ejpam-5960 74 16 πœ‡ πœ‡ ADP ejpam-5960 74 17 , , PUNCT ejpam-5960 74 18 if if SCONJ ejpam-5960 74 19 for for ADP ejpam-5960 74 20 each each DET ejpam-5960 74 21 π‘₯ π‘₯ PRON ejpam-5960 74 22 ∈ ∈ NOUN ejpam-5960 74 23 πœ‡π‘€π›Όβ€² πœ‡π‘€π›Όβ€² NOUN ejpam-5960 74 24 there there PRON ejpam-5960 74 25 is be VERB ejpam-5960 74 26 πœ† πœ† DET ejpam-5960 74 27 ∈ ∈ PROPN ejpam-5960 74 28 ψ ψ NOUN ejpam-5960 74 29 such such ADJ ejpam-5960 74 30 that that DET ejpam-5960 74 31 πœ†(π‘₯ πœ†(π‘₯ NOUN ejpam-5960 74 32 ) ) PUNCT ejpam-5960 74 33 β‰° β‰° PROPN ejpam-5960 74 34 𝛼. 𝛼. NOUN ejpam-5960 74 35 ( ( PUNCT ejpam-5960 74 36 ii ii NOUN ejpam-5960 74 37 ) ) PUNCT ejpam-5960 74 38 nearly nearly ADV ejpam-5960 74 39 𝛼–cover 𝛼–cover ADV ejpam-5960 74 40 of of ADP ejpam-5960 74 41 πœ‡ πœ‡ ADP ejpam-5960 74 42 , , PUNCT ejpam-5960 74 43 if if SCONJ ejpam-5960 74 44 for for ADP ejpam-5960 74 45 each each DET ejpam-5960 74 46 π‘₯ π‘₯ PRON ejpam-5960 74 47 ∈ ∈ NOUN ejpam-5960 74 48 πœ‡π‘€π›Όβ€² πœ‡π‘€π›Όβ€² NOUN ejpam-5960 74 49 there there PRON ejpam-5960 74 50 is be VERB ejpam-5960 74 51 πœ† πœ† DET ejpam-5960 74 52 ∈ ∈ PROPN ejpam-5960 74 53 ψ ψ NOUN ejpam-5960 74 54 such such ADJ ejpam-5960 74 55 that that DET ejpam-5960 74 56 int(cl(πœ† int(cl(πœ† NOUN ejpam-5960 74 57 ) ) PUNCT ejpam-5960 74 58 ) ) PUNCT ejpam-5960 75 1 ( ( PUNCT ejpam-5960 75 2 π‘₯ π‘₯ X ejpam-5960 75 3 ) ) PUNCT ejpam-5960 75 4 β‰° β‰° PROPN ejpam-5960 75 5 𝛼. 𝛼. NOUN ejpam-5960 75 6 definition definition NOUN ejpam-5960 75 7 2.7 2.7 NUM ejpam-5960 75 8 [ [ X ejpam-5960 75 9 17 17 NUM ejpam-5960 75 10 ] ] PUNCT ejpam-5960 75 11 . . PUNCT ejpam-5960 76 1 let let VERB ejpam-5960 76 2 ( ( PUNCT ejpam-5960 76 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 76 4 , , PUNCT ejpam-5960 76 5 𝜏 𝜏 NOUN ejpam-5960 76 6 ) ) PUNCT ejpam-5960 76 7 be be VERB ejpam-5960 76 8 an an DET ejpam-5960 76 9 𝐿-ts 𝐿-ts NOUN ejpam-5960 76 10 and and CCONJ ejpam-5960 76 11 πœ‡ πœ‡ X ejpam-5960 76 12 ∈ ∈ X ejpam-5960 76 13 𝐿𝑋. 𝐿𝑋. VERB ejpam-5960 76 14 then then ADV ejpam-5960 76 15 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 76 16 ∈ ∈ PROPN ejpam-5960 76 17 𝑀 𝑀 PROPN ejpam-5960 76 18 ( ( PUNCT ejpam-5960 76 19 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 76 20 ) ) PUNCT ejpam-5960 76 21 is be AUX ejpam-5960 76 22 called call VERB ejpam-5960 76 23 the the DET ejpam-5960 76 24 𝛿–adherent 𝛿–adherent NOUN ejpam-5960 76 25 point point NOUN ejpam-5960 76 26 of of ADP ejpam-5960 76 27 πœ‡ πœ‡ ADP ejpam-5960 76 28 and and CCONJ ejpam-5960 76 29 write write VERB ejpam-5960 76 30 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 76 31 ∈ ∈ PROPN ejpam-5960 76 32 𝛿𝑐𝑙 𝛿𝑐𝑙 X ejpam-5960 76 33 ( ( PUNCT ejpam-5960 76 34 πœ‡ πœ‡ NOUN ejpam-5960 76 35 ) ) PUNCT ejpam-5960 76 36 iff iff NOUN ejpam-5960 76 37 πœ‡ πœ‡ ADP ejpam-5960 76 38 β‰° β‰° PROPN ejpam-5960 76 39 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 76 40 ( ( PUNCT ejpam-5960 76 41 int(πœ† int(πœ† PROPN ejpam-5960 76 42 ) ) PUNCT ejpam-5960 76 43 ) ) PUNCT ejpam-5960 76 44 for for ADP ejpam-5960 76 45 each each DET ejpam-5960 76 46 πœ† πœ† DET ejpam-5960 76 47 ∈ ∈ PROPN ejpam-5960 76 48 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 76 49 . . PUNCT ejpam-5960 77 1 if if SCONJ ejpam-5960 77 2 πœ‡ πœ‡ ADP ejpam-5960 77 3 = = X ejpam-5960 77 4 𝛿𝑐𝑙 𝛿𝑐𝑙 X ejpam-5960 77 5 ( ( PUNCT ejpam-5960 77 6 πœ‡ πœ‡ NOUN ejpam-5960 77 7 ) ) PUNCT ejpam-5960 77 8 , , PUNCT ejpam-5960 77 9 then then ADV ejpam-5960 77 10 πœ‡ πœ‡ ADP ejpam-5960 77 11 is be AUX ejpam-5960 77 12 called call VERB ejpam-5960 77 13 a a DET ejpam-5960 77 14 𝛿–closed 𝛿–close VERB ejpam-5960 77 15 𝐿–subset 𝐿–subset PROPN ejpam-5960 77 16 . . PUNCT ejpam-5960 78 1 the the DET ejpam-5960 78 2 family family NOUN ejpam-5960 78 3 of of ADP ejpam-5960 78 4 all all DET ejpam-5960 78 5 𝛿–closed 𝛿–close VERB ejpam-5960 78 6 𝐿–subsets 𝐿–subset NOUN ejpam-5960 78 7 of of ADP ejpam-5960 78 8 𝑋 𝑋 NOUN ejpam-5960 78 9 is be AUX ejpam-5960 78 10 denoted denote VERB ejpam-5960 78 11 by by ADP ejpam-5960 78 12 𝛿𝐢 𝛿𝐢 PROPN ejpam-5960 78 13 ( ( PUNCT ejpam-5960 78 14 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 78 15 , , PUNCT ejpam-5960 78 16 𝜏 𝜏 NOUN ejpam-5960 78 17 ) ) PUNCT ejpam-5960 78 18 and and CCONJ ejpam-5960 78 19 its its PRON ejpam-5960 78 20 complement complement NOUN ejpam-5960 78 21 is be AUX ejpam-5960 78 22 called call VERB ejpam-5960 78 23 the the DET ejpam-5960 78 24 family family NOUN ejpam-5960 78 25 of of ADP ejpam-5960 78 26 all all PRON ejpam-5960 78 27 𝛿–open 𝛿–open VERB ejpam-5960 78 28 𝐿–subsets 𝐿–subset NOUN ejpam-5960 78 29 and and CCONJ ejpam-5960 78 30 denoted denote VERB ejpam-5960 78 31 by by ADP ejpam-5960 78 32 𝛿𝑂 𝛿𝑂 PROPN ejpam-5960 78 33 ( ( PUNCT ejpam-5960 78 34 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 78 35 , , PUNCT ejpam-5960 78 36 𝜏 𝜏 NOUN ejpam-5960 78 37 ) ) PUNCT ejpam-5960 78 38 . . PUNCT ejpam-5960 79 1 definition definition NOUN ejpam-5960 79 2 2.8 2.8 NUM ejpam-5960 80 1 [ [ X ejpam-5960 80 2 23 23 NUM ejpam-5960 80 3 ] ] PUNCT ejpam-5960 80 4 . . PUNCT ejpam-5960 81 1 let let VERB ejpam-5960 81 2 ( ( PUNCT ejpam-5960 81 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 81 4 , , PUNCT ejpam-5960 81 5 𝜏 𝜏 NOUN ejpam-5960 81 6 ) ) PUNCT ejpam-5960 81 7 be be VERB ejpam-5960 81 8 an an DET ejpam-5960 81 9 𝐿-ts 𝐿-ts NOUN ejpam-5960 81 10 , , PUNCT ejpam-5960 81 11 πœ‡ πœ‡ ADP ejpam-5960 81 12 ∈ ∈ ADP ejpam-5960 81 13 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 81 14 and and CCONJ ejpam-5960 81 15 𝛼 𝛼 NOUN ejpam-5960 81 16 ∈ ∈ PROPN ejpam-5960 81 17 𝑀 𝑀 PROPN ejpam-5960 81 18 ( ( PUNCT ejpam-5960 81 19 𝐿 𝐿 PROPN ejpam-5960 81 20 ) ) PUNCT ejpam-5960 81 21 . . PUNCT ejpam-5960 82 1 an an DET ejpam-5960 82 2 𝛼-rf 𝛼-rf PROPN ejpam-5960 82 3 ψ ψ X ejpam-5960 82 4 = = PUNCT ejpam-5960 82 5 { { PUNCT ejpam-5960 82 6 πœ‚ πœ‚ NOUN ejpam-5960 82 7 𝑗 𝑗 NOUN ejpam-5960 82 8 : : PUNCT ejpam-5960 82 9 𝑗 𝑗 PROPN ejpam-5960 82 10 ∈ ∈ PROPN ejpam-5960 82 11 𝐽 𝐽 PROPN ejpam-5960 82 12 } } PUNCT ejpam-5960 82 13 of of ADP ejpam-5960 82 14 πœ‡ πœ‡ ADV ejpam-5960 82 15 is be AUX ejpam-5960 82 16 called call VERB ejpam-5960 82 17 a a DET ejpam-5960 82 18 directed direct VERB ejpam-5960 82 19 if if SCONJ ejpam-5960 82 20 πœ‚1 πœ‚1 PROPN ejpam-5960 82 21 , , PUNCT ejpam-5960 82 22 πœ‚2 πœ‚2 NOUN ejpam-5960 82 23 ∈ ∈ PROPN ejpam-5960 82 24 ψ ψ ADP ejpam-5960 82 25 there there PRON ejpam-5960 82 26 is be VERB ejpam-5960 82 27 πœ‚3 πœ‚3 ADJ ejpam-5960 82 28 ∈ ∈ NOUN ejpam-5960 82 29 ψ ψ NOUN ejpam-5960 82 30 such such ADJ ejpam-5960 82 31 that that DET ejpam-5960 82 32 πœ‚3 πœ‚3 ADJ ejpam-5960 82 33 ≀ ≀ PUNCT ejpam-5960 82 34 πœ‚1 πœ‚1 ADJ ejpam-5960 82 35 ∧ ∧ PROPN ejpam-5960 82 36 πœ‚2 πœ‚2 NOUN ejpam-5960 82 37 . . PUNCT ejpam-5960 83 1 definition definition NOUN ejpam-5960 83 2 2.9 2.9 NUM ejpam-5960 84 1 [ [ X ejpam-5960 84 2 9 9 NUM ejpam-5960 84 3 ] ] PUNCT ejpam-5960 84 4 : : PUNCT ejpam-5960 84 5 let let VERB ejpam-5960 84 6 ( ( PUNCT ejpam-5960 84 7 𝐷 𝐷 NOUN ejpam-5960 84 8 , , PUNCT ejpam-5960 84 9 ≀ ≀ NUM ejpam-5960 84 10 ) ) PUNCT ejpam-5960 84 11 be be VERB ejpam-5960 84 12 a a DET ejpam-5960 84 13 directed direct VERB ejpam-5960 84 14 set set NOUN ejpam-5960 84 15 . . PUNCT ejpam-5960 85 1 then then ADV ejpam-5960 85 2 the the DET ejpam-5960 85 3 mapping mapping NOUN ejpam-5960 85 4 𝑆 𝑆 PROPN ejpam-5960 85 5 : : PUNCT ejpam-5960 85 6 𝐷 𝐷 PROPN ejpam-5960 85 7 β†’ β†’ SYM ejpam-5960 85 8 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 85 9 and and CCONJ ejpam-5960 85 10 denoted denote VERB ejpam-5960 85 11 by by ADP ejpam-5960 85 12 𝑆 𝑆 PROPN ejpam-5960 85 13 = = SYM ejpam-5960 85 14 { { PUNCT ejpam-5960 85 15 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 85 16 : : PUNCT ejpam-5960 85 17 𝑛 𝑛 PROPN ejpam-5960 85 18 ∈ ∈ PROPN ejpam-5960 85 19 𝐷 𝐷 PROPN ejpam-5960 85 20 } } PUNCT ejpam-5960 85 21 is be AUX ejpam-5960 85 22 called call VERB ejpam-5960 85 23 a a DET ejpam-5960 85 24 net net NOUN ejpam-5960 85 25 of of ADP ejpam-5960 85 26 𝐿-subsets 𝐿-subsets PROPN ejpam-5960 85 27 in in ADP ejpam-5960 85 28 𝑋. 𝑋. PROPN ejpam-5960 85 29 specifically specifically ADV ejpam-5960 85 30 , , PUNCT ejpam-5960 85 31 the the DET ejpam-5960 85 32 mapping mapping NOUN ejpam-5960 85 33 n. n. NOUN ejpam-5960 85 34 a. a. NOUN ejpam-5960 85 35 alsaedi alsaedi PROPN ejpam-5960 85 36 / / SYM ejpam-5960 85 37 eur eur PROPN ejpam-5960 85 38 . . PUNCT ejpam-5960 86 1 j. j. PROPN ejpam-5960 86 2 pure pure PROPN ejpam-5960 86 3 appl appl PROPN ejpam-5960 86 4 . . PROPN ejpam-5960 86 5 math math PROPN ejpam-5960 86 6 , , PUNCT ejpam-5960 86 7 18 18 NUM ejpam-5960 86 8 ( ( PUNCT ejpam-5960 86 9 4 4 NUM ejpam-5960 86 10 ) ) PUNCT ejpam-5960 86 11 ( ( PUNCT ejpam-5960 86 12 2025 2025 NUM ejpam-5960 86 13 ) ) PUNCT ejpam-5960 86 14 , , PUNCT ejpam-5960 86 15 5960 5960 NUM ejpam-5960 86 16 4 4 NUM ejpam-5960 86 17 of of ADP ejpam-5960 86 18 22 22 NUM ejpam-5960 86 19 𝑆 𝑆 PROPN ejpam-5960 86 20 : : PUNCT ejpam-5960 86 21 𝐷 𝐷 PROPN ejpam-5960 86 22 β†’ β†’ SYM ejpam-5960 86 23 𝑀 𝑀 PROPN ejpam-5960 86 24 ( ( PUNCT ejpam-5960 86 25 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 86 26 ) ) PUNCT ejpam-5960 86 27 is be AUX ejpam-5960 86 28 said say VERB ejpam-5960 86 29 to to PART ejpam-5960 86 30 be be AUX ejpam-5960 86 31 a a DET ejpam-5960 86 32 molecular molecular ADJ ejpam-5960 86 33 net net NOUN ejpam-5960 86 34 in in ADP ejpam-5960 86 35 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 86 36 if if SCONJ ejpam-5960 86 37 πœ‡ πœ‡ ADP ejpam-5960 86 38 ∈ ∈ PROPN ejpam-5960 86 39 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 86 40 and and CCONJ ejpam-5960 86 41 for for ADP ejpam-5960 86 42 each each DET ejpam-5960 86 43 𝑛 𝑛 PRON ejpam-5960 86 44 ∈ ∈ PROPN ejpam-5960 86 45 𝐷 𝐷 PROPN ejpam-5960 86 46 , , PUNCT ejpam-5960 86 47 𝑆 𝑆 PROPN ejpam-5960 86 48 ∈ ∈ PROPN ejpam-5960 86 49 πœ‡ πœ‡ SCONJ ejpam-5960 86 50 then then ADV ejpam-5960 86 51 𝑆 𝑆 PROPN ejpam-5960 86 52 is be AUX ejpam-5960 86 53 called call VERB ejpam-5960 86 54 a a DET ejpam-5960 86 55 net net NOUN ejpam-5960 86 56 in in ADP ejpam-5960 86 57 πœ‡. πœ‡. NOUN ejpam-5960 86 58 definition definition NOUN ejpam-5960 86 59 2.10 2.10 NUM ejpam-5960 86 60 [ [ X ejpam-5960 86 61 9 9 NUM ejpam-5960 86 62 ] ] PUNCT ejpam-5960 86 63 : : PUNCT ejpam-5960 86 64 let let VERB ejpam-5960 86 65 ( ( PUNCT ejpam-5960 86 66 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 86 67 , , PUNCT ejpam-5960 86 68 𝜏 𝜏 NOUN ejpam-5960 86 69 ) ) PUNCT ejpam-5960 86 70 be be VERB ejpam-5960 86 71 an an DET ejpam-5960 86 72 𝐿-ts 𝐿-ts NOUN ejpam-5960 86 73 and and CCONJ ejpam-5960 86 74 𝑆 𝑆 PROPN ejpam-5960 86 75 = = SYM ejpam-5960 86 76 { { PUNCT ejpam-5960 86 77 𝑆(𝑛 𝑆(𝑛 NOUN ejpam-5960 86 78 ) ) PUNCT ejpam-5960 86 79 : : PUNCT ejpam-5960 86 80 𝑛 𝑛 PROPN ejpam-5960 86 81 ∈ ∈ PROPN ejpam-5960 86 82 𝐷 𝐷 PROPN ejpam-5960 86 83 } } PUNCT ejpam-5960 86 84 be be AUX ejpam-5960 86 85 a a DET ejpam-5960 86 86 molecular molecular ADJ ejpam-5960 86 87 net net NOUN ejpam-5960 86 88 in in ADP ejpam-5960 86 89 𝐿𝑋. 𝐿𝑋. X ejpam-5960 86 90 𝑆 𝑆 PROPN ejpam-5960 86 91 is be AUX ejpam-5960 86 92 called call VERB ejpam-5960 86 93 a a DET ejpam-5960 86 94 molecular molecular ADJ ejpam-5960 86 95 𝛼-net 𝛼-net NOUN ejpam-5960 86 96 ( ( PUNCT ejpam-5960 86 97 𝛼 𝛼 PROPN ejpam-5960 86 98 ∈ ∈ PROPN ejpam-5960 86 99 𝑀 𝑀 PROPN ejpam-5960 86 100 ( ( PUNCT ejpam-5960 86 101 𝐿 𝐿 PROPN ejpam-5960 86 102 ) ) PUNCT ejpam-5960 86 103 ) ) PUNCT ejpam-5960 86 104 , , PUNCT ejpam-5960 86 105 if if SCONJ ejpam-5960 86 106 for for ADP ejpam-5960 86 107 each each PRON ejpam-5960 86 108 𝛾 𝛾 ADP ejpam-5960 86 109 ∈ ∈ PROPN ejpam-5960 86 110 π›½βˆ—(𝛼 π›½βˆ—(𝛼 PROPN ejpam-5960 86 111 ) ) PUNCT ejpam-5960 86 112 there there PRON ejpam-5960 86 113 exists exist VERB ejpam-5960 86 114 𝑛 𝑛 DET ejpam-5960 86 115 ∈ ∈ PROPN ejpam-5960 86 116 𝐷 𝐷 NOUN ejpam-5960 86 117 such such ADJ ejpam-5960 86 118 that that SCONJ ejpam-5960 86 119 ∨(𝑆(π‘š ∨(𝑆(π‘š NOUN ejpam-5960 86 120 ) ) PUNCT ejpam-5960 86 121 ) ) PUNCT ejpam-5960 86 122 β‰₯ β‰₯ X ejpam-5960 86 123 𝛾 𝛾 ADP ejpam-5960 86 124 whenever whenever SCONJ ejpam-5960 86 125 π‘š π‘š PROPN ejpam-5960 86 126 β‰₯ β‰₯ NUM ejpam-5960 86 127 𝑛 𝑛 NOUN ejpam-5960 86 128 , , PUNCT ejpam-5960 86 129 where where SCONJ ejpam-5960 86 130 ∨(𝑆(π‘š ∨(𝑆(π‘š NOUN ejpam-5960 86 131 ) ) PUNCT ejpam-5960 86 132 ) ) PUNCT ejpam-5960 86 133 is be AUX ejpam-5960 86 134 the the DET ejpam-5960 86 135 height height NOUN ejpam-5960 86 136 of of ADP ejpam-5960 86 137 the the DET ejpam-5960 86 138 molecular molecular ADJ ejpam-5960 86 139 𝑆(π‘š 𝑆(π‘š NOUN ejpam-5960 86 140 ) ) PUNCT ejpam-5960 86 141 . . PUNCT ejpam-5960 87 1 if if SCONJ ejpam-5960 87 2 ∨(𝑆(π‘š ∨(𝑆(π‘š NOUN ejpam-5960 87 3 ) ) PUNCT ejpam-5960 87 4 ) ) PUNCT ejpam-5960 88 1 = = SYM ejpam-5960 88 2 𝛼 𝛼 X ejpam-5960 88 3 for for ADP ejpam-5960 88 4 each each DET ejpam-5960 88 5 π‘š π‘š PROPN ejpam-5960 88 6 ∈ ∈ PROPN ejpam-5960 88 7 𝐷 𝐷 PROPN ejpam-5960 88 8 , , PUNCT ejpam-5960 88 9 then then ADV ejpam-5960 88 10 { { PUNCT ejpam-5960 88 11 𝑆(π‘š 𝑆(π‘š NOUN ejpam-5960 88 12 ) ) PUNCT ejpam-5960 88 13 : : PUNCT ejpam-5960 88 14 π‘š π‘š PROPN ejpam-5960 88 15 ∈ ∈ PROPN ejpam-5960 88 16 𝐷 𝐷 PROPN ejpam-5960 88 17 } } PUNCT ejpam-5960 88 18 is be AUX ejpam-5960 88 19 called call VERB ejpam-5960 88 20 a a DET ejpam-5960 88 21 constant constant ADJ ejpam-5960 88 22 molecular molecular ADJ ejpam-5960 88 23 𝛼-net 𝛼-net NOUN ejpam-5960 88 24 . . PUNCT ejpam-5960 89 1 definition definition NOUN ejpam-5960 89 2 2.11 2.11 NUM ejpam-5960 89 3 [ [ X ejpam-5960 89 4 9 9 NUM ejpam-5960 89 5 ] ] PUNCT ejpam-5960 89 6 : : PUNCT ejpam-5960 89 7 let let VERB ejpam-5960 89 8 𝑆 𝑆 PROPN ejpam-5960 89 9 = = SYM ejpam-5960 89 10 { { PUNCT ejpam-5960 89 11 𝑆(𝑛 𝑆(𝑛 NOUN ejpam-5960 89 12 ) ) PUNCT ejpam-5960 89 13 : : PUNCT ejpam-5960 89 14 𝑛 𝑛 PROPN ejpam-5960 89 15 ∈ ∈ PROPN ejpam-5960 89 16 𝐷 𝐷 PROPN ejpam-5960 89 17 } } PUNCT ejpam-5960 89 18 and and CCONJ ejpam-5960 89 19 𝑇 𝑇 PROPN ejpam-5960 89 20 = = SYM ejpam-5960 89 21 { { PUNCT ejpam-5960 89 22 𝑇 𝑇 PROPN ejpam-5960 89 23 ( ( PUNCT ejpam-5960 89 24 π‘š π‘š NOUN ejpam-5960 89 25 ) ) PUNCT ejpam-5960 89 26 : : PUNCT ejpam-5960 89 27 π‘š π‘š PROPN ejpam-5960 89 28 ∈ ∈ PROPN ejpam-5960 89 29 𝐸 𝐸 PROPN ejpam-5960 89 30 } } PUNCT ejpam-5960 89 31 be be AUX ejpam-5960 89 32 molecular molecular ADJ ejpam-5960 89 33 nets net NOUN ejpam-5960 89 34 in in ADP ejpam-5960 89 35 ( ( PUNCT ejpam-5960 89 36 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 89 37 , , PUNCT ejpam-5960 89 38 𝜏 𝜏 NOUN ejpam-5960 89 39 ) ) PUNCT ejpam-5960 89 40 . . PUNCT ejpam-5960 90 1 then then ADV ejpam-5960 90 2 𝑇 𝑇 PROPN ejpam-5960 90 3 is be AUX ejpam-5960 90 4 said say VERB ejpam-5960 90 5 to to PART ejpam-5960 90 6 be be AUX ejpam-5960 90 7 a a DET ejpam-5960 90 8 molecular molecular ADJ ejpam-5960 90 9 subnet subnet NOUN ejpam-5960 90 10 of of ADP ejpam-5960 90 11 𝑆 𝑆 PROPN ejpam-5960 90 12 if if SCONJ ejpam-5960 90 13 there there PRON ejpam-5960 90 14 is be VERB ejpam-5960 90 15 a a DET ejpam-5960 90 16 mapping mapping NOUN ejpam-5960 90 17 𝑓 𝑓 PRON ejpam-5960 90 18 : : PUNCT ejpam-5960 90 19 𝐸 𝐸 PROPN ejpam-5960 90 20 β†’ β†’ SYM ejpam-5960 90 21 𝐷 𝐷 PROPN ejpam-5960 90 22 that that PRON ejpam-5960 90 23 satisfies satisfy VERB ejpam-5960 90 24 the the DET ejpam-5960 90 25 following follow VERB ejpam-5960 90 26 conditions condition NOUN ejpam-5960 90 27 : : PUNCT ejpam-5960 90 28 ( ( PUNCT ejpam-5960 90 29 i i NOUN ejpam-5960 90 30 ) ) PUNCT ejpam-5960 90 31 𝑇 𝑇 PROPN ejpam-5960 90 32 = = SYM ejpam-5960 90 33 𝑆 𝑆 PROPN ejpam-5960 90 34 β—¦ β—¦ NOUN ejpam-5960 90 35 𝑓 𝑓 PROPN ejpam-5960 90 36 ( ( PUNCT ejpam-5960 90 37 ii ii NOUN ejpam-5960 90 38 ) ) PUNCT ejpam-5960 90 39 for for ADP ejpam-5960 90 40 each each DET ejpam-5960 90 41 𝑛 𝑛 PRON ejpam-5960 90 42 ∈ ∈ PROPN ejpam-5960 90 43 𝐷 𝐷 NOUN ejpam-5960 90 44 there there PRON ejpam-5960 90 45 is be VERB ejpam-5960 90 46 π‘š π‘š PRON ejpam-5960 90 47 ∈ ∈ PROPN ejpam-5960 90 48 𝐸 𝐸 PROPN ejpam-5960 90 49 such such ADJ ejpam-5960 90 50 that that SCONJ ejpam-5960 90 51 𝑓 𝑓 PROPN ejpam-5960 90 52 ( ( PUNCT ejpam-5960 90 53 𝑙 𝑙 NOUN ejpam-5960 90 54 ) ) PUNCT ejpam-5960 90 55 β‰₯ β‰₯ NOUN ejpam-5960 90 56 𝑛 𝑛 X ejpam-5960 90 57 for for ADP ejpam-5960 90 58 each each DET ejpam-5960 90 59 𝑙 𝑙 PRON ejpam-5960 90 60 ∈ ∈ PROPN ejpam-5960 90 61 𝐸 𝐸 PROPN ejpam-5960 90 62 , , PUNCT ejpam-5960 90 63 𝑙 𝑙 DET ejpam-5960 90 64 β‰₯ β‰₯ X ejpam-5960 90 65 π‘š. π‘š. ADJ ejpam-5960 90 66 definition definition NOUN ejpam-5960 90 67 2.12 2.12 NUM ejpam-5960 91 1 [ [ X ejpam-5960 91 2 21 21 NUM ejpam-5960 91 3 ] ] PUNCT ejpam-5960 91 4 : : PUNCT ejpam-5960 91 5 let let VERB ejpam-5960 91 6 ( ( PUNCT ejpam-5960 91 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 91 8 , , PUNCT ejpam-5960 91 9 𝜏 𝜏 NOUN ejpam-5960 91 10 ) ) PUNCT ejpam-5960 91 11 be be VERB ejpam-5960 91 12 an an DET ejpam-5960 91 13 𝐿–ts 𝐿–ts NOUN ejpam-5960 91 14 and and CCONJ ejpam-5960 91 15 Ξ΄ Ξ΄ NOUN ejpam-5960 91 16 = = PRON ejpam-5960 91 17 { { PUNCT ejpam-5960 91 18 πœ‡π‘› πœ‡π‘› INTJ ejpam-5960 91 19 : : PUNCT ejpam-5960 91 20 𝑛 𝑛 PROPN ejpam-5960 91 21 ∈ ∈ PROPN ejpam-5960 91 22 𝐷 𝐷 PROPN ejpam-5960 91 23 } } PUNCT ejpam-5960 91 24 be be AUX ejpam-5960 91 25 a a DET ejpam-5960 91 26 net net NOUN ejpam-5960 91 27 of of ADP ejpam-5960 91 28 𝐿–subsets 𝐿–subset NOUN ejpam-5960 91 29 in in ADP ejpam-5960 91 30 ( ( PUNCT ejpam-5960 91 31 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 91 32 , , PUNCT ejpam-5960 91 33 𝜏 𝜏 NOUN ejpam-5960 91 34 ) ) PUNCT ejpam-5960 91 35 and and CCONJ ejpam-5960 91 36 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 91 37 ∈ ∈ PROPN ejpam-5960 91 38 𝑀 𝑀 PROPN ejpam-5960 91 39 ( ( PUNCT ejpam-5960 91 40 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 91 41 ) ) PUNCT ejpam-5960 91 42 . . PUNCT ejpam-5960 92 1 then then ADV ejpam-5960 92 2 : : PUNCT ejpam-5960 92 3 ( ( PUNCT ejpam-5960 92 4 i i NOUN ejpam-5960 92 5 ) ) PUNCT ejpam-5960 92 6 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 92 7 is be AUX ejpam-5960 92 8 called call VERB ejpam-5960 92 9 the the DET ejpam-5960 92 10 𝛿–limit 𝛿–limit NOUN ejpam-5960 92 11 point point NOUN ejpam-5960 92 12 of of ADP ejpam-5960 92 13 Ξ΄ Ξ΄ PROPN ejpam-5960 92 14 ( ( PUNCT ejpam-5960 92 15 or or CCONJ ejpam-5960 92 16 𝛿–converges 𝛿–converge NOUN ejpam-5960 92 17 ) ) PUNCT ejpam-5960 92 18 to to ADP ejpam-5960 92 19 the the DET ejpam-5960 92 20 point point NOUN ejpam-5960 92 21 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 92 22 , , PUNCT ejpam-5960 92 23 in in ADP ejpam-5960 92 24 symbols symbol NOUN ejpam-5960 92 25 Ξ΄ Ξ΄ PROPN ejpam-5960 92 26 π›Ώβˆ’βˆ’β†’ π›Ώβˆ’βˆ’β†’ PRON ejpam-5960 92 27 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 92 28 if if SCONJ ejpam-5960 92 29 for for ADP ejpam-5960 92 30 every every DET ejpam-5960 92 31 πœ‚ πœ‚ PROPN ejpam-5960 92 32 ∈ ∈ PROPN ejpam-5960 92 33 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 92 34 there there PRON ejpam-5960 92 35 is be VERB ejpam-5960 92 36 an an DET ejpam-5960 92 37 𝑛 𝑛 PRON ejpam-5960 92 38 ∈ ∈ PROPN ejpam-5960 92 39 𝐷 𝐷 NOUN ejpam-5960 92 40 such such ADJ ejpam-5960 92 41 that that PRON ejpam-5960 92 42 for for ADP ejpam-5960 92 43 every every DET ejpam-5960 92 44 π‘š π‘š PROPN ejpam-5960 92 45 ∈ ∈ PROPN ejpam-5960 92 46 𝐷 𝐷 NOUN ejpam-5960 92 47 and and CCONJ ejpam-5960 92 48 π‘š π‘š ADP ejpam-5960 92 49 β‰₯ β‰₯ NOUN ejpam-5960 92 50 𝑛 𝑛 INTJ ejpam-5960 92 51 then then ADV ejpam-5960 92 52 πœ‡π‘› πœ‡π‘› INTJ ejpam-5960 92 53 βˆ‰ βˆ‰ PROPN ejpam-5960 92 54 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 92 55 ( ( PUNCT ejpam-5960 92 56 int(πœ‚ int(πœ‚ PROPN ejpam-5960 92 57 ) ) PUNCT ejpam-5960 92 58 ) ) PUNCT ejpam-5960 92 59 . . PUNCT ejpam-5960 93 1 the the DET ejpam-5960 93 2 union union NOUN ejpam-5960 93 3 of of ADP ejpam-5960 93 4 all all DET ejpam-5960 93 5 𝛿–limit 𝛿–limit NOUN ejpam-5960 93 6 points point NOUN ejpam-5960 93 7 of of ADP ejpam-5960 93 8 Ξ΄ Ξ΄ PROPN ejpam-5960 93 9 are be AUX ejpam-5960 93 10 denoted denote VERB ejpam-5960 93 11 by by ADP ejpam-5960 93 12 𝛿.lim(Ξ΄ 𝛿.lim(Ξ΄ NOUN ejpam-5960 93 13 ) ) PUNCT ejpam-5960 93 14 . . PUNCT ejpam-5960 94 1 ( ( PUNCT ejpam-5960 94 2 ii ii NOUN ejpam-5960 94 3 ) ) PUNCT ejpam-5960 94 4 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 94 5 is be AUX ejpam-5960 94 6 called call VERB ejpam-5960 94 7 a a DET ejpam-5960 94 8 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 94 9 ( ( PUNCT ejpam-5960 94 10 𝛿–adherent 𝛿–adherent NOUN ejpam-5960 94 11 ) ) PUNCT ejpam-5960 94 12 point point NOUN ejpam-5960 94 13 of of ADP ejpam-5960 94 14 Ξ΄ Ξ΄ PROPN ejpam-5960 94 15 , , PUNCT ejpam-5960 94 16 in in ADP ejpam-5960 94 17 symbols symbol NOUN ejpam-5960 94 18 Ξ΄ Ξ΄ PROPN ejpam-5960 94 19 π›Ώβˆ π›Ώβˆ NOUN ejpam-5960 94 20 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 94 21 if if SCONJ ejpam-5960 94 22 for for ADP ejpam-5960 94 23 every every DET ejpam-5960 94 24 πœ‚ πœ‚ PROPN ejpam-5960 94 25 ∈ ∈ PROPN ejpam-5960 94 26 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 94 27 and and CCONJ ejpam-5960 94 28 every every DET ejpam-5960 94 29 𝑛 𝑛 PROPN ejpam-5960 94 30 ∈ ∈ PROPN ejpam-5960 94 31 𝐷 𝐷 PROPN ejpam-5960 94 32 there there PRON ejpam-5960 94 33 exists exist VERB ejpam-5960 94 34 π‘š π‘š PROPN ejpam-5960 94 35 ∈ ∈ PROPN ejpam-5960 94 36 𝐷 𝐷 NOUN ejpam-5960 94 37 such such ADJ ejpam-5960 94 38 that that SCONJ ejpam-5960 94 39 π‘š π‘š PROPN ejpam-5960 94 40 β‰₯ β‰₯ PRON ejpam-5960 94 41 𝑛 𝑛 PROPN ejpam-5960 94 42 and and CCONJ ejpam-5960 94 43 πœ‡π‘› πœ‡π‘› PROPN ejpam-5960 94 44 βˆ‰ βˆ‰ PROPN ejpam-5960 94 45 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 94 46 ( ( PUNCT ejpam-5960 94 47 int(πœ‚ int(πœ‚ PROPN ejpam-5960 94 48 ) ) PUNCT ejpam-5960 94 49 ) ) PUNCT ejpam-5960 94 50 . . PUNCT ejpam-5960 95 1 the the DET ejpam-5960 95 2 union union NOUN ejpam-5960 95 3 of of ADP ejpam-5960 95 4 all all DET ejpam-5960 95 5 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 95 6 points point NOUN ejpam-5960 95 7 of of ADP ejpam-5960 95 8 Ξ΄ Ξ΄ PROPN ejpam-5960 95 9 are be AUX ejpam-5960 95 10 denoted denote VERB ejpam-5960 95 11 by by ADP ejpam-5960 95 12 𝛿.lim(Ξ΄ 𝛿.lim(Ξ΄ NOUN ejpam-5960 95 13 ) ) PUNCT ejpam-5960 95 14 . . PUNCT ejpam-5960 96 1 if if SCONJ ejpam-5960 96 2 𝛿.lim(Ξ΄)=𝛿.lim(Ξ΄ 𝛿.lim(Ξ΄)=𝛿.lim(Ξ΄ PRON ejpam-5960 96 3 ) ) PUNCT ejpam-5960 96 4 = = SYM ejpam-5960 96 5 πœ‡ πœ‡ ADP ejpam-5960 96 6 , , PUNCT ejpam-5960 96 7 then then ADV ejpam-5960 96 8 we we PRON ejpam-5960 96 9 say say VERB ejpam-5960 96 10 that that SCONJ ejpam-5960 96 11 πœ‡ πœ‡ ADP ejpam-5960 96 12 is be AUX ejpam-5960 96 13 the the DET ejpam-5960 96 14 𝛿–limit 𝛿–limit NOUN ejpam-5960 96 15 of of ADP ejpam-5960 96 16 Ξ΄ Ξ΄ PROPN ejpam-5960 96 17 , , PUNCT ejpam-5960 96 18 or or CCONJ ejpam-5960 96 19 we we PRON ejpam-5960 96 20 say say VERB ejpam-5960 96 21 that that SCONJ ejpam-5960 96 22 Ξ΄ Ξ΄ PROPN ejpam-5960 96 23 𝛿–converges 𝛿–converge VERB ejpam-5960 96 24 to to AUX ejpam-5960 96 25 πœ‡ πœ‡ VERB ejpam-5960 96 26 , , PUNCT ejpam-5960 96 27 in in ADP ejpam-5960 96 28 symbol symbol NOUN ejpam-5960 96 29 𝛿.lim(Ξ΄ 𝛿.lim(Ξ΄ X ejpam-5960 96 30 ) ) PUNCT ejpam-5960 96 31 = = SYM ejpam-5960 96 32 πœ‡. πœ‡. NOUN ejpam-5960 96 33 the the DET ejpam-5960 96 34 𝛿–limit 𝛿–limit PROPN ejpam-5960 96 35 and and CCONJ ejpam-5960 96 36 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 96 37 points point NOUN ejpam-5960 96 38 of of ADP ejpam-5960 96 39 a a DET ejpam-5960 96 40 molecular molecular ADJ ejpam-5960 96 41 net net NOUN ejpam-5960 96 42 are be AUX ejpam-5960 96 43 defined define VERB ejpam-5960 96 44 similarly similarly ADV ejpam-5960 96 45 in in ADP ejpam-5960 96 46 [ [ X ejpam-5960 96 47 16 16 NUM ejpam-5960 96 48 ] ] PUNCT ejpam-5960 96 49 . . PUNCT ejpam-5960 97 1 definition definition NOUN ejpam-5960 97 2 2.13 2.13 NUM ejpam-5960 98 1 [ [ X ejpam-5960 98 2 17 17 NUM ejpam-5960 98 3 ] ] PUNCT ejpam-5960 98 4 : : PUNCT ejpam-5960 98 5 let let VERB ejpam-5960 98 6 ( ( PUNCT ejpam-5960 98 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 98 8 , , PUNCT ejpam-5960 98 9 𝜏 𝜏 NOUN ejpam-5960 98 10 ) ) PUNCT ejpam-5960 98 11 be be VERB ejpam-5960 98 12 an an DET ejpam-5960 98 13 𝐿-ts 𝐿-ts NOUN ejpam-5960 98 14 and and CCONJ ejpam-5960 98 15 𝑆 𝑆 PROPN ejpam-5960 98 16 be be VERB ejpam-5960 98 17 a a DET ejpam-5960 98 18 molecular molecular ADJ ejpam-5960 98 19 𝛼-net 𝛼-net NOUN ejpam-5960 98 20 in in ADP ejpam-5960 98 21 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 98 22 then then ADV ejpam-5960 98 23 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 98 24 ∈ ∈ PROPN ejpam-5960 98 25 𝑀 𝑀 PROPN ejpam-5960 98 26 ( ( PUNCT ejpam-5960 98 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 98 28 ) ) PUNCT ejpam-5960 98 29 is be AUX ejpam-5960 98 30 called call VERB ejpam-5960 98 31 the the DET ejpam-5960 98 32 𝛿–limit 𝛿–limit NOUN ejpam-5960 98 33 point point NOUN ejpam-5960 98 34 of of ADP ejpam-5960 98 35 𝑆 𝑆 PROPN ejpam-5960 98 36 , , PUNCT ejpam-5960 98 37 ( ( PUNCT ejpam-5960 98 38 or or CCONJ ejpam-5960 98 39 𝑆 𝑆 PROPN ejpam-5960 98 40 𝛿–converges 𝛿–converge NOUN ejpam-5960 98 41 to to ADP ejpam-5960 98 42 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 98 43 ) ) PUNCT ejpam-5960 98 44 in in ADP ejpam-5960 98 45 symbol symbol NOUN ejpam-5960 98 46 𝑆 𝑆 PROPN ejpam-5960 98 47 π›Ώβˆ’β†’ π›Ώβˆ’β†’ NOUN ejpam-5960 98 48 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 98 49 if if SCONJ ejpam-5960 98 50 for for ADP ejpam-5960 98 51 every every DET ejpam-5960 98 52 πœ‡ πœ‡ DET ejpam-5960 98 53 ∈ ∈ PROPN ejpam-5960 98 54 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 98 55 there there PRON ejpam-5960 98 56 is be VERB ejpam-5960 98 57 an an DET ejpam-5960 98 58 𝑛 𝑛 PRON ejpam-5960 98 59 ∈ ∈ PROPN ejpam-5960 98 60 𝐷 𝐷 NOUN ejpam-5960 98 61 such such ADJ ejpam-5960 98 62 that that PRON ejpam-5960 98 63 for for ADP ejpam-5960 98 64 each each DET ejpam-5960 98 65 π‘š π‘š PROPN ejpam-5960 98 66 ∈ ∈ PROPN ejpam-5960 98 67 𝐷 𝐷 NOUN ejpam-5960 98 68 and and CCONJ ejpam-5960 98 69 π‘š π‘š ADP ejpam-5960 98 70 β‰₯ β‰₯ NOUN ejpam-5960 98 71 𝑛 𝑛 ADP ejpam-5960 98 72 we we PRON ejpam-5960 98 73 have have VERB ejpam-5960 98 74 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 98 75 ) ) PUNCT ejpam-5960 98 76 β‰° β‰° PROPN ejpam-5960 98 77 𝑐𝑙 𝑐𝑙 X ejpam-5960 98 78 ( ( PUNCT ejpam-5960 98 79 int(πœ‡ int(πœ‡ NOUN ejpam-5960 98 80 ) ) PUNCT ejpam-5960 98 81 ) ) PUNCT ejpam-5960 98 82 . . PUNCT ejpam-5960 99 1 the the DET ejpam-5960 99 2 union union NOUN ejpam-5960 99 3 of of ADP ejpam-5960 99 4 all all DET ejpam-5960 99 5 limit limit NOUN ejpam-5960 99 6 points point NOUN ejpam-5960 99 7 of of ADP ejpam-5960 99 8 𝑆 𝑆 PROPN ejpam-5960 99 9 is be AUX ejpam-5960 99 10 denoted denote VERB ejpam-5960 99 11 by by ADP ejpam-5960 99 12 𝛿.lim(𝑆 𝛿.lim(𝑆 PROPN ejpam-5960 99 13 ) ) PUNCT ejpam-5960 99 14 . . PUNCT ejpam-5960 100 1 definition definition NOUN ejpam-5960 100 2 2.14 2.14 NUM ejpam-5960 101 1 [ [ X ejpam-5960 101 2 17 17 NUM ejpam-5960 101 3 ] ] PUNCT ejpam-5960 101 4 : : PUNCT ejpam-5960 101 5 let let VERB ejpam-5960 101 6 ( ( PUNCT ejpam-5960 101 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 101 8 , , PUNCT ejpam-5960 101 9 𝜏 𝜏 NOUN ejpam-5960 101 10 ) ) PUNCT ejpam-5960 101 11 be be VERB ejpam-5960 101 12 an an DET ejpam-5960 101 13 𝐿–ts 𝐿–ts NOUN ejpam-5960 101 14 and and CCONJ ejpam-5960 101 15 𝑆 𝑆 PROPN ejpam-5960 101 16 be be VERB ejpam-5960 101 17 a a DET ejpam-5960 101 18 molecular molecular ADJ ejpam-5960 101 19 𝛼–net 𝛼–net NUM ejpam-5960 101 20 in in ADP ejpam-5960 101 21 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 101 22 then then ADV ejpam-5960 101 23 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 101 24 ∈ ∈ PROPN ejpam-5960 101 25 𝑀 𝑀 PROPN ejpam-5960 101 26 ( ( PUNCT ejpam-5960 101 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 101 28 ) ) PUNCT ejpam-5960 101 29 is be AUX ejpam-5960 101 30 called call VERB ejpam-5960 101 31 a a DET ejpam-5960 101 32 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 101 33 point point NOUN ejpam-5960 101 34 of of ADP ejpam-5960 101 35 𝑆 𝑆 PROPN ejpam-5960 101 36 , , PUNCT ejpam-5960 101 37 in in ADP ejpam-5960 101 38 symbol symbol NOUN ejpam-5960 101 39 𝑆 𝑆 PROPN ejpam-5960 101 40 π›Ώβˆ π›Ώβˆ PROPN ejpam-5960 101 41 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 101 42 if if SCONJ ejpam-5960 101 43 for for ADP ejpam-5960 101 44 every every DET ejpam-5960 101 45 πœ‡ πœ‡ PRON ejpam-5960 101 46 ∈ ∈ PROPN ejpam-5960 101 47 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 101 48 and and CCONJ ejpam-5960 101 49 every every DET ejpam-5960 101 50 𝑛 𝑛 PROPN ejpam-5960 101 51 ∈ ∈ PROPN ejpam-5960 101 52 𝐷 𝐷 NOUN ejpam-5960 101 53 there there PRON ejpam-5960 101 54 is be VERB ejpam-5960 101 55 π‘š π‘š PRON ejpam-5960 101 56 ∈ ∈ PROPN ejpam-5960 101 57 𝐷 𝐷 NOUN ejpam-5960 101 58 such such ADJ ejpam-5960 101 59 that that SCONJ ejpam-5960 101 60 π‘š π‘š PROPN ejpam-5960 101 61 β‰₯ β‰₯ PRON ejpam-5960 101 62 𝑛 𝑛 PROPN ejpam-5960 101 63 and and CCONJ ejpam-5960 101 64 𝑆(π‘š 𝑆(π‘š SYM ejpam-5960 101 65 ) ) PUNCT ejpam-5960 101 66 βˆ‰ βˆ‰ PROPN ejpam-5960 101 67 𝑐𝑙 𝑐𝑙 X ejpam-5960 101 68 ( ( PUNCT ejpam-5960 101 69 int(πœ‡ int(πœ‡ NOUN ejpam-5960 101 70 ) ) PUNCT ejpam-5960 101 71 ) ) PUNCT ejpam-5960 101 72 . . PUNCT ejpam-5960 102 1 the the DET ejpam-5960 102 2 union union NOUN ejpam-5960 102 3 of of ADP ejpam-5960 102 4 all all DET ejpam-5960 102 5 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 102 6 points point NOUN ejpam-5960 102 7 of of ADP ejpam-5960 102 8 𝑆 𝑆 PROPN ejpam-5960 102 9 is be AUX ejpam-5960 102 10 denoted denote VERB ejpam-5960 102 11 by by ADP ejpam-5960 102 12 π›Ώπ‘Žπ‘‘β„Ž(𝑆 π›Ώπ‘Žπ‘‘β„Ž(𝑆 PROPN ejpam-5960 102 13 ) ) PUNCT ejpam-5960 102 14 . . PUNCT ejpam-5960 103 1 theorem theorem VERB ejpam-5960 103 2 2.15 2.15 NUM ejpam-5960 104 1 [ [ X ejpam-5960 104 2 22 22 NUM ejpam-5960 104 3 ] ] PUNCT ejpam-5960 104 4 : : PUNCT ejpam-5960 104 5 assume assume VERB ejpam-5960 104 6 that that SCONJ ejpam-5960 104 7 𝑆 𝑆 PROPN ejpam-5960 104 8 = = SYM ejpam-5960 104 9 { { PUNCT ejpam-5960 104 10 𝑆(𝑛 𝑆(𝑛 NOUN ejpam-5960 104 11 ) ) PUNCT ejpam-5960 104 12 : : PUNCT ejpam-5960 104 13 𝑛 𝑛 PROPN ejpam-5960 104 14 ∈ ∈ PROPN ejpam-5960 104 15 𝐷 𝐷 PROPN ejpam-5960 104 16 } } PUNCT ejpam-5960 104 17 is be AUX ejpam-5960 104 18 a a DET ejpam-5960 104 19 molecular molecular ADJ ejpam-5960 104 20 net net NOUN ejpam-5960 104 21 in in ADP ejpam-5960 104 22 an an DET ejpam-5960 104 23 𝐿–ts 𝐿–ts NOUN ejpam-5960 104 24 ( ( PUNCT ejpam-5960 104 25 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 104 26 , , PUNCT ejpam-5960 104 27 𝜏 𝜏 NOUN ejpam-5960 104 28 ) ) PUNCT ejpam-5960 104 29 and and CCONJ ejpam-5960 104 30 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 104 31 ∈ ∈ PROPN ejpam-5960 104 32 𝑀 𝑀 PROPN ejpam-5960 104 33 ( ( PUNCT ejpam-5960 104 34 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 104 35 ) ) PUNCT ejpam-5960 104 36 . . PUNCT ejpam-5960 105 1 then then ADV ejpam-5960 105 2 the the DET ejpam-5960 105 3 following follow VERB ejpam-5960 105 4 results result NOUN ejpam-5960 105 5 are be AUX ejpam-5960 105 6 true true ADJ ejpam-5960 105 7 : : PUNCT ejpam-5960 105 8 ( ( PUNCT ejpam-5960 105 9 i i NOUN ejpam-5960 105 10 ) ) PUNCT ejpam-5960 105 11 𝑆 𝑆 PROPN ejpam-5960 105 12 π›Ώβˆ π›Ώβˆ PROPN ejpam-5960 105 13 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 105 14 iff iff PROPN ejpam-5960 105 15 there there PRON ejpam-5960 105 16 exists exist VERB ejpam-5960 105 17 a a DET ejpam-5960 105 18 subnet subnet NOUN ejpam-5960 105 19 𝑇 𝑇 PROPN ejpam-5960 105 20 of of ADP ejpam-5960 105 21 𝑆 𝑆 PROPN ejpam-5960 105 22 such such ADJ ejpam-5960 105 23 that that SCONJ ejpam-5960 105 24 𝑇 𝑇 PROPN ejpam-5960 105 25 π›Ώβˆ’βˆ’β†’ π›Ώβˆ’βˆ’β†’ PRON ejpam-5960 105 26 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 105 27 ( ( PUNCT ejpam-5960 105 28 ii ii NOUN ejpam-5960 105 29 ) ) PUNCT ejpam-5960 105 30 if if SCONJ ejpam-5960 105 31 𝑆 𝑆 PROPN ejpam-5960 105 32 π›Ώβˆ’βˆ’β†’ π›Ώβˆ’βˆ’β†’ PRON ejpam-5960 105 33 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 105 34 , , PUNCT ejpam-5960 105 35 then then ADV ejpam-5960 105 36 𝑇 𝑇 PROPN ejpam-5960 105 37 π›Ώβˆ’βˆ’β†’ π›Ώβˆ’βˆ’β†’ PRON ejpam-5960 105 38 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 105 39 for for ADP ejpam-5960 105 40 each each DET ejpam-5960 105 41 subnet subnet NOUN ejpam-5960 105 42 𝑇 𝑇 PROPN ejpam-5960 105 43 of of ADP ejpam-5960 105 44 𝑆. 𝑆. PROPN ejpam-5960 105 45 definition definition NOUN ejpam-5960 105 46 2.16 2.16 NUM ejpam-5960 106 1 [ [ X ejpam-5960 106 2 8 8 NUM ejpam-5960 106 3 ] ] PUNCT ejpam-5960 106 4 : : PUNCT ejpam-5960 106 5 let let VERB ejpam-5960 106 6 ( ( PUNCT ejpam-5960 106 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 106 8 , , PUNCT ejpam-5960 106 9 𝜏 𝜏 NOUN ejpam-5960 106 10 ) ) PUNCT ejpam-5960 106 11 be be VERB ejpam-5960 106 12 an an DET ejpam-5960 106 13 𝐿-ts 𝐿-ts NOUN ejpam-5960 106 14 , , PUNCT ejpam-5960 106 15 πœ‡ πœ‡ ADP ejpam-5960 106 16 ∈ ∈ X ejpam-5960 106 17 𝐿𝑋. 𝐿𝑋. X ejpam-5960 106 18 then then ADV ejpam-5960 106 19 πœ‡ πœ‡ ADP ejpam-5960 106 20 is be AUX ejpam-5960 106 21 called call VERB ejpam-5960 106 22 nearly nearly ADV ejpam-5960 106 23 𝑄𝛼–compact 𝑄𝛼–compact PROPN ejpam-5960 106 24 ( ( PUNCT ejpam-5960 106 25 or or CCONJ ejpam-5960 106 26 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 106 27 ) ) PUNCT ejpam-5960 106 28 in in ADP ejpam-5960 106 29 ( ( PUNCT ejpam-5960 106 30 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 106 31 , , PUNCT ejpam-5960 106 32 𝜏 𝜏 NOUN ejpam-5960 106 33 ) ) PUNCT ejpam-5960 106 34 if if SCONJ ejpam-5960 106 35 for for ADP ejpam-5960 106 36 each each DET ejpam-5960 106 37 𝛼 𝛼 PROPN ejpam-5960 106 38 ∈ ∈ PROPN ejpam-5960 106 39 𝑀 𝑀 PROPN ejpam-5960 106 40 ( ( PUNCT ejpam-5960 106 41 𝐿 𝐿 PROPN ejpam-5960 106 42 ) ) PUNCT ejpam-5960 106 43 and and CCONJ ejpam-5960 107 1 every every DET ejpam-5960 107 2 𝛼–rf 𝛼–rf X ejpam-5960 107 3 ψ ψ NOUN ejpam-5960 107 4 of of ADP ejpam-5960 107 5 πœ‡ πœ‡ ADP ejpam-5960 107 6 there there PRON ejpam-5960 107 7 is be VERB ejpam-5960 107 8 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 107 9 ∈ ∈ PROPN ejpam-5960 107 10 2(ψ 2(ψ NUM ejpam-5960 107 11 ) ) PUNCT ejpam-5960 107 12 such such ADJ ejpam-5960 107 13 that that SCONJ ejpam-5960 107 14 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 107 15 is be AUX ejpam-5960 107 16 an an DET ejpam-5960 107 17 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 107 18 of of ADP ejpam-5960 107 19 πœ‡. πœ‡. NOUN ejpam-5960 107 20 n. n. PROPN ejpam-5960 107 21 a. a. PROPN ejpam-5960 107 22 alsaedi alsaedi PROPN ejpam-5960 107 23 / / SYM ejpam-5960 107 24 eur eur PROPN ejpam-5960 107 25 . . PUNCT ejpam-5960 108 1 j. j. PROPN ejpam-5960 108 2 pure pure PROPN ejpam-5960 108 3 appl appl PROPN ejpam-5960 108 4 . . PROPN ejpam-5960 108 5 math math PROPN ejpam-5960 108 6 , , PUNCT ejpam-5960 108 7 18 18 NUM ejpam-5960 108 8 ( ( PUNCT ejpam-5960 108 9 4 4 NUM ejpam-5960 108 10 ) ) PUNCT ejpam-5960 108 11 ( ( PUNCT ejpam-5960 108 12 2025 2025 NUM ejpam-5960 108 13 ) ) PUNCT ejpam-5960 108 14 , , PUNCT ejpam-5960 108 15 5960 5960 NUM ejpam-5960 108 16 5 5 NUM ejpam-5960 108 17 of of ADP ejpam-5960 108 18 22 22 NUM ejpam-5960 108 19 if if SCONJ ejpam-5960 108 20 1𝑋 1𝑋 PROPN ejpam-5960 108 21 is be AUX ejpam-5960 108 22 nearly nearly ADV ejpam-5960 108 23 𝑄𝛼–compact 𝑄𝛼–compact NUM ejpam-5960 108 24 , , PUNCT ejpam-5960 108 25 then then ADV ejpam-5960 108 26 ( ( PUNCT ejpam-5960 108 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 108 28 , , PUNCT ejpam-5960 108 29 𝜏 𝜏 NOUN ejpam-5960 108 30 ) ) PUNCT ejpam-5960 108 31 is be AUX ejpam-5960 108 32 called call VERB ejpam-5960 108 33 a a DET ejpam-5960 108 34 nearly nearly ADV ejpam-5960 108 35 𝑄𝛼–compact 𝑄𝛼–compact PROPN ejpam-5960 108 36 space space NOUN ejpam-5960 108 37 . . PUNCT ejpam-5960 109 1 definition definition NOUN ejpam-5960 109 2 2.17 2.17 NUM ejpam-5960 110 1 [ [ X ejpam-5960 110 2 23 23 NUM ejpam-5960 110 3 ] ] X ejpam-5960 110 4 : : PUNCT ejpam-5960 110 5 an an DET ejpam-5960 110 6 𝐿–ts 𝐿–ts NOUN ejpam-5960 110 7 ( ( PUNCT ejpam-5960 110 8 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 110 9 , , PUNCT ejpam-5960 110 10 𝜏 𝜏 NOUN ejpam-5960 110 11 ) ) PUNCT ejpam-5960 110 12 is be AUX ejpam-5960 110 13 said say VERB ejpam-5960 110 14 to to PART ejpam-5960 110 15 be be AUX ejpam-5960 110 16 : : PUNCT ejpam-5960 110 17 ( ( PUNCT ejpam-5960 110 18 i i NOUN ejpam-5960 110 19 ) ) PUNCT ejpam-5960 110 20 𝐿𝑇1 𝐿𝑇1 PROPN ejpam-5960 110 21 – – PUNCT ejpam-5960 110 22 space space NOUN ejpam-5960 110 23 iff iff PROPN ejpam-5960 110 24 for for ADP ejpam-5960 110 25 any any DET ejpam-5960 110 26 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 110 27 , , PUNCT ejpam-5960 110 28 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 110 29 ∈ ∈ PROPN ejpam-5960 110 30 𝑀 𝑀 PROPN ejpam-5960 110 31 ( ( PUNCT ejpam-5960 110 32 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 110 33 ) ) PUNCT ejpam-5960 110 34 , , PUNCT ejpam-5960 110 35 π‘₯ π‘₯ PROPN ejpam-5960 110 36 β‰  β‰  PROPN ejpam-5960 110 37 𝑦 𝑦 NOUN ejpam-5960 110 38 there there PRON ejpam-5960 110 39 is be VERB ejpam-5960 110 40 πœ† πœ† DET ejpam-5960 110 41 ∈ ∈ PROPN ejpam-5960 110 42 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 110 43 such such ADJ ejpam-5960 110 44 that that SCONJ ejpam-5960 110 45 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 110 46 ∈ ∈ PROPN ejpam-5960 110 47 πœ†. πœ†. NOUN ejpam-5960 110 48 ( ( PUNCT ejpam-5960 110 49 ii ii PROPN ejpam-5960 110 50 ) ) PUNCT ejpam-5960 110 51 𝐿𝑇2 𝐿𝑇2 PROPN ejpam-5960 110 52 – – PUNCT ejpam-5960 110 53 space space NOUN ejpam-5960 110 54 iff iff NOUN ejpam-5960 110 55 for for ADP ejpam-5960 110 56 any any DET ejpam-5960 110 57 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 110 58 , , PUNCT ejpam-5960 110 59 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 110 60 ∈ ∈ PROPN ejpam-5960 110 61 𝑀 𝑀 PROPN ejpam-5960 110 62 ( ( PUNCT ejpam-5960 110 63 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 110 64 ) ) PUNCT ejpam-5960 110 65 , , PUNCT ejpam-5960 110 66 π‘₯ π‘₯ PROPN ejpam-5960 110 67 β‰  β‰  PROPN ejpam-5960 110 68 𝑦 𝑦 NOUN ejpam-5960 110 69 there there PRON ejpam-5960 110 70 is be VERB ejpam-5960 110 71 πœ† πœ† DET ejpam-5960 110 72 ∈ ∈ PROPN ejpam-5960 110 73 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 110 74 , , PUNCT ejpam-5960 110 75 πœ‚ πœ‚ PROPN ejpam-5960 110 76 ∈ ∈ PROPN ejpam-5960 110 77 𝑅𝑦𝛾 𝑅𝑦𝛾 PROPN ejpam-5960 110 78 such such ADJ ejpam-5960 110 79 that that SCONJ ejpam-5960 110 80 πœ† πœ† DET ejpam-5960 110 81 ∨ ∨ NUM ejpam-5960 110 82 πœ‚ πœ‚ NOUN ejpam-5960 110 83 = = SYM ejpam-5960 110 84 1𝑋. 1𝑋. NUM ejpam-5960 110 85 ( ( PUNCT ejpam-5960 110 86 iii iii NOUN ejpam-5960 110 87 ) ) PUNCT ejpam-5960 110 88 𝐿𝑇2 𝐿𝑇2 PROPN ejpam-5960 110 89 1 1 NUM ejpam-5960 110 90 2 2 NUM ejpam-5960 110 91 – – PUNCT ejpam-5960 110 92 space space NOUN ejpam-5960 110 93 iff iff NOUN ejpam-5960 110 94 for for ADP ejpam-5960 110 95 any any DET ejpam-5960 110 96 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 110 97 , , PUNCT ejpam-5960 110 98 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 110 99 ∈ ∈ PROPN ejpam-5960 110 100 𝑀 𝑀 PROPN ejpam-5960 110 101 ( ( PUNCT ejpam-5960 110 102 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 110 103 ) ) PUNCT ejpam-5960 110 104 , , PUNCT ejpam-5960 110 105 π‘₯ π‘₯ PROPN ejpam-5960 110 106 β‰  β‰  PROPN ejpam-5960 110 107 𝑦 𝑦 NOUN ejpam-5960 110 108 there there PRON ejpam-5960 110 109 is be VERB ejpam-5960 110 110 πœ† πœ† DET ejpam-5960 110 111 ∈ ∈ PROPN ejpam-5960 110 112 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 110 113 , , PUNCT ejpam-5960 110 114 πœ‚ πœ‚ PROPN ejpam-5960 110 115 ∈ ∈ PROPN ejpam-5960 110 116 𝑅𝑦𝛾 𝑅𝑦𝛾 PROPN ejpam-5960 110 117 such such ADJ ejpam-5960 110 118 that that DET ejpam-5960 110 119 int(πœ† int(πœ† PROPN ejpam-5960 110 120 ) ) PUNCT ejpam-5960 110 121 ∨ ∨ NUM ejpam-5960 110 122 int(πœ‚ int(πœ‚ PROPN ejpam-5960 110 123 ) ) PUNCT ejpam-5960 110 124 = = SYM ejpam-5960 110 125 1𝑋. 1𝑋. NUM ejpam-5960 110 126 ( ( PUNCT ejpam-5960 110 127 iv iv X ejpam-5960 110 128 ) ) PUNCT ejpam-5960 110 129 𝐿𝑅2 𝐿𝑅2 PROPN ejpam-5960 110 130 – – PUNCT ejpam-5960 110 131 space space NOUN ejpam-5960 110 132 ( ( PUNCT ejpam-5960 110 133 regular regular ADJ ejpam-5960 110 134 space space NOUN ejpam-5960 110 135 ) ) PUNCT ejpam-5960 110 136 iff iff NOUN ejpam-5960 110 137 for for ADP ejpam-5960 110 138 all all DET ejpam-5960 110 139 𝛼 𝛼 PROPN ejpam-5960 110 140 ∈ ∈ PROPN ejpam-5960 110 141 𝑀 𝑀 PROPN ejpam-5960 110 142 ( ( PUNCT ejpam-5960 110 143 𝐿 𝐿 PROPN ejpam-5960 110 144 ) ) PUNCT ejpam-5960 110 145 , , PUNCT ejpam-5960 110 146 π‘₯ π‘₯ PROPN ejpam-5960 110 147 ∈ ∈ PROPN ejpam-5960 110 148 𝑋 𝑋 NOUN ejpam-5960 110 149 and and CCONJ ejpam-5960 110 150 for for ADP ejpam-5960 110 151 each each DET ejpam-5960 110 152 πœ† πœ† DET ejpam-5960 110 153 ∈ ∈ PROPN ejpam-5960 110 154 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 110 155 there there PRON ejpam-5960 110 156 is be VERB ejpam-5960 110 157 πœ‚ πœ‚ DET ejpam-5960 110 158 ∈ ∈ PROPN ejpam-5960 110 159 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 110 160 , , PUNCT ejpam-5960 110 161 𝜌 𝜌 X ejpam-5960 110 162 ∈ ∈ NOUN ejpam-5960 110 163 πœβ€² πœβ€² NOUN ejpam-5960 110 164 such such ADJ ejpam-5960 110 165 that that SCONJ ejpam-5960 110 166 πœ‚ πœ‚ PROPN ejpam-5960 110 167 ∨ ∨ NUM ejpam-5960 110 168 𝜌 𝜌 ADP ejpam-5960 110 169 = = SYM ejpam-5960 110 170 1𝑋 1𝑋 PROPN ejpam-5960 110 171 and and CCONJ ejpam-5960 110 172 πœ† πœ† PROPN ejpam-5960 110 173 ∧ ∧ PROPN ejpam-5960 110 174 𝜌 𝜌 ADP ejpam-5960 110 175 = = SYM ejpam-5960 110 176 0𝑋. 0𝑋. NUM ejpam-5960 110 177 ( ( PUNCT ejpam-5960 110 178 v v NOUN ejpam-5960 110 179 ) ) PUNCT ejpam-5960 110 180 𝐿𝑆𝑅2 𝐿𝑆𝑅2 NOUN ejpam-5960 110 181 – – PUNCT ejpam-5960 110 182 space space NOUN ejpam-5960 110 183 ( ( PUNCT ejpam-5960 110 184 semi semi ADJ ejpam-5960 110 185 - - ADJ ejpam-5960 110 186 regular regular ADJ ejpam-5960 110 187 space space NOUN ejpam-5960 110 188 ) ) PUNCT ejpam-5960 110 189 iff iff NOUN ejpam-5960 110 190 for for ADP ejpam-5960 110 191 all all PRON ejpam-5960 110 192 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 110 193 ∈ ∈ PROPN ejpam-5960 110 194 𝑀 𝑀 PROPN ejpam-5960 110 195 ( ( PUNCT ejpam-5960 110 196 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 110 197 ) ) PUNCT ejpam-5960 110 198 and and CCONJ ejpam-5960 110 199 for for ADP ejpam-5960 110 200 each each DET ejpam-5960 110 201 πœ† πœ† DET ejpam-5960 110 202 ∈ ∈ PROPN ejpam-5960 110 203 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 110 204 there there PRON ejpam-5960 110 205 is be VERB ejpam-5960 110 206 πœ‚ πœ‚ DET ejpam-5960 110 207 ∈ ∈ PROPN ejpam-5960 110 208 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 110 209 such such ADJ ejpam-5960 110 210 that that SCONJ ejpam-5960 110 211 πœ† πœ† DET ejpam-5960 110 212 ≀ ≀ ADJ ejpam-5960 110 213 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 110 214 ( ( PUNCT ejpam-5960 110 215 int(πœ‚ int(πœ‚ PROPN ejpam-5960 110 216 ) ) PUNCT ejpam-5960 110 217 ) ) PUNCT ejpam-5960 110 218 . . PUNCT ejpam-5960 111 1 ( ( PUNCT ejpam-5960 111 2 vi vi NOUN ejpam-5960 111 3 ) ) PUNCT ejpam-5960 111 4 𝐿𝑇3 𝐿𝑇3 PROPN ejpam-5960 111 5 – – PUNCT ejpam-5960 111 6 space space NOUN ejpam-5960 111 7 iff iff NOUN ejpam-5960 111 8 it it PRON ejpam-5960 111 9 is be AUX ejpam-5960 111 10 𝐿𝑅2 𝐿𝑅2 PROPN ejpam-5960 111 11 – – PUNCT ejpam-5960 111 12 space space NOUN ejpam-5960 111 13 and and CCONJ ejpam-5960 111 14 𝐿𝑇1 𝐿𝑇1 PROPN ejpam-5960 111 15 – – PUNCT ejpam-5960 111 16 space space NOUN ejpam-5960 111 17 . . PUNCT ejpam-5960 112 1 theorem theorem VERB ejpam-5960 112 2 2.18 2.18 NUM ejpam-5960 112 3 [ [ NOUN ejpam-5960 112 4 8 8 NUM ejpam-5960 112 5 ] ] PUNCT ejpam-5960 112 6 : : PUNCT ejpam-5960 112 7 let let VERB ejpam-5960 112 8 ( ( PUNCT ejpam-5960 112 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 112 10 , , PUNCT ejpam-5960 112 11 𝜏 𝜏 NOUN ejpam-5960 112 12 ) ) PUNCT ejpam-5960 112 13 be be VERB ejpam-5960 112 14 an an DET ejpam-5960 112 15 𝐿–ts 𝐿–ts NOUN ejpam-5960 112 16 and and CCONJ ejpam-5960 112 17 πœ‡ πœ‡ X ejpam-5960 112 18 ∈ ∈ X ejpam-5960 112 19 𝐿𝑋. 𝐿𝑋. X ejpam-5960 112 20 then then ADV ejpam-5960 112 21 the the DET ejpam-5960 112 22 following follow VERB ejpam-5960 112 23 properties property NOUN ejpam-5960 112 24 are be AUX ejpam-5960 112 25 true true ADJ ejpam-5960 112 26 : : PUNCT ejpam-5960 112 27 ( ( PUNCT ejpam-5960 112 28 i i NOUN ejpam-5960 112 29 ) ) PUNCT ejpam-5960 112 30 every every DET ejpam-5960 112 31 set set NOUN ejpam-5960 112 32 with with ADP ejpam-5960 112 33 finite finite ADJ ejpam-5960 112 34 support support NOUN ejpam-5960 112 35 is be AUX ejpam-5960 112 36 nearly nearly ADV ejpam-5960 112 37 𝑄𝛼–compact 𝑄𝛼–compact NUM ejpam-5960 112 38 . . PUNCT ejpam-5960 113 1 ( ( PUNCT ejpam-5960 113 2 ii ii NOUN ejpam-5960 113 3 ) ) PUNCT ejpam-5960 113 4 every every DET ejpam-5960 113 5 nearly nearly ADV ejpam-5960 113 6 𝑄𝛼–compact 𝑄𝛼–compact PROPN ejpam-5960 113 7 set set VERB ejpam-5960 113 8 in in ADP ejpam-5960 113 9 a a DET ejpam-5960 113 10 fully fully ADV ejpam-5960 113 11 stratified stratify VERB ejpam-5960 113 12 and and CCONJ ejpam-5960 113 13 𝐿𝑇2 𝐿𝑇2 PROPN ejpam-5960 113 14 – – PUNCT ejpam-5960 113 15 space space NOUN ejpam-5960 113 16 , , PUNCT ejpam-5960 113 17 then then ADV ejpam-5960 113 18 it it PRON ejpam-5960 113 19 is be AUX ejpam-5960 113 20 𝛿–closed 𝛿–close VERB ejpam-5960 113 21 . . PUNCT ejpam-5960 114 1 theorem theorem VERB ejpam-5960 114 2 2.19 2.19 NUM ejpam-5960 114 3 [ [ NOUN ejpam-5960 114 4 8 8 NUM ejpam-5960 114 5 ] ] PUNCT ejpam-5960 114 6 : : PUNCT ejpam-5960 114 7 let let VERB ejpam-5960 114 8 ( ( PUNCT ejpam-5960 114 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 114 10 , , PUNCT ejpam-5960 114 11 𝜏 𝜏 NOUN ejpam-5960 114 12 ) ) PUNCT ejpam-5960 114 13 be be VERB ejpam-5960 114 14 an an DET ejpam-5960 114 15 𝐿–ts 𝐿–ts NOUN ejpam-5960 114 16 , , PUNCT ejpam-5960 114 17 𝛼 𝛼 PROPN ejpam-5960 114 18 ∈ ∈ PROPN ejpam-5960 114 19 𝑀 𝑀 PROPN ejpam-5960 114 20 ( ( PUNCT ejpam-5960 114 21 𝐿 𝐿 PROPN ejpam-5960 114 22 ) ) PUNCT ejpam-5960 114 23 and and CCONJ ejpam-5960 114 24 πœ‡ πœ‡ X ejpam-5960 114 25 ∈ ∈ X ejpam-5960 114 26 𝐿𝑋. 𝐿𝑋. X ejpam-5960 114 27 then then ADV ejpam-5960 114 28 πœ‡ πœ‡ ADP ejpam-5960 114 29 is be AUX ejpam-5960 114 30 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 114 31 iff iff NOUN ejpam-5960 114 32 for for ADP ejpam-5960 114 33 each each DET ejpam-5960 114 34 constant constant ADJ ejpam-5960 114 35 molecular molecular ADJ ejpam-5960 114 36 𝛼–net 𝛼–net NOUN ejpam-5960 114 37 𝑆 𝑆 PROPN ejpam-5960 114 38 contained contain VERB ejpam-5960 114 39 in in ADP ejpam-5960 114 40 πœ‡ πœ‡ ADP ejpam-5960 114 41 has have AUX ejpam-5960 114 42 a a DET ejpam-5960 114 43 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 114 44 point point NOUN ejpam-5960 114 45 with with ADP ejpam-5960 114 46 height height NOUN ejpam-5960 114 47 𝛼 𝛼 NOUN ejpam-5960 114 48 in in ADP ejpam-5960 114 49 πœ‡. πœ‡. NOUN ejpam-5960 114 50 theorem theorem ADJ ejpam-5960 114 51 2.20 2.20 NUM ejpam-5960 114 52 [ [ X ejpam-5960 114 53 21 21 NUM ejpam-5960 114 54 ] ] PUNCT ejpam-5960 114 55 . . PUNCT ejpam-5960 115 1 if if SCONJ ejpam-5960 115 2 𝐿–ts 𝐿–ts NOUN ejpam-5960 115 3 ( ( PUNCT ejpam-5960 115 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 115 5 , , PUNCT ejpam-5960 115 6 𝜏 𝜏 NOUN ejpam-5960 115 7 ) ) PUNCT ejpam-5960 115 8 is be AUX ejpam-5960 115 9 𝐿𝑅2 𝐿𝑅2 PROPN ejpam-5960 115 10 – – PUNCT ejpam-5960 115 11 space space NOUN ejpam-5960 115 12 , , PUNCT ejpam-5960 115 13 then then ADV ejpam-5960 115 14 it it PRON ejpam-5960 115 15 is be AUX ejpam-5960 115 16 𝐿𝑆𝑅2 𝐿𝑆𝑅2 NOUN ejpam-5960 115 17 – – PUNCT ejpam-5960 115 18 space space NOUN ejpam-5960 115 19 . . PUNCT ejpam-5960 116 1 theorem theorem VERB ejpam-5960 116 2 2.21 2.21 NUM ejpam-5960 116 3 [ [ X ejpam-5960 116 4 21 21 NUM ejpam-5960 116 5 ] ] X ejpam-5960 116 6 : : PUNCT ejpam-5960 116 7 an an DET ejpam-5960 116 8 𝐿–ts 𝐿–ts NOUN ejpam-5960 116 9 ( ( PUNCT ejpam-5960 116 10 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 116 11 , , PUNCT ejpam-5960 116 12 𝜏 𝜏 NOUN ejpam-5960 116 13 ) ) PUNCT ejpam-5960 116 14 is be AUX ejpam-5960 116 15 𝐿𝑆𝑅2 𝐿𝑆𝑅2 NOUN ejpam-5960 116 16 – – PUNCT ejpam-5960 116 17 space space NOUN ejpam-5960 116 18 iff iff NOUN ejpam-5960 116 19 for for ADP ejpam-5960 116 20 any any DET ejpam-5960 116 21 πœ‡ πœ‡ ADP ejpam-5960 116 22 ∈ ∈ PROPN ejpam-5960 116 23 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 116 24 , , PUNCT ejpam-5960 116 25 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 116 26 ( ( PUNCT ejpam-5960 116 27 πœ‡ πœ‡ ADP ejpam-5960 116 28 ) ) PUNCT ejpam-5960 116 29 = = SYM ejpam-5960 116 30 𝛿𝑐𝑙 𝛿𝑐𝑙 X ejpam-5960 116 31 ( ( PUNCT ejpam-5960 116 32 πœ‡ πœ‡ NOUN ejpam-5960 116 33 ) ) PUNCT ejpam-5960 116 34 . . PUNCT ejpam-5960 117 1 corollary corollary NOUN ejpam-5960 117 2 2.22 2.22 NUM ejpam-5960 118 1 [ [ X ejpam-5960 118 2 19 19 NUM ejpam-5960 118 3 ] ] PUNCT ejpam-5960 118 4 . . PUNCT ejpam-5960 119 1 if if SCONJ ejpam-5960 119 2 𝐿–ts 𝐿–ts NOUN ejpam-5960 119 3 ( ( PUNCT ejpam-5960 119 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 119 5 , , PUNCT ejpam-5960 119 6 𝜏 𝜏 NOUN ejpam-5960 119 7 ) ) PUNCT ejpam-5960 119 8 is be AUX ejpam-5960 119 9 𝐿𝑆𝑅2 𝐿𝑆𝑅2 NOUN ejpam-5960 119 10 – – PUNCT ejpam-5960 119 11 space space NOUN ejpam-5960 119 12 , , PUNCT ejpam-5960 119 13 then then ADV ejpam-5960 119 14 a a DET ejpam-5960 119 15 closed closed ADJ ejpam-5960 119 16 𝐿–subset 𝐿–subset NOUN ejpam-5960 119 17 is be AUX ejpam-5960 119 18 a a DET ejpam-5960 119 19 𝛿–closed 𝛿–closed ADJ ejpam-5960 119 20 𝐿–subset 𝐿–subset NOUN ejpam-5960 119 21 and and CCONJ ejpam-5960 119 22 hence hence ADV ejpam-5960 119 23 𝛿𝑐𝑙 𝛿𝑐𝑙 ADP ejpam-5960 119 24 ( ( PUNCT ejpam-5960 119 25 πœ‡ πœ‡ NOUN ejpam-5960 119 26 ) ) PUNCT ejpam-5960 119 27 is be AUX ejpam-5960 119 28 a a DET ejpam-5960 119 29 𝛿–closed 𝛿–closed ADJ ejpam-5960 119 30 𝐿–subset 𝐿–subset PROPN ejpam-5960 119 31 . . PUNCT ejpam-5960 120 1 definition definition NOUN ejpam-5960 120 2 2.23 2.23 NUM ejpam-5960 121 1 [ [ X ejpam-5960 121 2 15 15 NUM ejpam-5960 121 3 ] ] X ejpam-5960 121 4 : : PUNCT ejpam-5960 122 1 the the DET ejpam-5960 122 2 nonempty nonempty ADJ ejpam-5960 122 3 family family NOUN ejpam-5960 122 4 f f PROPN ejpam-5960 122 5 βŠ‚ βŠ‚ PROPN ejpam-5960 123 1 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 123 2 is be AUX ejpam-5960 123 3 called call VERB ejpam-5960 123 4 an an DET ejpam-5960 123 5 𝐿–filter 𝐿–filter NOUN ejpam-5960 123 6 if if SCONJ ejpam-5960 123 7 the the DET ejpam-5960 123 8 following follow VERB ejpam-5960 123 9 conditions condition NOUN ejpam-5960 123 10 are be AUX ejpam-5960 123 11 satisfied satisfied ADJ ejpam-5960 123 12 , , PUNCT ejpam-5960 123 13 for for ADP ejpam-5960 123 14 each each DET ejpam-5960 123 15 πœ‡1 πœ‡1 NOUN ejpam-5960 123 16 , , PUNCT ejpam-5960 124 1 πœ‡2 πœ‡2 PROPN ejpam-5960 124 2 ∈ ∈ PROPN ejpam-5960 124 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 124 4 ( ( PUNCT ejpam-5960 124 5 i i NOUN ejpam-5960 124 6 ) ) PUNCT ejpam-5960 124 7 0𝑋 0𝑋 PROPN ejpam-5960 124 8 βˆ‰ βˆ‰ PROPN ejpam-5960 124 9 f f PROPN ejpam-5960 124 10 ( ( PUNCT ejpam-5960 124 11 ii ii PROPN ejpam-5960 124 12 ) ) PUNCT ejpam-5960 124 13 if if SCONJ ejpam-5960 124 14 πœ‡1 πœ‡1 PROPN ejpam-5960 124 15 ≀ ≀ PROPN ejpam-5960 124 16 πœ‡2 πœ‡2 PROPN ejpam-5960 124 17 and and CCONJ ejpam-5960 124 18 πœ‡1 πœ‡1 PROPN ejpam-5960 124 19 ∈ ∈ PROPN ejpam-5960 124 20 f f PROPN ejpam-5960 124 21 , , PUNCT ejpam-5960 124 22 then then ADV ejpam-5960 124 23 πœ‡2 πœ‡2 PROPN ejpam-5960 124 24 ∈ ∈ PROPN ejpam-5960 125 1 f f X ejpam-5960 125 2 . . PUNCT ejpam-5960 126 1 ( ( PUNCT ejpam-5960 126 2 iii iii X ejpam-5960 126 3 ) ) PUNCT ejpam-5960 126 4 if if SCONJ ejpam-5960 126 5 πœ‡1 πœ‡1 ADJ ejpam-5960 126 6 , , PUNCT ejpam-5960 126 7 πœ‡2 πœ‡2 PROPN ejpam-5960 126 8 ∈ ∈ PROPN ejpam-5960 126 9 f f X ejpam-5960 126 10 , , PUNCT ejpam-5960 126 11 then then ADV ejpam-5960 126 12 πœ‡1 πœ‡1 PROPN ejpam-5960 126 13 ∧ ∧ PROPN ejpam-5960 126 14 πœ‡2 πœ‡2 PROPN ejpam-5960 126 15 ∈ ∈ PROPN ejpam-5960 126 16 f f X ejpam-5960 126 17 . . PUNCT ejpam-5960 127 1 definition definition NOUN ejpam-5960 127 2 2.24 2.24 NUM ejpam-5960 128 1 [ [ X ejpam-5960 128 2 15 15 NUM ejpam-5960 128 3 ] ] X ejpam-5960 128 4 : : PUNCT ejpam-5960 128 5 a a DET ejpam-5960 128 6 filter filter NOUN ejpam-5960 128 7 f f NOUN ejpam-5960 128 8 in in ADP ejpam-5960 128 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 128 10 is be AUX ejpam-5960 128 11 called call VERB ejpam-5960 128 12 an an DET ejpam-5960 128 13 𝛼–filter 𝛼–filter NOUN ejpam-5960 128 14 ( ( PUNCT ejpam-5960 128 15 𝛼 𝛼 PROPN ejpam-5960 128 16 ∈ ∈ PROPN ejpam-5960 128 17 𝑀 𝑀 PROPN ejpam-5960 128 18 ( ( PUNCT ejpam-5960 128 19 𝐿 𝐿 PROPN ejpam-5960 128 20 ) ) PUNCT ejpam-5960 128 21 ) ) PUNCT ejpam-5960 129 1 , , PUNCT ejpam-5960 129 2 if if SCONJ ejpam-5960 129 3 for for ADP ejpam-5960 129 4 every every DET ejpam-5960 129 5 πœ† πœ† PROPN ejpam-5960 129 6 ∈ ∈ PROPN ejpam-5960 129 7 f f PROPN ejpam-5960 129 8 , , PUNCT ejpam-5960 129 9 ∨ ∨ NUM ejpam-5960 129 10 π‘₯βˆˆπ‘‹ π‘₯βˆˆπ‘‹ PROPN ejpam-5960 129 11 πœ†(π‘₯ πœ†(π‘₯ PROPN ejpam-5960 129 12 ) ) PUNCT ejpam-5960 129 13 β‰₯ β‰₯ NOUN ejpam-5960 129 14 𝛼. 𝛼. NOUN ejpam-5960 129 15 definition definition NOUN ejpam-5960 129 16 2.25 2.25 NUM ejpam-5960 129 17 [ [ X ejpam-5960 129 18 15 15 NUM ejpam-5960 129 19 ] ] PUNCT ejpam-5960 129 20 : : PUNCT ejpam-5960 129 21 let let VERB ejpam-5960 129 22 ( ( PUNCT ejpam-5960 129 23 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 129 24 , , PUNCT ejpam-5960 129 25 𝜏 𝜏 NOUN ejpam-5960 129 26 ) ) PUNCT ejpam-5960 129 27 be be VERB ejpam-5960 129 28 an an DET ejpam-5960 129 29 𝐿–ts 𝐿–ts NOUN ejpam-5960 129 30 and and CCONJ ejpam-5960 129 31 f f PROPN ejpam-5960 129 32 be be AUX ejpam-5960 129 33 an an DET ejpam-5960 129 34 𝐿–filter 𝐿–filter NOUN ejpam-5960 129 35 in in ADP ejpam-5960 129 36 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 129 37 then then ADV ejpam-5960 129 38 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 129 39 ∈ ∈ PROPN ejpam-5960 129 40 𝑀 𝑀 PROPN ejpam-5960 129 41 ( ( PUNCT ejpam-5960 129 42 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 129 43 ) ) PUNCT ejpam-5960 129 44 is be AUX ejpam-5960 129 45 called call VERB ejpam-5960 129 46 the the DET ejpam-5960 129 47 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 129 48 point point NOUN ejpam-5960 129 49 of of ADP ejpam-5960 129 50 f f PROPN ejpam-5960 129 51 , , PUNCT ejpam-5960 129 52 in in ADP ejpam-5960 129 53 symbol symbol NOUN ejpam-5960 129 54 f f PROPN ejpam-5960 129 55 π›Ώβˆ π›Ώβˆ NOUN ejpam-5960 129 56 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 129 57 if if SCONJ ejpam-5960 129 58 for for ADP ejpam-5960 129 59 each each DET ejpam-5960 129 60 πœ† πœ† PROPN ejpam-5960 129 61 ∈ ∈ PROPN ejpam-5960 129 62 f f NOUN ejpam-5960 129 63 and and CCONJ ejpam-5960 129 64 each each DET ejpam-5960 129 65 πœ‡ πœ‡ ADP ejpam-5960 129 66 ∈ ∈ PROPN ejpam-5960 129 67 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 129 68 , , PUNCT ejpam-5960 129 69 πœ† πœ† PRON ejpam-5960 129 70 ⊈ ⊈ PRON ejpam-5960 129 71 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 129 72 ( ( PUNCT ejpam-5960 129 73 int(πœ‡ int(πœ‡ NOUN ejpam-5960 129 74 ) ) PUNCT ejpam-5960 129 75 ) ) PUNCT ejpam-5960 129 76 . . PUNCT ejpam-5960 130 1 the the DET ejpam-5960 130 2 union union NOUN ejpam-5960 130 3 of of ADP ejpam-5960 130 4 all all DET ejpam-5960 130 5 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 130 6 points point NOUN ejpam-5960 130 7 of of ADP ejpam-5960 130 8 f f PROPN ejpam-5960 130 9 is be AUX ejpam-5960 130 10 denoted denote VERB ejpam-5960 130 11 by by ADP ejpam-5960 130 12 π›Ώπ‘Žπ‘‘β„Ž(f π›Ώπ‘Žπ‘‘β„Ž(f PROPN ejpam-5960 130 13 ) ) PUNCT ejpam-5960 130 14 . . PUNCT ejpam-5960 131 1 definition definition NOUN ejpam-5960 131 2 2.26 2.26 NUM ejpam-5960 131 3 [ [ X ejpam-5960 131 4 27 27 NUM ejpam-5960 131 5 ] ] X ejpam-5960 131 6 : : PUNCT ejpam-5960 131 7 the the DET ejpam-5960 131 8 nonempty nonempty ADJ ejpam-5960 131 9 family family NOUN ejpam-5960 131 10 𝐼 𝐼 PROPN ejpam-5960 131 11 βŠ‚ βŠ‚ PROPN ejpam-5960 131 12 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 131 13 is be AUX ejpam-5960 131 14 called call VERB ejpam-5960 131 15 an an DET ejpam-5960 131 16 𝐿–ideal 𝐿–ideal NOUN ejpam-5960 131 17 if if SCONJ ejpam-5960 131 18 the the DET ejpam-5960 131 19 following follow VERB ejpam-5960 131 20 conditions condition NOUN ejpam-5960 131 21 are be AUX ejpam-5960 131 22 satisfied satisfied ADJ ejpam-5960 131 23 , , PUNCT ejpam-5960 131 24 for for ADP ejpam-5960 131 25 each each DET ejpam-5960 131 26 πœ‡1 πœ‡1 NOUN ejpam-5960 131 27 , , PUNCT ejpam-5960 132 1 πœ‡2 πœ‡2 PROPN ejpam-5960 132 2 ∈ ∈ PROPN ejpam-5960 132 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 132 4 ( ( PUNCT ejpam-5960 132 5 i i NOUN ejpam-5960 132 6 ) ) PUNCT ejpam-5960 132 7 1𝑋 1𝑋 PROPN ejpam-5960 132 8 βˆ‰ βˆ‰ PROPN ejpam-5960 133 1 𝐼 𝐼 PROPN ejpam-5960 133 2 ( ( PUNCT ejpam-5960 133 3 ii ii PROPN ejpam-5960 133 4 ) ) PUNCT ejpam-5960 133 5 if if SCONJ ejpam-5960 133 6 πœ‡1 πœ‡1 PROPN ejpam-5960 133 7 ≀ ≀ PROPN ejpam-5960 133 8 πœ‡2 πœ‡2 PROPN ejpam-5960 133 9 and and CCONJ ejpam-5960 133 10 πœ‡2 πœ‡2 PROPN ejpam-5960 133 11 ∈ ∈ PROPN ejpam-5960 133 12 𝐼 𝐼 PROPN ejpam-5960 133 13 , , PUNCT ejpam-5960 133 14 then then ADV ejpam-5960 133 15 πœ‡1 πœ‡1 PROPN ejpam-5960 133 16 ∈ ∈ PROPN ejpam-5960 133 17 𝐼. 𝐼. PROPN ejpam-5960 133 18 n. n. PROPN ejpam-5960 133 19 a. a. NOUN ejpam-5960 133 20 alsaedi alsaedi PROPN ejpam-5960 133 21 / / SYM ejpam-5960 133 22 eur eur PROPN ejpam-5960 133 23 . . PUNCT ejpam-5960 134 1 j. j. PROPN ejpam-5960 134 2 pure pure PROPN ejpam-5960 134 3 appl appl PROPN ejpam-5960 134 4 . . PROPN ejpam-5960 134 5 math math PROPN ejpam-5960 134 6 , , PUNCT ejpam-5960 134 7 18 18 NUM ejpam-5960 134 8 ( ( PUNCT ejpam-5960 134 9 4 4 NUM ejpam-5960 134 10 ) ) PUNCT ejpam-5960 134 11 ( ( PUNCT ejpam-5960 134 12 2025 2025 NUM ejpam-5960 134 13 ) ) PUNCT ejpam-5960 134 14 , , PUNCT ejpam-5960 134 15 5960 5960 NUM ejpam-5960 134 16 6 6 NUM ejpam-5960 134 17 of of ADP ejpam-5960 134 18 22 22 NUM ejpam-5960 134 19 ( ( PUNCT ejpam-5960 134 20 iii iii NOUN ejpam-5960 134 21 ) ) PUNCT ejpam-5960 134 22 if if SCONJ ejpam-5960 134 23 πœ‡1 πœ‡1 ADJ ejpam-5960 134 24 , , PUNCT ejpam-5960 134 25 πœ‡2 πœ‡2 PROPN ejpam-5960 134 26 ∈ ∈ PROPN ejpam-5960 134 27 𝐼 𝐼 PROPN ejpam-5960 134 28 , , PUNCT ejpam-5960 134 29 then then ADV ejpam-5960 134 30 πœ‡1 πœ‡1 PROPN ejpam-5960 134 31 ∨ ∨ PROPN ejpam-5960 134 32 πœ‡2 πœ‡2 PROPN ejpam-5960 134 33 ∈ ∈ PROPN ejpam-5960 134 34 𝐼. 𝐼. PROPN ejpam-5960 134 35 definition definition NOUN ejpam-5960 134 36 2.27 2.27 NUM ejpam-5960 135 1 [ [ X ejpam-5960 135 2 27 27 NUM ejpam-5960 135 3 ] ] PUNCT ejpam-5960 135 4 : : PUNCT ejpam-5960 135 5 let let VERB ejpam-5960 135 6 𝐼 𝐼 PRON ejpam-5960 135 7 be be AUX ejpam-5960 135 8 an an DET ejpam-5960 135 9 𝐿–ideal 𝐿–ideal NOUN ejpam-5960 135 10 in in ADP ejpam-5960 135 11 an an DET ejpam-5960 135 12 𝐿–ts 𝐿–ts NOUN ejpam-5960 135 13 ( ( PUNCT ejpam-5960 135 14 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 135 15 , , PUNCT ejpam-5960 135 16 𝜏 𝜏 NOUN ejpam-5960 135 17 ) ) PUNCT ejpam-5960 135 18 and and CCONJ ejpam-5960 135 19 𝛼 𝛼 PROPN ejpam-5960 135 20 ∈ ∈ PROPN ejpam-5960 135 21 𝑀 𝑀 PROPN ejpam-5960 135 22 ( ( PUNCT ejpam-5960 135 23 𝐿 𝐿 PROPN ejpam-5960 135 24 ) ) PUNCT ejpam-5960 135 25 . . PUNCT ejpam-5960 136 1 then then ADV ejpam-5960 136 2 𝐼 𝐼 PROPN ejpam-5960 136 3 is be AUX ejpam-5960 136 4 said say VERB ejpam-5960 136 5 to to PART ejpam-5960 136 6 be be AUX ejpam-5960 136 7 an an DET ejpam-5960 136 8 𝛼–ideal 𝛼–ideal NOUN ejpam-5960 136 9 , , PUNCT ejpam-5960 136 10 if if SCONJ ejpam-5960 136 11 βˆ¨π‘›βˆˆπ‘‹πœ‚(π‘₯ βˆ¨π‘›βˆˆπ‘‹πœ‚(π‘₯ PROPN ejpam-5960 136 12 ) ) PUNCT ejpam-5960 136 13 < < X ejpam-5960 136 14 𝛼 𝛼 X ejpam-5960 136 15 for for ADP ejpam-5960 136 16 each each DET ejpam-5960 136 17 πœ‚ πœ‚ PROPN ejpam-5960 136 18 ∈ ∈ PROPN ejpam-5960 136 19 𝐼. 𝐼. PROPN ejpam-5960 136 20 theorem theorem VERB ejpam-5960 136 21 2.28 2.28 NUM ejpam-5960 137 1 [ [ X ejpam-5960 137 2 22 22 NUM ejpam-5960 137 3 ] ] PUNCT ejpam-5960 137 4 : : PUNCT ejpam-5960 137 5 let let VERB ejpam-5960 137 6 f f PRON ejpam-5960 137 7 be be AUX ejpam-5960 137 8 a a DET ejpam-5960 137 9 𝐿–filter 𝐿–filter NOUN ejpam-5960 137 10 in in ADP ejpam-5960 137 11 an an DET ejpam-5960 137 12 𝐿–ts 𝐿–ts NOUN ejpam-5960 137 13 ( ( PUNCT ejpam-5960 137 14 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 137 15 , , PUNCT ejpam-5960 137 16 𝜏 𝜏 NOUN ejpam-5960 137 17 ) ) PUNCT ejpam-5960 137 18 and and CCONJ ejpam-5960 137 19 𝑆(f 𝑆(f PROPN ejpam-5960 137 20 ) ) PUNCT ejpam-5960 137 21 be be AUX ejpam-5960 137 22 the the DET ejpam-5960 137 23 𝐿–molecular 𝐿–molecular ADJ ejpam-5960 137 24 net net NOUN ejpam-5960 137 25 induced induce VERB ejpam-5960 137 26 by by ADP ejpam-5960 137 27 f f PROPN ejpam-5960 137 28 . . PUNCT ejpam-5960 138 1 then then ADV ejpam-5960 138 2 π›Ώπ‘Žπ‘‘β„Ž(f π›Ώπ‘Žπ‘‘β„Ž(f VERB ejpam-5960 138 3 ) ) PUNCT ejpam-5960 139 1 = = SYM ejpam-5960 139 2 π›Ώπ‘Žπ‘‘β„Ž(𝑆(f π›Ώπ‘Žπ‘‘β„Ž(𝑆(f ADJ ejpam-5960 139 3 ) ) PUNCT ejpam-5960 139 4 ) ) PUNCT ejpam-5960 139 5 . . PUNCT ejpam-5960 140 1 theorem theorem VERB ejpam-5960 140 2 2.29 2.29 NUM ejpam-5960 140 3 [ [ NOUN ejpam-5960 140 4 22 22 NUM ejpam-5960 140 5 ] ] PUNCT ejpam-5960 140 6 : : PUNCT ejpam-5960 140 7 suppose suppose VERB ejpam-5960 140 8 that that SCONJ ejpam-5960 140 9 𝑆 𝑆 PROPN ejpam-5960 140 10 is be AUX ejpam-5960 140 11 a a DET ejpam-5960 140 12 𝐿–net 𝐿–net NOUN ejpam-5960 140 13 in in ADP ejpam-5960 140 14 an an DET ejpam-5960 140 15 𝐿–ts 𝐿–ts NOUN ejpam-5960 140 16 ( ( PUNCT ejpam-5960 140 17 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 140 18 , , PUNCT ejpam-5960 140 19 𝜏 𝜏 NOUN ejpam-5960 140 20 ) ) PUNCT ejpam-5960 140 21 and and CCONJ ejpam-5960 140 22 f f PROPN ejpam-5960 140 23 ( ( PUNCT ejpam-5960 140 24 𝑆 𝑆 PROPN ejpam-5960 140 25 ) ) PUNCT ejpam-5960 140 26 is be AUX ejpam-5960 140 27 the the DET ejpam-5960 140 28 𝐿–filter 𝐿–filter NOUN ejpam-5960 140 29 induced induce VERB ejpam-5960 140 30 by by ADP ejpam-5960 140 31 𝑆. 𝑆. PROPN ejpam-5960 140 32 then then ADV ejpam-5960 140 33 π›Ώπ‘Žπ‘‘β„Ž(𝑆 π›Ώπ‘Žπ‘‘β„Ž(𝑆 NOUN ejpam-5960 140 34 ) ) PUNCT ejpam-5960 140 35 = = SYM ejpam-5960 140 36 π›Ώπ‘Žπ‘‘β„Ž(f π›Ώπ‘Žπ‘‘β„Ž(f PROPN ejpam-5960 140 37 ( ( PUNCT ejpam-5960 140 38 𝑆 𝑆 PROPN ejpam-5960 140 39 ) ) PUNCT ejpam-5960 140 40 ) ) PUNCT ejpam-5960 140 41 . . PUNCT ejpam-5960 141 1 theorem theorem VERB ejpam-5960 141 2 2.30 2.30 NUM ejpam-5960 142 1 [ [ X ejpam-5960 142 2 22 22 NUM ejpam-5960 142 3 ] ] PUNCT ejpam-5960 142 4 : : PUNCT ejpam-5960 142 5 suppose suppose VERB ejpam-5960 142 6 that that SCONJ ejpam-5960 142 7 𝐼 𝐼 PROPN ejpam-5960 142 8 is be AUX ejpam-5960 142 9 an an DET ejpam-5960 142 10 𝐿–ideal 𝐿–ideal NOUN ejpam-5960 142 11 in in ADP ejpam-5960 142 12 an an DET ejpam-5960 142 13 𝐿–ts 𝐿–ts NOUN ejpam-5960 142 14 ( ( PUNCT ejpam-5960 142 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 142 16 , , PUNCT ejpam-5960 142 17 𝜏 𝜏 NOUN ejpam-5960 142 18 ) ) PUNCT ejpam-5960 142 19 , , PUNCT ejpam-5960 142 20 and and CCONJ ejpam-5960 142 21 𝑆(𝐼 𝑆(𝐼 ADJ ejpam-5960 142 22 ) ) PUNCT ejpam-5960 142 23 is be AUX ejpam-5960 142 24 the the DET ejpam-5960 142 25 𝐿–molecular 𝐿–molecular ADJ ejpam-5960 142 26 net net NOUN ejpam-5960 142 27 induced induce VERB ejpam-5960 142 28 by by ADP ejpam-5960 142 29 𝐼. 𝐼. PROPN ejpam-5960 142 30 then then ADV ejpam-5960 142 31 π›Ώπ‘Žπ‘‘β„Ž(𝐼 π›Ώπ‘Žπ‘‘β„Ž(𝐼 PUNCT ejpam-5960 142 32 ) ) PUNCT ejpam-5960 142 33 = = SYM ejpam-5960 142 34 π›Ώπ‘Žπ‘‘β„Ž(𝑆(𝐼 π›Ώπ‘Žπ‘‘β„Ž(𝑆(𝐼 NOUN ejpam-5960 142 35 ) ) PUNCT ejpam-5960 142 36 ) ) PUNCT ejpam-5960 142 37 . . PUNCT ejpam-5960 143 1 3 3 X ejpam-5960 143 2 . . X ejpam-5960 143 3 nearly nearly ADV ejpam-5960 143 4 𝛼–boundedness 𝛼–boundedness PROPN ejpam-5960 143 5 in in ADP ejpam-5960 143 6 𝐿–topological 𝐿–topological ADJ ejpam-5960 143 7 spaces space NOUN ejpam-5960 143 8 in in ADP ejpam-5960 143 9 this this DET ejpam-5960 143 10 section section NOUN ejpam-5960 143 11 , , PUNCT ejpam-5960 143 12 we we PRON ejpam-5960 143 13 introduce introduce VERB ejpam-5960 143 14 the the DET ejpam-5960 143 15 concept concept NOUN ejpam-5960 143 16 of of ADP ejpam-5960 143 17 nearly nearly ADV ejpam-5960 143 18 𝛼–bounded 𝛼–bounde VERB ejpam-5960 143 19 sets set NOUN ejpam-5960 143 20 in in ADP ejpam-5960 143 21 𝐿–topological 𝐿–topological ADJ ejpam-5960 143 22 spaces space NOUN ejpam-5960 143 23 . . PUNCT ejpam-5960 144 1 then then ADV ejpam-5960 144 2 we we PRON ejpam-5960 144 3 obtain obtain VERB ejpam-5960 144 4 several several ADJ ejpam-5960 144 5 characterizations characterization NOUN ejpam-5960 144 6 of of ADP ejpam-5960 144 7 nearly nearly ADV ejpam-5960 144 8 𝛼–bounded 𝛼–bounde VERB ejpam-5960 144 9 sets set NOUN ejpam-5960 144 10 . . PUNCT ejpam-5960 145 1 definition definition NOUN ejpam-5960 145 2 3.1 3.1 NUM ejpam-5960 145 3 . . PUNCT ejpam-5960 146 1 let let VERB ejpam-5960 146 2 ( ( PUNCT ejpam-5960 146 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 146 4 , , PUNCT ejpam-5960 146 5 𝜏 𝜏 NOUN ejpam-5960 146 6 ) ) PUNCT ejpam-5960 146 7 be be VERB ejpam-5960 146 8 an an DET ejpam-5960 146 9 πΏβˆ’ts πΏβˆ’t NOUN ejpam-5960 146 10 , , PUNCT ejpam-5960 146 11 πœ‡ πœ‡ ADP ejpam-5960 146 12 ∈ ∈ ADP ejpam-5960 146 13 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 146 14 and and CCONJ ejpam-5960 146 15 𝛼 𝛼 NOUN ejpam-5960 146 16 ∈ ∈ PROPN ejpam-5960 146 17 𝑀 𝑀 PROPN ejpam-5960 146 18 ( ( PUNCT ejpam-5960 146 19 𝐿 𝐿 PROPN ejpam-5960 146 20 ) ) PUNCT ejpam-5960 146 21 , , PUNCT ejpam-5960 146 22 then then ADV ejpam-5960 146 23 πœ‡ πœ‡ SCONJ ejpam-5960 146 24 ∈ ∈ PROPN ejpam-5960 146 25 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 146 26 is be AUX ejpam-5960 146 27 called call VERB ejpam-5960 146 28 a a DET ejpam-5960 146 29 nearly nearly ADV ejpam-5960 146 30 π›Όβˆ’bounded π›Όβˆ’bounde VERB ejpam-5960 146 31 ( ( PUNCT ejpam-5960 146 32 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 146 33 , , PUNCT ejpam-5960 146 34 for for ADP ejpam-5960 146 35 short short ADJ ejpam-5960 146 36 ) ) PUNCT ejpam-5960 146 37 set set VERB ejpam-5960 146 38 in in ADP ejpam-5960 146 39 ( ( PUNCT ejpam-5960 146 40 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 146 41 , , PUNCT ejpam-5960 146 42 𝜏 𝜏 NOUN ejpam-5960 146 43 ) ) PUNCT ejpam-5960 146 44 iff iff NOUN ejpam-5960 146 45 for for ADP ejpam-5960 146 46 each each DET ejpam-5960 146 47 𝛼 𝛼 NOUN ejpam-5960 146 48 βˆ’ βˆ’ PROPN ejpam-5960 146 49 𝑅𝐹 𝑅𝐹 PROPN ejpam-5960 146 50 ψ ψ PROPN ejpam-5960 146 51 of of ADP ejpam-5960 146 52 1𝑋 1𝑋 NOUN ejpam-5960 146 53 , , PUNCT ejpam-5960 146 54 there there PRON ejpam-5960 146 55 exists exist VERB ejpam-5960 146 56 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 146 57 ∈ ∈ PROPN ejpam-5960 146 58 2(ψ 2(ψ NUM ejpam-5960 146 59 ) ) PUNCT ejpam-5960 146 60 such such ADJ ejpam-5960 146 61 that that SCONJ ejpam-5960 146 62 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 146 63 is be AUX ejpam-5960 146 64 an an DET ejpam-5960 146 65 𝛼 𝛼 NOUN ejpam-5960 146 66 βˆ’ βˆ’ NOUN ejpam-5960 146 67 𝑅𝐢𝑅𝐹 𝑅𝐢𝑅𝐹 PROPN ejpam-5960 146 68 of of ADP ejpam-5960 146 69 πœ‡. πœ‡. NOUN ejpam-5960 146 70 theorem theorem ADJ ejpam-5960 146 71 3.2 3.2 NUM ejpam-5960 146 72 . . PUNCT ejpam-5960 147 1 suppose suppose VERB ejpam-5960 147 2 that that SCONJ ejpam-5960 147 3 𝑓𝐿 𝑓𝐿 NOUN ejpam-5960 147 4 : : PUNCT ejpam-5960 147 5 ( ( PUNCT ejpam-5960 147 6 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 147 7 , , PUNCT ejpam-5960 147 8 𝜏 𝜏 NOUN ejpam-5960 147 9 ) ) PUNCT ejpam-5960 147 10 β†’ β†’ SYM ejpam-5960 147 11 ( ( PUNCT ejpam-5960 147 12 πΏπ‘Œ πΏπ‘Œ PROPN ejpam-5960 147 13 , , PUNCT ejpam-5960 147 14 Ξ΄ Ξ΄ PROPN ejpam-5960 147 15 ) ) PUNCT ejpam-5960 147 16 is be AUX ejpam-5960 147 17 a a DET ejpam-5960 147 18 πΏβˆ’continuous πΏβˆ’continuous ADJ ejpam-5960 147 19 and and CCONJ ejpam-5960 147 20 πœ‡ πœ‡ X ejpam-5960 147 21 ∈ ∈ PROPN ejpam-5960 147 22 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 147 23 is be AUX ejpam-5960 147 24 a a DET ejpam-5960 147 25 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 147 26 πΏβˆ’subset πΏβˆ’subset ADJ ejpam-5960 147 27 in in ADP ejpam-5960 147 28 ( ( PUNCT ejpam-5960 147 29 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 147 30 , , PUNCT ejpam-5960 147 31 𝜏 𝜏 NOUN ejpam-5960 147 32 ) ) PUNCT ejpam-5960 147 33 , , PUNCT ejpam-5960 147 34 then then ADV ejpam-5960 147 35 𝑓𝐿 𝑓𝐿 ADV ejpam-5960 147 36 ( ( PUNCT ejpam-5960 147 37 πœ‡ πœ‡ NOUN ejpam-5960 147 38 ) ) PUNCT ejpam-5960 147 39 is be AUX ejpam-5960 147 40 a a DET ejpam-5960 147 41 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 147 42 πΏβˆ’subset πΏβˆ’subset ADJ ejpam-5960 147 43 in in ADP ejpam-5960 147 44 ( ( PUNCT ejpam-5960 147 45 πΏπ‘Œ πΏπ‘Œ PROPN ejpam-5960 147 46 , , PUNCT ejpam-5960 147 47 Ξ΄ Ξ΄ PROPN ejpam-5960 147 48 ) ) PUNCT ejpam-5960 147 49 . . PUNCT ejpam-5960 148 1 proof proof NOUN ejpam-5960 148 2 . . PUNCT ejpam-5960 149 1 let let VERB ejpam-5960 149 2 πœ‡ πœ‡ PART ejpam-5960 149 3 be be AUX ejpam-5960 149 4 a a DET ejpam-5960 149 5 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 149 6 in in ADP ejpam-5960 149 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 149 8 and and CCONJ ejpam-5960 149 9 let let VERB ejpam-5960 149 10 ψ ψ X ejpam-5960 149 11 βŠ‚ βŠ‚ PRON ejpam-5960 149 12 Ξ΄β€² Ξ΄β€² NOUN ejpam-5960 149 13 be be VERB ejpam-5960 149 14 an an DET ejpam-5960 149 15 𝛼–rf 𝛼–rf NOUN ejpam-5960 149 16 of of ADP ejpam-5960 149 17 1π‘Œ 1π‘Œ NOUN ejpam-5960 149 18 ( ( PUNCT ejpam-5960 149 19 𝛼 𝛼 PROPN ejpam-5960 149 20 ∈ ∈ PROPN ejpam-5960 149 21 𝑀 𝑀 PROPN ejpam-5960 149 22 ( ( PUNCT ejpam-5960 149 23 𝐿 𝐿 PROPN ejpam-5960 149 24 ) ) PUNCT ejpam-5960 149 25 ) ) PUNCT ejpam-5960 149 26 . . PUNCT ejpam-5960 150 1 to to PART ejpam-5960 150 2 begin begin VERB ejpam-5960 150 3 with with ADP ejpam-5960 150 4 , , PUNCT ejpam-5960 150 5 let let VERB ejpam-5960 150 6 us we PRON ejpam-5960 150 7 show show VERB ejpam-5960 150 8 that that SCONJ ejpam-5960 150 9 𝑓 𝑓 DET ejpam-5960 150 10 βˆ’1 βˆ’1 NOUN ejpam-5960 150 11 𝐿 𝐿 PROPN ejpam-5960 150 12 ( ( PUNCT ejpam-5960 150 13 ψ ψ NOUN ejpam-5960 150 14 ) ) PUNCT ejpam-5960 150 15 = = PRON ejpam-5960 150 16 { { PUNCT ejpam-5960 150 17 𝑓 𝑓 DET ejpam-5960 150 18 βˆ’1 βˆ’1 NOUN ejpam-5960 150 19 𝐿 𝐿 PROPN ejpam-5960 150 20 ( ( PUNCT ejpam-5960 150 21 πœ† πœ† NOUN ejpam-5960 150 22 ) ) PUNCT ejpam-5960 150 23 : : PUNCT ejpam-5960 151 1 πœ† πœ† X ejpam-5960 151 2 ∈ ∈ PROPN ejpam-5960 151 3 ψ ψ AUX ejpam-5960 151 4 } } PUNCT ejpam-5960 151 5 is be AUX ejpam-5960 151 6 an an DET ejpam-5960 151 7 𝛼–rf 𝛼–rf NUM ejpam-5960 151 8 of of ADP ejpam-5960 151 9 1𝑋. 1𝑋. NUM ejpam-5960 151 10 since since SCONJ ejpam-5960 151 11 𝑓𝐿 𝑓𝐿 NOUN ejpam-5960 151 12 is be AUX ejpam-5960 151 13 a a DET ejpam-5960 151 14 𝐿–continuous 𝐿–continuous ADJ ejpam-5960 151 15 , , PUNCT ejpam-5960 151 16 then then ADV ejpam-5960 151 17 𝑓 𝑓 DET ejpam-5960 151 18 βˆ’1 βˆ’1 NOUN ejpam-5960 151 19 𝐿 𝐿 PROPN ejpam-5960 151 20 ( ( PUNCT ejpam-5960 151 21 ψ ψ NOUN ejpam-5960 151 22 ) ) PUNCT ejpam-5960 151 23 βŠ‚ βŠ‚ NOUN ejpam-5960 151 24 πœβ€². πœβ€². NOUN ejpam-5960 151 25 let let VERB ejpam-5960 151 26 π‘₯ π‘₯ X ejpam-5960 151 27 ∈ ∈ PROPN ejpam-5960 151 28 𝑋 𝑋 PROPN ejpam-5960 151 29 , , PUNCT ejpam-5960 151 30 then then ADV ejpam-5960 151 31 𝑓𝐿 𝑓𝐿 ADV ejpam-5960 151 32 ( ( PUNCT ejpam-5960 151 33 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 151 34 ) ) PUNCT ejpam-5960 151 35 = = SYM ejpam-5960 152 1 ( ( PUNCT ejpam-5960 152 2 𝑓 𝑓 PRON ejpam-5960 152 3 ( ( PUNCT ejpam-5960 152 4 π‘₯))𝛼 π‘₯))𝛼 NOUN ejpam-5960 152 5 ∈ ∈ PROPN ejpam-5960 152 6 𝑓𝐿 𝑓𝐿 NOUN ejpam-5960 152 7 ( ( PUNCT ejpam-5960 152 8 1𝑋 1𝑋 NOUN ejpam-5960 152 9 ) ) PUNCT ejpam-5960 152 10 and and CCONJ ejpam-5960 152 11 by by ADP ejpam-5960 152 12 ψ ψ X ejpam-5960 152 13 βŠ‚ βŠ‚ X ejpam-5960 152 14 Ξ΄β€² Ξ΄β€² PROPN ejpam-5960 152 15 is be AUX ejpam-5960 152 16 an an DET ejpam-5960 152 17 𝛼–rf 𝛼–rf NUM ejpam-5960 152 18 of of ADP ejpam-5960 152 19 1π‘Œ 1π‘Œ NOUN ejpam-5960 152 20 there there PRON ejpam-5960 152 21 exists exist VERB ejpam-5960 152 22 πœ† πœ† DET ejpam-5960 152 23 ∈ ∈ PROPN ejpam-5960 152 24 ψ ψ NOUN ejpam-5960 152 25 with with ADP ejpam-5960 152 26 πœ† πœ† DET ejpam-5960 152 27 ∈ ∈ PROPN ejpam-5960 152 28 𝑅 𝑅 PROPN ejpam-5960 152 29 ( ( PUNCT ejpam-5960 152 30 𝑓 𝑓 PROPN ejpam-5960 152 31 ( ( PUNCT ejpam-5960 152 32 π‘₯ π‘₯ PROPN ejpam-5960 152 33 ) ) PUNCT ejpam-5960 152 34 ) ) PUNCT ejpam-5960 152 35 𝛼 𝛼 X ejpam-5960 152 36 , , PUNCT ejpam-5960 152 37 i.e i.e X ejpam-5960 152 38 , , PUNCT ejpam-5960 152 39 ( ( PUNCT ejpam-5960 152 40 𝑓 𝑓 PRON ejpam-5960 152 41 ( ( PUNCT ejpam-5960 152 42 π‘₯))𝛼 π‘₯))𝛼 PROPN ejpam-5960 152 43 βˆ‰ βˆ‰ PROPN ejpam-5960 152 44 πœ† πœ† X ejpam-5960 152 45 or or CCONJ ejpam-5960 152 46 , , PUNCT ejpam-5960 152 47 equivalently equivalently ADV ejpam-5960 152 48 , , PUNCT ejpam-5960 152 49 πœ† πœ† X ejpam-5960 152 50 ( ( PUNCT ejpam-5960 152 51 𝑓 𝑓 PROPN ejpam-5960 152 52 ( ( PUNCT ejpam-5960 152 53 π‘₯ π‘₯ NOUN ejpam-5960 152 54 ) ) PUNCT ejpam-5960 152 55 ) ) PUNCT ejpam-5960 153 1 ≱ ≱ PROPN ejpam-5960 154 1 𝛼. 𝛼. NOUN ejpam-5960 154 2 by by ADP ejpam-5960 154 3 the the DET ejpam-5960 154 4 definition definition NOUN ejpam-5960 154 5 of of ADP ejpam-5960 154 6 inverse inverse NOUN ejpam-5960 154 7 mapping mapping NOUN ejpam-5960 154 8 , , PUNCT ejpam-5960 154 9 𝑓 𝑓 DET ejpam-5960 154 10 βˆ’1 βˆ’1 NOUN ejpam-5960 154 11 𝐿 𝐿 PROPN ejpam-5960 154 12 ( ( PUNCT ejpam-5960 154 13 πœ† πœ† NOUN ejpam-5960 154 14 ) ) PUNCT ejpam-5960 154 15 ( ( PUNCT ejpam-5960 154 16 π‘₯ π‘₯ NOUN ejpam-5960 154 17 ) ) PUNCT ejpam-5960 154 18 = = SYM ejpam-5960 155 1 πœ† πœ† X ejpam-5960 155 2 ( ( PUNCT ejpam-5960 155 3 𝑓 𝑓 DET ejpam-5960 155 4 ( ( PUNCT ejpam-5960 155 5 π‘₯ π‘₯ NOUN ejpam-5960 155 6 ) ) PUNCT ejpam-5960 155 7 ) ) PUNCT ejpam-5960 155 8 ≱ ≱ PROPN ejpam-5960 155 9 𝛼 𝛼 VERB ejpam-5960 155 10 , , PUNCT ejpam-5960 155 11 hence hence ADV ejpam-5960 155 12 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 155 13 βˆ‰ βˆ‰ PROPN ejpam-5960 155 14 𝑓 𝑓 DET ejpam-5960 155 15 βˆ’1 βˆ’1 NOUN ejpam-5960 155 16 𝐿 𝐿 PROPN ejpam-5960 155 17 ( ( PUNCT ejpam-5960 155 18 πœ† πœ† NOUN ejpam-5960 155 19 ) ) PUNCT ejpam-5960 155 20 . . PUNCT ejpam-5960 156 1 it it PRON ejpam-5960 156 2 follows follow VERB ejpam-5960 156 3 that that SCONJ ejpam-5960 156 4 𝑓 𝑓 DET ejpam-5960 156 5 βˆ’1 βˆ’1 NOUN ejpam-5960 156 6 𝐿 𝐿 PROPN ejpam-5960 156 7 ( ( PUNCT ejpam-5960 156 8 πœ† πœ† NOUN ejpam-5960 156 9 ) ) PUNCT ejpam-5960 156 10 ∈ ∈ PROPN ejpam-5960 156 11 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 156 12 . . PUNCT ejpam-5960 157 1 therefore therefore ADV ejpam-5960 157 2 𝑓 𝑓 PRON ejpam-5960 157 3 βˆ’1 βˆ’1 NOUN ejpam-5960 157 4 𝐿 𝐿 PROPN ejpam-5960 157 5 ( ( PUNCT ejpam-5960 157 6 ψ ψ NOUN ejpam-5960 157 7 ) ) PUNCT ejpam-5960 157 8 is be AUX ejpam-5960 157 9 an an DET ejpam-5960 157 10 𝛼–rf 𝛼–rf NUM ejpam-5960 157 11 of of ADP ejpam-5960 157 12 1𝑋. 1𝑋. NUM ejpam-5960 157 13 from from ADP ejpam-5960 157 14 the the DET ejpam-5960 157 15 𝑁.𝛼–boundedness 𝑁.𝛼–boundedness PROPN ejpam-5960 157 16 of of ADP ejpam-5960 157 17 πœ‡ πœ‡ ADP ejpam-5960 157 18 there there PRON ejpam-5960 157 19 exists exist VERB ejpam-5960 157 20 ψ ψ VERB ejpam-5960 157 21 β—¦ β—¦ NOUN ejpam-5960 157 22 ∈ ∈ NOUN ejpam-5960 157 23 2(ψ 2(ψ NUM ejpam-5960 157 24 ) ) PUNCT ejpam-5960 157 25 such such ADJ ejpam-5960 157 26 that that SCONJ ejpam-5960 157 27 𝑓 𝑓 DET ejpam-5960 157 28 βˆ’1 βˆ’1 NOUN ejpam-5960 157 29 𝐿 𝐿 PROPN ejpam-5960 157 30 ( ( PUNCT ejpam-5960 157 31 ψ ψ NOUN ejpam-5960 157 32 β—¦ β—¦ NOUN ejpam-5960 157 33 ) ) PUNCT ejpam-5960 157 34 is be AUX ejpam-5960 157 35 an an DET ejpam-5960 157 36 𝛼–rcrf 𝛼–rcrf PROPN ejpam-5960 157 37 of of ADP ejpam-5960 157 38 πœ‡ πœ‡ ADP ejpam-5960 157 39 , , PUNCT ejpam-5960 157 40 that that ADV ejpam-5960 157 41 is is ADV ejpam-5960 157 42 , , PUNCT ejpam-5960 157 43 for for SCONJ ejpam-5960 157 44 each each PRON ejpam-5960 157 45 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 157 46 ∈ ∈ PROPN ejpam-5960 157 47 πœ‡ πœ‡ SCONJ ejpam-5960 157 48 there there PRON ejpam-5960 157 49 exists exist VERB ejpam-5960 157 50 πœ† πœ† DET ejpam-5960 157 51 ∈ ∈ NOUN ejpam-5960 157 52 ψ ψ ADP ejpam-5960 157 53 such such ADJ ejpam-5960 157 54 that that SCONJ ejpam-5960 157 55 𝑓 𝑓 DET ejpam-5960 157 56 βˆ’1 βˆ’1 NOUN ejpam-5960 157 57 𝐿 𝐿 PROPN ejpam-5960 157 58 ( ( PUNCT ejpam-5960 157 59 πœ† πœ† NOUN ejpam-5960 157 60 ) ) PUNCT ejpam-5960 157 61 ∈ ∈ PROPN ejpam-5960 157 62 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 157 63 , , PUNCT ejpam-5960 157 64 i.e. i.e. X ejpam-5960 157 65 , , PUNCT ejpam-5960 157 66 𝑓 𝑓 DET ejpam-5960 157 67 βˆ’1 βˆ’1 NOUN ejpam-5960 157 68 𝐿 𝐿 PROPN ejpam-5960 157 69 ( ( PUNCT ejpam-5960 157 70 πœ† πœ† NOUN ejpam-5960 157 71 ) ) PUNCT ejpam-5960 157 72 ( ( PUNCT ejpam-5960 157 73 π‘₯ π‘₯ X ejpam-5960 157 74 ) ) PUNCT ejpam-5960 157 75 ≱ ≱ PROPN ejpam-5960 157 76 𝛼. 𝛼. NOUN ejpam-5960 157 77 hence hence ADV ejpam-5960 157 78 πœ†(𝑦 πœ†(𝑦 NOUN ejpam-5960 157 79 ) ) PUNCT ejpam-5960 157 80 = = PUNCT ejpam-5960 158 1 πœ† πœ† X ejpam-5960 158 2 ( ( PUNCT ejpam-5960 158 3 𝑓 𝑓 DET ejpam-5960 158 4 ( ( PUNCT ejpam-5960 158 5 π‘₯ π‘₯ NOUN ejpam-5960 158 6 ) ) PUNCT ejpam-5960 158 7 ) ) PUNCT ejpam-5960 158 8 ≱ ≱ PROPN ejpam-5960 158 9 𝛼 𝛼 PROPN ejpam-5960 158 10 and and CCONJ ejpam-5960 158 11 so so ADV ejpam-5960 158 12 for for SCONJ ejpam-5960 158 13 each each PRON ejpam-5960 158 14 𝑦𝛼 𝑦𝛼 PROPN ejpam-5960 158 15 ∈ ∈ PROPN ejpam-5960 158 16 𝑓𝐿 𝑓𝐿 NOUN ejpam-5960 158 17 ( ( PUNCT ejpam-5960 158 18 πœ‡ πœ‡ NOUN ejpam-5960 158 19 ) ) PUNCT ejpam-5960 158 20 , , PUNCT ejpam-5960 158 21 there there PRON ejpam-5960 158 22 exists exist VERB ejpam-5960 158 23 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 158 24 ∈ ∈ PROPN ejpam-5960 158 25 πœ‡ πœ‡ ADP ejpam-5960 158 26 and and CCONJ ejpam-5960 158 27 πœ† πœ† DET ejpam-5960 158 28 ∈ ∈ PROPN ejpam-5960 158 29 ψ ψ ADP ejpam-5960 158 30 β—¦ β—¦ NOUN ejpam-5960 158 31 satisfying satisfy VERB ejpam-5960 158 32 𝑦𝛼 𝑦𝛼 X ejpam-5960 158 33 = = SYM ejpam-5960 158 34 𝑓𝐿 𝑓𝐿 ADV ejpam-5960 158 35 ( ( PUNCT ejpam-5960 158 36 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 158 37 ) ) PUNCT ejpam-5960 158 38 βˆ‰ βˆ‰ PROPN ejpam-5960 158 39 πœ†. πœ†. PROPN ejpam-5960 158 40 hence hence ADV ejpam-5960 158 41 πœ†(𝑦 πœ†(𝑦 NOUN ejpam-5960 158 42 ) ) PUNCT ejpam-5960 158 43 ≱ ≱ PROPN ejpam-5960 158 44 𝛼 𝛼 NOUN ejpam-5960 158 45 , , PUNCT ejpam-5960 158 46 i.e. i.e. X ejpam-5960 158 47 , , PUNCT ejpam-5960 158 48 πœ† πœ† DET ejpam-5960 158 49 ∈ ∈ PROPN ejpam-5960 158 50 𝑅𝑦𝛼 𝑅𝑦𝛼 NOUN ejpam-5960 158 51 . . PUNCT ejpam-5960 159 1 this this PRON ejpam-5960 159 2 implies imply VERB ejpam-5960 159 3 that that SCONJ ejpam-5960 159 4 ψ ψ AUX ejpam-5960 159 5 β—¦ β—¦ NOUN ejpam-5960 159 6 ∈ ∈ NOUN ejpam-5960 159 7 2(ψ 2(ψ NUM ejpam-5960 159 8 ) ) PUNCT ejpam-5960 159 9 is be AUX ejpam-5960 159 10 an an DET ejpam-5960 159 11 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 159 12 of of ADP ejpam-5960 159 13 𝑓𝐿 𝑓𝐿 NOUN ejpam-5960 159 14 ( ( PUNCT ejpam-5960 159 15 πœ‡ πœ‡ NOUN ejpam-5960 159 16 ) ) PUNCT ejpam-5960 159 17 . . PUNCT ejpam-5960 160 1 by by ADP ejpam-5960 160 2 definition definition NOUN ejpam-5960 160 3 3.1 3.1 NUM ejpam-5960 160 4 , , PUNCT ejpam-5960 160 5 we we PRON ejpam-5960 160 6 have have AUX ejpam-5960 160 7 𝑓𝐿 𝑓𝐿 ADV ejpam-5960 160 8 ( ( PUNCT ejpam-5960 160 9 πœ‡ πœ‡ NOUN ejpam-5960 160 10 ) ) PUNCT ejpam-5960 160 11 is be AUX ejpam-5960 160 12 a a DET ejpam-5960 160 13 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 160 14 𝐿–subset 𝐿–subset PROPN ejpam-5960 160 15 in in ADP ejpam-5960 160 16 ( ( PUNCT ejpam-5960 160 17 πΏπ‘Œ πΏπ‘Œ PROPN ejpam-5960 160 18 , , PUNCT ejpam-5960 160 19 Ξ΄ Ξ΄ PROPN ejpam-5960 160 20 ) ) PUNCT ejpam-5960 160 21 . . PUNCT ejpam-5960 161 1 theorem theorem VERB ejpam-5960 161 2 3.3 3.3 NUM ejpam-5960 161 3 . . PUNCT ejpam-5960 162 1 let let VERB ejpam-5960 162 2 ( ( PUNCT ejpam-5960 162 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 162 4 , , PUNCT ejpam-5960 162 5 𝜏 𝜏 NOUN ejpam-5960 162 6 ) ) PUNCT ejpam-5960 162 7 be be VERB ejpam-5960 162 8 an an DET ejpam-5960 162 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 162 10 and and CCONJ ejpam-5960 162 11 𝛼′ 𝛼′ NUM ejpam-5960 162 12 ∈ ∈ PROPN ejpam-5960 162 13 𝑀 𝑀 PROPN ejpam-5960 162 14 ( ( PUNCT ejpam-5960 162 15 𝐿 𝐿 PROPN ejpam-5960 162 16 ) ) PUNCT ejpam-5960 162 17 , , PUNCT ejpam-5960 162 18 then then ADV ejpam-5960 162 19 the the DET ejpam-5960 162 20 set set NOUN ejpam-5960 162 21 πœ‡ πœ‡ ADP ejpam-5960 162 22 ∈ ∈ NOUN ejpam-5960 162 23 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 162 24 is be AUX ejpam-5960 162 25 𝑁𝛼–bounded 𝑁𝛼–bounde VERB ejpam-5960 162 26 iff iff PROPN ejpam-5960 162 27 for for ADP ejpam-5960 162 28 every every DET ejpam-5960 162 29 𝛼–cover 𝛼–cover NOUN ejpam-5960 162 30 ψ ψ ADP ejpam-5960 162 31 βŠ† βŠ† NUM ejpam-5960 162 32 𝜏 𝜏 NOUN ejpam-5960 162 33 of of ADP ejpam-5960 162 34 1𝑋 1𝑋 NOUN ejpam-5960 162 35 there there PRON ejpam-5960 162 36 exists exist VERB ejpam-5960 162 37 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 162 38 ∈ ∈ PROPN ejpam-5960 162 39 2(ψ 2(ψ NUM ejpam-5960 162 40 ) ) PUNCT ejpam-5960 162 41 such such ADJ ejpam-5960 162 42 that that SCONJ ejpam-5960 162 43 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 162 44 is be AUX ejpam-5960 162 45 a a DET ejpam-5960 162 46 nearly nearly ADV ejpam-5960 162 47 𝛼–cover 𝛼–cover NOUN ejpam-5960 162 48 of of ADP ejpam-5960 162 49 πœ‡. πœ‡. NOUN ejpam-5960 162 50 proof proof NOUN ejpam-5960 162 51 . . PUNCT ejpam-5960 163 1 let let VERB ejpam-5960 163 2 πœ‡ πœ‡ SCONJ ejpam-5960 163 3 ∈ ∈ PROPN ejpam-5960 163 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 163 5 be be AUX ejpam-5960 163 6 a a DET ejpam-5960 163 7 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 163 8 set set NOUN ejpam-5960 163 9 and and CCONJ ejpam-5960 163 10 let let VERB ejpam-5960 163 11 ψ ψ PRON ejpam-5960 163 12 βŠ† βŠ† NUM ejpam-5960 163 13 𝜏 𝜏 NOUN ejpam-5960 163 14 is be AUX ejpam-5960 163 15 any any DET ejpam-5960 163 16 𝛼–cover 𝛼–cover NOUN ejpam-5960 163 17 of of ADP ejpam-5960 163 18 1𝑋. 1𝑋. NUM ejpam-5960 163 19 let let VERB ejpam-5960 163 20 πœ‘ πœ‘ NOUN ejpam-5960 163 21 = = PUNCT ejpam-5960 163 22 Οˆβ€² Οˆβ€² PROPN ejpam-5960 163 23 = = NOUN ejpam-5960 163 24 { { PUNCT ejpam-5960 163 25 πœ†β€² πœ†β€² NOUN ejpam-5960 163 26 : : PUNCT ejpam-5960 163 27 πœ† πœ† X ejpam-5960 163 28 ∈ ∈ PROPN ejpam-5960 163 29 ψ ψ AUX ejpam-5960 163 30 } } PUNCT ejpam-5960 163 31 and and CCONJ ejpam-5960 163 32 let let VERB ejpam-5960 163 33 𝛾 𝛾 AUX ejpam-5960 163 34 = = SYM ejpam-5960 163 35 𝛼′. 𝛼′. ADP ejpam-5960 163 36 one one PRON ejpam-5960 163 37 can can AUX ejpam-5960 163 38 see see VERB ejpam-5960 163 39 that that SCONJ ejpam-5960 163 40 πœ‘ πœ‘ PROPN ejpam-5960 163 41 is be AUX ejpam-5960 163 42 an an DET ejpam-5960 163 43 𝛾–rf 𝛾–rf NOUN ejpam-5960 163 44 of of ADP ejpam-5960 163 45 1𝑋. 1𝑋. NUM ejpam-5960 163 46 since since SCONJ ejpam-5960 163 47 1𝑋 1𝑋 PROPN ejpam-5960 163 48 ( ( PUNCT ejpam-5960 163 49 π‘₯ π‘₯ NOUN ejpam-5960 163 50 ) ) PUNCT ejpam-5960 163 51 β‰₯ β‰₯ NOUN ejpam-5960 163 52 𝛾 𝛾 NOUN ejpam-5960 163 53 for for ADP ejpam-5960 163 54 each each DET ejpam-5960 163 55 π‘₯𝛾 π‘₯𝛾 CCONJ ejpam-5960 163 56 ∈ ∈ PROPN ejpam-5960 163 57 1𝑋 1𝑋 NOUN ejpam-5960 163 58 , , PUNCT ejpam-5960 163 59 i.e. i.e. X ejpam-5960 163 60 , , PUNCT ejpam-5960 163 61 π‘₯ π‘₯ DET ejpam-5960 163 62 ∈ ∈ PROPN ejpam-5960 163 63 𝑋 𝑋 NOUN ejpam-5960 163 64 for for ADP ejpam-5960 163 65 each each DET ejpam-5960 163 66 π‘₯𝛾 π‘₯𝛾 CCONJ ejpam-5960 163 67 ∈ ∈ PROPN ejpam-5960 163 68 1𝑋 1𝑋 NOUN ejpam-5960 163 69 , , PUNCT ejpam-5960 163 70 there there PRON ejpam-5960 163 71 exists exist VERB ejpam-5960 163 72 πœ† πœ† DET ejpam-5960 163 73 ∈ ∈ PROPN ejpam-5960 163 74 ψ ψ X ejpam-5960 163 75 satisfying satisfy VERB ejpam-5960 163 76 πœ†(π‘₯ πœ†(π‘₯ NOUN ejpam-5960 163 77 ) ) PUNCT ejpam-5960 163 78 ≱ ≱ PROPN ejpam-5960 163 79 𝛼 𝛼 NOUN ejpam-5960 163 80 = = SYM ejpam-5960 163 81 𝛾′ 𝛾′ PROPN ejpam-5960 163 82 , , PUNCT ejpam-5960 163 83 this this PRON ejpam-5960 163 84 equivalently equivalently ADV ejpam-5960 163 85 , , PUNCT ejpam-5960 163 86 there there PRON ejpam-5960 163 87 exists exist VERB ejpam-5960 163 88 πœ†β€² πœ†β€² NOUN ejpam-5960 163 89 ∈ ∈ NOUN ejpam-5960 163 90 πœ‘ πœ‘ NOUN ejpam-5960 163 91 with with ADP ejpam-5960 163 92 𝛾 𝛾 ADP ejpam-5960 163 93 β‰° β‰° PROPN ejpam-5960 163 94 πœ†β€²(π‘₯ πœ†β€²(π‘₯ PUNCT ejpam-5960 163 95 ) ) PUNCT ejpam-5960 163 96 , , PUNCT ejpam-5960 163 97 and and CCONJ ejpam-5960 163 98 so so ADV ejpam-5960 163 99 πœ†β€² πœ†β€² PUNCT ejpam-5960 163 100 ∈ ∈ PROPN ejpam-5960 163 101 𝑅π‘₯𝛾 𝑅π‘₯𝛾 PROPN ejpam-5960 163 102 . . PUNCT ejpam-5960 164 1 this this PRON ejpam-5960 164 2 implies imply VERB ejpam-5960 164 3 that that SCONJ ejpam-5960 164 4 πœ‘ πœ‘ PROPN ejpam-5960 164 5 is be AUX ejpam-5960 164 6 an an DET ejpam-5960 164 7 𝛾–rf 𝛾–rf NOUN ejpam-5960 164 8 of of ADP ejpam-5960 164 9 1𝑋. 1𝑋. NUM ejpam-5960 164 10 being be AUX ejpam-5960 164 11 πœ‡ πœ‡ ADV ejpam-5960 164 12 is be AUX ejpam-5960 164 13 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 164 14 , , PUNCT ejpam-5960 164 15 then then ADV ejpam-5960 164 16 there there PRON ejpam-5960 164 17 exists exist VERB ejpam-5960 164 18 πœ‘ πœ‘ ADP ejpam-5960 164 19 β—¦ β—¦ NOUN ejpam-5960 164 20 ∈ ∈ PRON ejpam-5960 164 21 2(πœ‘ 2(πœ‘ NUM ejpam-5960 164 22 ) ) PUNCT ejpam-5960 164 23 such such ADJ ejpam-5960 164 24 that that SCONJ ejpam-5960 164 25 πœ‘ πœ‘ X ejpam-5960 164 26 β—¦ β—¦ NOUN ejpam-5960 164 27 is be AUX ejpam-5960 164 28 an an DET ejpam-5960 164 29 𝛾–rcrf 𝛾–rcrf NOUN ejpam-5960 164 30 of of ADP ejpam-5960 164 31 πœ‡. πœ‡. NOUN ejpam-5960 164 32 we we PRON ejpam-5960 164 33 assert assert VERB ejpam-5960 164 34 πœ‘β€² πœ‘β€² NOUN ejpam-5960 164 35 β—¦ β—¦ NOUN ejpam-5960 164 36 ∈ ∈ PROPN ejpam-5960 164 37 2(ψ 2(ψ NUM ejpam-5960 164 38 ) ) PUNCT ejpam-5960 164 39 is be AUX ejpam-5960 164 40 a a DET ejpam-5960 164 41 nearly nearly ADV ejpam-5960 164 42 𝛼–cover 𝛼–cover NOUN ejpam-5960 164 43 of of ADP ejpam-5960 164 44 πœ‡. πœ‡. NOUN ejpam-5960 164 45 in in ADP ejpam-5960 164 46 fact fact NOUN ejpam-5960 164 47 , , PUNCT ejpam-5960 164 48 for for SCONJ ejpam-5960 164 49 each each DET ejpam-5960 164 50 π‘₯𝛾 π‘₯𝛾 ADP ejpam-5960 164 51 ∈ ∈ PROPN ejpam-5960 164 52 πœ‡ πœ‡ SCONJ ejpam-5960 164 53 there there PRON ejpam-5960 164 54 is be VERB ejpam-5960 164 55 πœ†β€² πœ†β€² NUM ejpam-5960 164 56 ∈ ∈ NOUN ejpam-5960 164 57 πœ‘ πœ‘ DET ejpam-5960 164 58 β—¦ β—¦ NOUN ejpam-5960 164 59 satisfying satisfy VERB ejpam-5960 164 60 𝑐𝑙 𝑐𝑙 ADP ejpam-5960 164 61 ( ( PUNCT ejpam-5960 164 62 int(πœ†β€² int(πœ†β€² PROPN ejpam-5960 164 63 ) ) PUNCT ejpam-5960 164 64 ) ) PUNCT ejpam-5960 165 1 ∈ ∈ PROPN ejpam-5960 165 2 𝑅π‘₯𝛾 𝑅π‘₯𝛾 PROPN ejpam-5960 165 3 , , PUNCT ejpam-5960 165 4 that that PRON ejpam-5960 165 5 is be AUX ejpam-5960 165 6 𝛾 𝛾 ADP ejpam-5960 165 7 β‰° β‰° PROPN ejpam-5960 165 8 𝑐𝑙 𝑐𝑙 ADJ ejpam-5960 165 9 ( ( PUNCT ejpam-5960 165 10 int(πœ†β€²(π‘₯ int(πœ†β€²(π‘₯ NOUN ejpam-5960 165 11 ) ) PUNCT ejpam-5960 165 12 ) ) PUNCT ejpam-5960 165 13 ) ) PUNCT ejpam-5960 165 14 , , PUNCT ejpam-5960 165 15 equivalently equivalently ADV ejpam-5960 165 16 , , PUNCT ejpam-5960 165 17 for for ADP ejpam-5960 165 18 each each DET ejpam-5960 165 19 π‘₯ π‘₯ PROPN ejpam-5960 165 20 ∈ ∈ PROPN ejpam-5960 165 21 πœ‡π‘€π›Ύ πœ‡π‘€π›Ύ NOUN ejpam-5960 165 22 we we PRON ejpam-5960 165 23 have have VERB ejpam-5960 165 24 πœ† πœ† DET ejpam-5960 165 25 ∈ ∈ PROPN ejpam-5960 165 26 πœ‘β€² πœ‘β€² NOUN ejpam-5960 165 27 β—¦ β—¦ NOUN ejpam-5960 165 28 with with ADP ejpam-5960 165 29 int(𝑐𝑙 int(𝑐𝑙 NOUN ejpam-5960 165 30 ( ( PUNCT ejpam-5960 165 31 πœ†(π‘₯ πœ†(π‘₯ NUM ejpam-5960 165 32 ) ) PUNCT ejpam-5960 165 33 ) ) PUNCT ejpam-5960 165 34 ) ) PUNCT ejpam-5960 166 1 = = PRON ejpam-5960 166 2 ( ( PUNCT ejpam-5960 166 3 𝑐𝑙 𝑐𝑙 X ejpam-5960 166 4 ( ( PUNCT ejpam-5960 166 5 int(πœ†β€² int(πœ†β€² PROPN ejpam-5960 166 6 ) ) PUNCT ejpam-5960 166 7 ) ) PUNCT ejpam-5960 167 1 ( ( PUNCT ejpam-5960 167 2 π‘₯))β€² π‘₯))β€² X ejpam-5960 167 3 β‰° β‰° PROPN ejpam-5960 167 4 𝛼. 𝛼. NOUN ejpam-5960 167 5 therefore therefore ADV ejpam-5960 167 6 , , PUNCT ejpam-5960 167 7 πœ‘β€² πœ‘β€² NOUN ejpam-5960 167 8 β—¦ β—¦ NOUN ejpam-5960 167 9 is be AUX ejpam-5960 167 10 a a DET ejpam-5960 167 11 nearly nearly ADV ejpam-5960 167 12 𝛼–cover 𝛼–cover NOUN ejpam-5960 167 13 of of ADP ejpam-5960 167 14 πœ‡. πœ‡. NOUN ejpam-5960 167 15 n. n. PROPN ejpam-5960 167 16 a. a. PROPN ejpam-5960 167 17 alsaedi alsaedi PROPN ejpam-5960 167 18 / / SYM ejpam-5960 167 19 eur eur PROPN ejpam-5960 167 20 . . PUNCT ejpam-5960 168 1 j. j. PROPN ejpam-5960 168 2 pure pure PROPN ejpam-5960 168 3 appl appl PROPN ejpam-5960 168 4 . . PROPN ejpam-5960 168 5 math math PROPN ejpam-5960 168 6 , , PUNCT ejpam-5960 168 7 18 18 NUM ejpam-5960 168 8 ( ( PUNCT ejpam-5960 168 9 4 4 NUM ejpam-5960 168 10 ) ) PUNCT ejpam-5960 168 11 ( ( PUNCT ejpam-5960 168 12 2025 2025 NUM ejpam-5960 168 13 ) ) PUNCT ejpam-5960 168 14 , , PUNCT ejpam-5960 168 15 5960 5960 NUM ejpam-5960 168 16 7 7 NUM ejpam-5960 168 17 of of ADP ejpam-5960 168 18 22 22 NUM ejpam-5960 168 19 conversely conversely ADV ejpam-5960 168 20 , , PUNCT ejpam-5960 168 21 suppose suppose VERB ejpam-5960 168 22 that that SCONJ ejpam-5960 168 23 the the DET ejpam-5960 168 24 condition condition NOUN ejpam-5960 168 25 is be AUX ejpam-5960 168 26 satisfied satisfied ADJ ejpam-5960 168 27 and and CCONJ ejpam-5960 168 28 let let VERB ejpam-5960 168 29 that that PRON ejpam-5960 168 30 ψ ψ NOUN ejpam-5960 168 31 is be AUX ejpam-5960 168 32 an an DET ejpam-5960 168 33 𝛾–rf 𝛾–rf NOUN ejpam-5960 168 34 of of ADP ejpam-5960 168 35 1𝑋. 1𝑋. NUM ejpam-5960 168 36 put put VERB ejpam-5960 168 37 Οˆβ€² Οˆβ€² PUNCT ejpam-5960 168 38 = = SYM ejpam-5960 168 39 πœ‘ πœ‘ PROPN ejpam-5960 168 40 and and CCONJ ejpam-5960 168 41 𝛾′ 𝛾′ NUM ejpam-5960 168 42 = = SYM ejpam-5960 168 43 𝛼 𝛼 PROPN ejpam-5960 168 44 , , PUNCT ejpam-5960 168 45 then then ADV ejpam-5960 168 46 πœ‘ πœ‘ PROPN ejpam-5960 168 47 is be AUX ejpam-5960 168 48 an an DET ejpam-5960 168 49 𝛼–cover 𝛼–cover NOUN ejpam-5960 168 50 of of ADP ejpam-5960 168 51 1𝑋 1𝑋 NOUN ejpam-5960 168 52 , , PUNCT ejpam-5960 168 53 and and CCONJ ejpam-5960 168 54 then then ADV ejpam-5960 168 55 there there PRON ejpam-5960 168 56 exists exist VERB ejpam-5960 168 57 πœ‘ πœ‘ ADP ejpam-5960 168 58 β—¦ β—¦ NOUN ejpam-5960 168 59 ∈ ∈ PRON ejpam-5960 168 60 2(πœ‘ 2(πœ‘ NUM ejpam-5960 168 61 ) ) PUNCT ejpam-5960 168 62 such such ADJ ejpam-5960 168 63 that that SCONJ ejpam-5960 168 64 πœ‘ πœ‘ X ejpam-5960 168 65 β—¦ β—¦ NOUN ejpam-5960 168 66 is be AUX ejpam-5960 168 67 a a DET ejpam-5960 168 68 nearly nearly ADV ejpam-5960 168 69 𝛼–cover 𝛼–cover NOUN ejpam-5960 168 70 of of ADP ejpam-5960 168 71 πœ‡. πœ‡. NOUN ejpam-5960 168 72 evidently evidently ADV ejpam-5960 168 73 , , PUNCT ejpam-5960 168 74 πœ‘β€² πœ‘β€² PROPN ejpam-5960 168 75 β—¦ β—¦ NOUN ejpam-5960 168 76 ∈ ∈ PROPN ejpam-5960 168 77 2(ψ 2(ψ NUM ejpam-5960 168 78 ) ) PUNCT ejpam-5960 168 79 is be AUX ejpam-5960 168 80 an an DET ejpam-5960 168 81 𝛾–rcrf 𝛾–rcrf PROPN ejpam-5960 168 82 of of ADP ejpam-5960 168 83 πœ‡. πœ‡. NOUN ejpam-5960 168 84 hence hence ADV ejpam-5960 168 85 πœ‡ πœ‡ ADV ejpam-5960 168 86 is be AUX ejpam-5960 168 87 a a DET ejpam-5960 168 88 𝑁.𝛼–bounded 𝑁.𝛼–bounded ADJ ejpam-5960 168 89 . . PUNCT ejpam-5960 168 90 theorem theorem VERB ejpam-5960 168 91 3.4 3.4 NUM ejpam-5960 168 92 . . PUNCT ejpam-5960 169 1 let let VERB ejpam-5960 169 2 ( ( PUNCT ejpam-5960 169 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 169 4 , , PUNCT ejpam-5960 169 5 𝜏 𝜏 NOUN ejpam-5960 169 6 ) ) PUNCT ejpam-5960 169 7 be be VERB ejpam-5960 169 8 an an DET ejpam-5960 169 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 169 10 , , PUNCT ejpam-5960 169 11 πœ‡ πœ‡ ADP ejpam-5960 169 12 ∈ ∈ PROPN ejpam-5960 169 13 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 169 14 is be AUX ejpam-5960 169 15 a a DET ejpam-5960 169 16 𝛼–bounded 𝛼–bounde VERB ejpam-5960 169 17 [ [ X ejpam-5960 169 18 23 23 NUM ejpam-5960 169 19 ] ] PUNCT ejpam-5960 169 20 , , PUNCT ejpam-5960 169 21 then then ADV ejpam-5960 169 22 πœ‡ πœ‡ SCONJ ejpam-5960 169 23 is be AUX ejpam-5960 169 24 a a DET ejpam-5960 169 25 𝑁𝛼–bounded 𝑁𝛼–bounded PROPN ejpam-5960 169 26 . . PUNCT ejpam-5960 170 1 proof proof NOUN ejpam-5960 170 2 it it PRON ejpam-5960 170 3 follows follow VERB ejpam-5960 170 4 directly directly ADV ejpam-5960 170 5 from from ADP ejpam-5960 170 6 the the DET ejpam-5960 170 7 fact fact NOUN ejpam-5960 170 8 that that SCONJ ejpam-5960 170 9 𝑅𝐢 𝑅𝐢 PROPN ejpam-5960 170 10 ( ( PUNCT ejpam-5960 170 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 170 12 , , PUNCT ejpam-5960 170 13 𝜏 𝜏 NOUN ejpam-5960 170 14 ) ) PUNCT ejpam-5960 170 15 βŠ† βŠ† NUM ejpam-5960 170 16 πœβ€². πœβ€². ADP ejpam-5960 170 17 the the DET ejpam-5960 170 18 following follow VERB ejpam-5960 170 19 example example NOUN ejpam-5960 170 20 shows show VERB ejpam-5960 170 21 that that SCONJ ejpam-5960 170 22 the the DET ejpam-5960 170 23 converse converse NOUN ejpam-5960 170 24 is be AUX ejpam-5960 170 25 not not PART ejpam-5960 170 26 true true ADJ ejpam-5960 170 27 in in ADP ejpam-5960 170 28 general general ADJ ejpam-5960 170 29 . . PUNCT ejpam-5960 171 1 example example NOUN ejpam-5960 171 2 3.5 3.5 NUM ejpam-5960 171 3 . . PUNCT ejpam-5960 172 1 let let VERB ejpam-5960 172 2 𝐿 𝐿 PROPN ejpam-5960 172 3 = = SYM ejpam-5960 173 1 [ [ X ejpam-5960 173 2 0 0 NUM ejpam-5960 173 3 , , PUNCT ejpam-5960 173 4 1 1 NUM ejpam-5960 173 5 ] ] PUNCT ejpam-5960 173 6 , , PUNCT ejpam-5960 173 7 𝑋 𝑋 NOUN ejpam-5960 173 8 = = SYM ejpam-5960 173 9 𝑁 𝑁 PROPN ejpam-5960 173 10 and and CCONJ ejpam-5960 173 11 let let VERB ejpam-5960 173 12 𝜏 𝜏 NOUN ejpam-5960 173 13 = = SYM ejpam-5960 173 14 { { PUNCT ejpam-5960 173 15 0𝑋 0𝑋 PROPN ejpam-5960 173 16 , , PUNCT ejpam-5960 173 17 π‘₯.5 π‘₯.5 ADV ejpam-5960 173 18 , , PUNCT ejpam-5960 173 19 1𝑋 1𝑋 PROPN ejpam-5960 173 20 } } PUNCT ejpam-5960 173 21 . . PUNCT ejpam-5960 174 1 then then ADV ejpam-5960 174 2 ( ( PUNCT ejpam-5960 174 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 174 4 , , PUNCT ejpam-5960 174 5 𝜏 𝜏 NOUN ejpam-5960 174 6 ) ) PUNCT ejpam-5960 174 7 is be AUX ejpam-5960 174 8 𝐿–ts 𝐿–ts NOUN ejpam-5960 174 9 . . PUNCT ejpam-5960 175 1 firstly firstly ADV ejpam-5960 175 2 , , PUNCT ejpam-5960 175 3 we we PRON ejpam-5960 175 4 show show VERB ejpam-5960 175 5 that that SCONJ ejpam-5960 175 6 1𝑋 1𝑋 PROPN ejpam-5960 175 7 is be AUX ejpam-5960 175 8 not not PART ejpam-5960 175 9 𝛼–bounded 𝛼–bounde VERB ejpam-5960 175 10 set set NOUN ejpam-5960 175 11 . . PUNCT ejpam-5960 176 1 in in ADP ejpam-5960 176 2 fact fact NOUN ejpam-5960 176 3 , , PUNCT ejpam-5960 176 4 we we PRON ejpam-5960 176 5 suppose suppose VERB ejpam-5960 176 6 that that SCONJ ejpam-5960 176 7 constant constant ADJ ejpam-5960 176 8 0.5 0.5 NUM ejpam-5960 176 9 – – PUNCT ejpam-5960 176 10 net net ADJ ejpam-5960 176 11 𝑆 𝑆 PROPN ejpam-5960 176 12 = = SYM ejpam-5960 176 13 { { PUNCT ejpam-5960 176 14 π‘₯.5 π‘₯.5 X ejpam-5960 176 15 : : PUNCT ejpam-5960 176 16 π‘₯ π‘₯ X ejpam-5960 176 17 ∈ ∈ PROPN ejpam-5960 176 18 𝑋 𝑋 PROPN ejpam-5960 176 19 } } PUNCT ejpam-5960 176 20 in in ADP ejpam-5960 176 21 1𝑋 1𝑋 NOUN ejpam-5960 176 22 , , PUNCT ejpam-5960 176 23 let let VERB ejpam-5960 176 24 𝑦0.3 𝑦0.3 PROPN ejpam-5960 176 25 ∈ ∈ PROPN ejpam-5960 176 26 𝑀 𝑀 PROPN ejpam-5960 176 27 ( ( PUNCT ejpam-5960 176 28 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 176 29 ) ) PUNCT ejpam-5960 176 30 , , PUNCT ejpam-5960 176 31 π‘₯ π‘₯ PROPN ejpam-5960 176 32 β‰  β‰  PROPN ejpam-5960 176 33 𝑦 𝑦 NOUN ejpam-5960 176 34 , , PUNCT ejpam-5960 176 35 then then ADV ejpam-5960 176 36 𝑅𝑦0.3 𝑅𝑦0.3 PRON ejpam-5960 176 37 = = PUNCT ejpam-5960 176 38 { { PUNCT ejpam-5960 176 39 0𝑋 0𝑋 PROPN ejpam-5960 176 40 , , PUNCT ejpam-5960 176 41 π‘₯0.5 π‘₯0.5 NOUN ejpam-5960 176 42 } } PUNCT ejpam-5960 176 43 . . PUNCT ejpam-5960 177 1 since since SCONJ ejpam-5960 177 2 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 177 3 ) ) PUNCT ejpam-5960 177 4 = = PUNCT ejpam-5960 178 1 π‘₯0.5 π‘₯0.5 NUM ejpam-5960 178 2 ≀ ≀ NUM ejpam-5960 178 3 π‘₯0.5 π‘₯0.5 NUM ejpam-5960 178 4 ∈ ∈ NOUN ejpam-5960 178 5 𝑅𝑦0.3 𝑅𝑦0.3 X ejpam-5960 178 6 . . PUNCT ejpam-5960 179 1 so so ADV ejpam-5960 179 2 𝑦0.3 𝑦0.3 PROPN ejpam-5960 179 3 is be AUX ejpam-5960 179 4 not not PART ejpam-5960 179 5 cluster cluster NOUN ejpam-5960 179 6 point point NOUN ejpam-5960 179 7 of of ADP ejpam-5960 179 8 𝑆 𝑆 PROPN ejpam-5960 179 9 in in ADP ejpam-5960 179 10 1𝑋. 1𝑋. NUM ejpam-5960 179 11 on on ADP ejpam-5960 179 12 account account NOUN ejpam-5960 179 13 of of ADP ejpam-5960 179 14 the the DET ejpam-5960 179 15 arbitrariness arbitrariness NOUN ejpam-5960 179 16 of of ADP ejpam-5960 179 17 𝑦 𝑦 PRON ejpam-5960 179 18 it it PRON ejpam-5960 179 19 follows follow VERB ejpam-5960 179 20 that that SCONJ ejpam-5960 179 21 the the DET ejpam-5960 179 22 𝑆 𝑆 PROPN ejpam-5960 179 23 has have VERB ejpam-5960 179 24 no no DET ejpam-5960 179 25 cluster cluster NOUN ejpam-5960 179 26 point point NOUN ejpam-5960 179 27 in in ADP ejpam-5960 179 28 1𝑋 1𝑋 NOUN ejpam-5960 179 29 with with ADP ejpam-5960 179 30 height height NOUN ejpam-5960 179 31 0.3 0.3 NUM ejpam-5960 179 32 . . PUNCT ejpam-5960 180 1 thus thus ADV ejpam-5960 180 2 πœ‡ πœ‡ PRON ejpam-5960 180 3 is be AUX ejpam-5960 180 4 not not PART ejpam-5960 180 5 𝛼–bounded 𝛼–bounde VERB ejpam-5960 180 6 set set NOUN ejpam-5960 180 7 . . PUNCT ejpam-5960 181 1 now now ADV ejpam-5960 181 2 , , PUNCT ejpam-5960 181 3 we we PRON ejpam-5960 181 4 show show VERB ejpam-5960 181 5 that that SCONJ ejpam-5960 181 6 1𝑋 1𝑋 PROPN ejpam-5960 181 7 is be AUX ejpam-5960 181 8 𝑁𝛼–bounded 𝑁𝛼–bounde VERB ejpam-5960 181 9 set set NOUN ejpam-5960 181 10 . . PUNCT ejpam-5960 182 1 let let VERB ejpam-5960 182 2 𝑆 𝑆 PROPN ejpam-5960 182 3 = = PRON ejpam-5960 182 4 { { PUNCT ejpam-5960 182 5 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 182 6 : : PUNCT ejpam-5960 182 7 π‘₯ π‘₯ PROPN ejpam-5960 182 8 ∈ ∈ PROPN ejpam-5960 182 9 𝑋 𝑋 PROPN ejpam-5960 182 10 , , PUNCT ejpam-5960 182 11 𝛼 𝛼 PROPN ejpam-5960 182 12 ∈ ∈ PROPN ejpam-5960 182 13 𝐿 𝐿 PROPN ejpam-5960 182 14 } } PUNCT ejpam-5960 182 15 is be AUX ejpam-5960 182 16 any any DET ejpam-5960 182 17 constant constant ADJ ejpam-5960 182 18 𝛼–net 𝛼–net NUM ejpam-5960 182 19 in in ADP ejpam-5960 182 20 1𝑋. 1𝑋. NUM ejpam-5960 182 21 ( ( PUNCT ejpam-5960 182 22 i i NOUN ejpam-5960 182 23 ) ) PUNCT ejpam-5960 182 24 if if SCONJ ejpam-5960 182 25 𝛼 𝛼 X ejpam-5960 182 26 > > X ejpam-5960 182 27 0.5 0.5 NUM ejpam-5960 182 28 then then ADV ejpam-5960 182 29 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 182 30 = = SYM ejpam-5960 182 31 { { PUNCT ejpam-5960 182 32 0𝑋 0𝑋 PROPN ejpam-5960 182 33 , , PUNCT ejpam-5960 182 34 π‘₯0.5 π‘₯0.5 NOUN ejpam-5960 182 35 } } PUNCT ejpam-5960 182 36 , , PUNCT ejpam-5960 182 37 where where SCONJ ejpam-5960 182 38 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 182 39 ( ( PUNCT ejpam-5960 182 40 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 182 41 ( ( PUNCT ejpam-5960 182 42 π‘₯0.5 π‘₯0.5 NOUN ejpam-5960 182 43 ) ) PUNCT ejpam-5960 182 44 ) ) PUNCT ejpam-5960 183 1 = = PUNCT ejpam-5960 184 1 π‘₯0.5 π‘₯0.5 NOUN ejpam-5960 184 2 . . NOUN ejpam-5960 184 3 since since SCONJ ejpam-5960 184 4 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 184 5 ) ) PUNCT ejpam-5960 184 6 = = PUNCT ejpam-5960 184 7 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 184 8 βˆ‰ βˆ‰ PROPN ejpam-5960 184 9 0𝑋 0𝑋 PROPN ejpam-5960 184 10 βˆ€ βˆ€ X ejpam-5960 184 11 𝑛 𝑛 PRON ejpam-5960 184 12 ∈ ∈ PROPN ejpam-5960 184 13 𝑁 𝑁 PROPN ejpam-5960 184 14 , , PUNCT ejpam-5960 184 15 βˆ€π›Ό βˆ€π›Ό NOUN ejpam-5960 184 16 > > X ejpam-5960 184 17 0.5 0.5 NUM ejpam-5960 184 18 . . PUNCT ejpam-5960 185 1 hence hence ADV ejpam-5960 185 2 , , PUNCT ejpam-5960 185 3 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 185 4 is be AUX ejpam-5960 185 5 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 185 6 point point NOUN ejpam-5960 185 7 of of ADP ejpam-5960 185 8 𝑆 𝑆 PROPN ejpam-5960 185 9 with with ADP ejpam-5960 185 10 height height NOUN ejpam-5960 185 11 𝛼 𝛼 NOUN ejpam-5960 185 12 in in ADP ejpam-5960 185 13 1𝑋. 1𝑋. NUM ejpam-5960 185 14 ( ( PUNCT ejpam-5960 185 15 ii ii NOUN ejpam-5960 185 16 ) ) PUNCT ejpam-5960 185 17 if if SCONJ ejpam-5960 185 18 𝛼 𝛼 PRON ejpam-5960 185 19 ≀ ≀ NUM ejpam-5960 185 20 0.5 0.5 NUM ejpam-5960 185 21 , , PUNCT ejpam-5960 185 22 then then ADV ejpam-5960 185 23 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 185 24 = = SYM ejpam-5960 185 25 { { PUNCT ejpam-5960 185 26 0𝑋 0𝑋 PROPN ejpam-5960 185 27 } } PUNCT ejpam-5960 185 28 where where SCONJ ejpam-5960 185 29 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 185 30 ( ( PUNCT ejpam-5960 185 31 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 185 32 ( ( PUNCT ejpam-5960 185 33 0𝑋 0𝑋 PROPN ejpam-5960 185 34 ) ) PUNCT ejpam-5960 185 35 ) ) PUNCT ejpam-5960 186 1 = = PUNCT ejpam-5960 186 2 0𝑋. 0𝑋. NOUN ejpam-5960 186 3 since since SCONJ ejpam-5960 186 4 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 186 5 ) ) PUNCT ejpam-5960 186 6 = = PUNCT ejpam-5960 186 7 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 186 8 βˆ‰ βˆ‰ PROPN ejpam-5960 186 9 0𝑋 0𝑋 PROPN ejpam-5960 186 10 βˆ€ βˆ€ X ejpam-5960 186 11 𝑛 𝑛 PRON ejpam-5960 186 12 ∈ ∈ PROPN ejpam-5960 186 13 𝑁 𝑁 PROPN ejpam-5960 186 14 , , PUNCT ejpam-5960 186 15 βˆ€π›Ό βˆ€π›Ό NOUN ejpam-5960 186 16 ≀ ≀ NUM ejpam-5960 186 17 0.5 0.5 NUM ejpam-5960 186 18 . . PUNCT ejpam-5960 187 1 so so ADV ejpam-5960 187 2 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 187 3 is be AUX ejpam-5960 187 4 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 187 5 point point NOUN ejpam-5960 187 6 of of ADP ejpam-5960 187 7 𝑆 𝑆 PROPN ejpam-5960 187 8 with with ADP ejpam-5960 187 9 height height NOUN ejpam-5960 187 10 𝛼 𝛼 NOUN ejpam-5960 187 11 in in ADP ejpam-5960 187 12 1𝑋. 1𝑋. NUM ejpam-5960 187 13 thus thus ADV ejpam-5960 187 14 1𝑋 1𝑋 PROPN ejpam-5960 187 15 is be AUX ejpam-5960 187 16 𝑁𝛼–bounded 𝑁𝛼–bounde VERB ejpam-5960 187 17 set set NOUN ejpam-5960 187 18 . . PUNCT ejpam-5960 188 1 theorem theorem VERB ejpam-5960 188 2 3.6 3.6 NUM ejpam-5960 188 3 . . PUNCT ejpam-5960 189 1 ( ( PUNCT ejpam-5960 189 2 the the DET ejpam-5960 189 3 goodness goodness NOUN ejpam-5960 189 4 of of ADP ejpam-5960 189 5 π›Όβˆ’boundedness π›Όβˆ’boundedness NOUN ejpam-5960 189 6 ) ) PUNCT ejpam-5960 189 7 let let VERB ejpam-5960 189 8 ( ( PUNCT ejpam-5960 189 9 πΏπ‘‹π‘–πœ”πΏ πΏπ‘‹π‘–πœ”πΏ PROPN ejpam-5960 189 10 ( ( PUNCT ejpam-5960 189 11 𝑇 𝑇 PROPN ejpam-5960 189 12 ) ) PUNCT ejpam-5960 189 13 ) ) PUNCT ejpam-5960 189 14 be be AUX ejpam-5960 189 15 the the DET ejpam-5960 189 16 induced induced ADJ ejpam-5960 189 17 𝐿-ts 𝐿-ts X ejpam-5960 189 18 by by ADP ejpam-5960 189 19 the the DET ejpam-5960 189 20 ordinary ordinary ADJ ejpam-5960 189 21 space space NOUN ejpam-5960 189 22 ( ( PUNCT ejpam-5960 189 23 𝑋,𝑇 𝑋,𝑇 NOUN ejpam-5960 189 24 ) ) PUNCT ejpam-5960 189 25 , , PUNCT ejpam-5960 189 26 𝛼 𝛼 PROPN ejpam-5960 189 27 ∈ ∈ PROPN ejpam-5960 189 28 𝑀 𝑀 PROPN ejpam-5960 189 29 ( ( PUNCT ejpam-5960 189 30 𝐿 𝐿 PROPN ejpam-5960 189 31 ) ) PUNCT ejpam-5960 189 32 and and CCONJ ejpam-5960 189 33 πœ‡ πœ‡ X ejpam-5960 189 34 ∈ ∈ X ejpam-5960 189 35 𝐿𝑋. 𝐿𝑋. X ejpam-5960 189 36 then then ADV ejpam-5960 189 37 πœ‡ πœ‡ ADP ejpam-5960 189 38 is be AUX ejpam-5960 189 39 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 189 40 in in ADP ejpam-5960 189 41 ( ( PUNCT ejpam-5960 189 42 𝐿𝑋𝑖 𝐿𝑋𝑖 NOUN ejpam-5960 189 43 , , PUNCT ejpam-5960 189 44 πœ”πΏ πœ”πΏ PROPN ejpam-5960 189 45 ( ( PUNCT ejpam-5960 189 46 𝑇 𝑇 PROPN ejpam-5960 189 47 ) ) PUNCT ejpam-5960 189 48 ) ) PUNCT ejpam-5960 190 1 iff iff PROPN ejpam-5960 190 2 πœ‡πœ”π›Ό πœ‡πœ”π›Ό NOUN ejpam-5960 190 3 = = PUNCT ejpam-5960 190 4 { { PUNCT ejpam-5960 190 5 π‘₯ π‘₯ PUNCT ejpam-5960 190 6 ∈ ∈ PROPN ejpam-5960 190 7 𝑋 𝑋 NOUN ejpam-5960 190 8 : : PUNCT ejpam-5960 190 9 πœ‡(π‘₯ πœ‡(π‘₯ PROPN ejpam-5960 190 10 ) ) PUNCT ejpam-5960 190 11 β‰₯ β‰₯ PRON ejpam-5960 191 1 𝛼 𝛼 X ejpam-5960 191 2 } } PUNCT ejpam-5960 191 3 is be AUX ejpam-5960 191 4 nearly nearly ADV ejpam-5960 191 5 bounded bound VERB ejpam-5960 191 6 in in ADP ejpam-5960 191 7 ( ( PUNCT ejpam-5960 191 8 𝑋,𝑇 𝑋,𝑇 NOUN ejpam-5960 191 9 ) ) PUNCT ejpam-5960 191 10 . . PUNCT ejpam-5960 192 1 proof proof NOUN ejpam-5960 192 2 let let VERB ejpam-5960 192 3 πœ‡ πœ‡ SCONJ ejpam-5960 192 4 ∈ ∈ PROPN ejpam-5960 192 5 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 192 6 be be AUX ejpam-5960 192 7 a a DET ejpam-5960 192 8 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 192 9 and and CCONJ ejpam-5960 192 10 { { PUNCT ejpam-5960 192 11 π‘ˆ π‘ˆ NOUN ejpam-5960 192 12 𝑗 𝑗 NOUN ejpam-5960 192 13 : : PUNCT ejpam-5960 192 14 𝑗 𝑗 PROPN ejpam-5960 192 15 ∈ ∈ PROPN ejpam-5960 192 16 𝐽 𝐽 PROPN ejpam-5960 192 17 } } PUNCT ejpam-5960 192 18 be be AUX ejpam-5960 192 19 an an DET ejpam-5960 192 20 open open ADJ ejpam-5960 192 21 cover cover NOUN ejpam-5960 192 22 of of ADP ejpam-5960 192 23 𝑋 𝑋 PROPN ejpam-5960 192 24 in in ADP ejpam-5960 192 25 ( ( PUNCT ejpam-5960 192 26 𝑋,𝑇 𝑋,𝑇 NOUN ejpam-5960 192 27 ) ) PUNCT ejpam-5960 192 28 . . PUNCT ejpam-5960 193 1 then then ADV ejpam-5960 193 2 the the DET ejpam-5960 193 3 family family NOUN ejpam-5960 193 4 { { PUNCT ejpam-5960 193 5 1π‘ˆ 1π‘ˆ NOUN ejpam-5960 193 6 𝑗 𝑗 X ejpam-5960 193 7 : : PUNCT ejpam-5960 193 8 𝑗 𝑗 PROPN ejpam-5960 193 9 ∈ ∈ PROPN ejpam-5960 193 10 𝐽 𝐽 PROPN ejpam-5960 193 11 } } PUNCT ejpam-5960 193 12 is be AUX ejpam-5960 193 13 a a DET ejpam-5960 193 14 π›Όβˆ’covr π›Όβˆ’covr NOUN ejpam-5960 193 15 of of ADP ejpam-5960 193 16 1𝑋 1𝑋 NOUN ejpam-5960 193 17 in in ADP ejpam-5960 193 18 ( ( PUNCT ejpam-5960 193 19 𝐿𝑋𝑖 𝐿𝑋𝑖 NOUN ejpam-5960 193 20 , , PUNCT ejpam-5960 193 21 πœ”πΏ πœ”πΏ PROPN ejpam-5960 193 22 ( ( PUNCT ejpam-5960 193 23 𝑇 𝑇 PROPN ejpam-5960 193 24 ) ) PUNCT ejpam-5960 193 25 ) ) PUNCT ejpam-5960 193 26 . . PUNCT ejpam-5960 194 1 since since SCONJ ejpam-5960 194 2 πœ‡ πœ‡ ADV ejpam-5960 194 3 is be AUX ejpam-5960 194 4 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 194 5 , , PUNCT ejpam-5960 194 6 there there PRON ejpam-5960 194 7 is be VERB ejpam-5960 194 8 a a DET ejpam-5960 194 9 finite finite NOUN ejpam-5960 194 10 subset subset NOUN ejpam-5960 195 1 π½π‘œ π½π‘œ PROPN ejpam-5960 195 2 of of ADP ejpam-5960 195 3 𝐽 𝐽 PROPN ejpam-5960 195 4 such such ADJ ejpam-5960 195 5 that that SCONJ ejpam-5960 195 6 { { PUNCT ejpam-5960 195 7 1π‘ˆ 1π‘ˆ NOUN ejpam-5960 195 8 𝑗 𝑗 X ejpam-5960 195 9 : : PUNCT ejpam-5960 195 10 𝑗 𝑗 X ejpam-5960 195 11 ∈ ∈ PROPN ejpam-5960 195 12 π½π‘œ π½π‘œ NOUN ejpam-5960 195 13 } } PUNCT ejpam-5960 195 14 is be AUX ejpam-5960 195 15 an an DET ejpam-5960 195 16 nearly nearly ADV ejpam-5960 195 17 π›Όβˆ’cover π›Όβˆ’cov ADJ ejpam-5960 195 18 of of ADP ejpam-5960 195 19 πœ‡ πœ‡ X ejpam-5960 195 20 in in ADP ejpam-5960 195 21 ( ( PUNCT ejpam-5960 195 22 πΏπ‘‹π‘–πœ”πΏ πΏπ‘‹π‘–πœ”πΏ PROPN ejpam-5960 195 23 ( ( PUNCT ejpam-5960 195 24 𝑇 𝑇 PROPN ejpam-5960 195 25 ) ) PUNCT ejpam-5960 195 26 ) ) PUNCT ejpam-5960 195 27 in in ADP ejpam-5960 195 28 line line NOUN ejpam-5960 195 29 with with ADP ejpam-5960 195 30 theorem theorem ADJ ejpam-5960 195 31 3.3 3.3 NUM ejpam-5960 195 32 , , PUNCT ejpam-5960 195 33 i.e i.e PROPN ejpam-5960 195 34 , , PUNCT ejpam-5960 195 35 for for ADP ejpam-5960 195 36 each each DET ejpam-5960 195 37 π‘₯ π‘₯ DET ejpam-5960 195 38 ∈ ∈ PROPN ejpam-5960 195 39 πœ‡πœ”π›Ό πœ‡πœ”π›Ό NOUN ejpam-5960 195 40 there there PRON ejpam-5960 195 41 exists exist VERB ejpam-5960 195 42 𝑗 𝑗 PRON ejpam-5960 195 43 ≀ ≀ NOUN ejpam-5960 195 44 𝑛 𝑛 PRON ejpam-5960 195 45 and and CCONJ ejpam-5960 195 46 1π‘ˆ 1π‘ˆ VERB ejpam-5960 195 47 𝑗 𝑗 PROPN ejpam-5960 195 48 ∈ ∈ NOUN ejpam-5960 195 49 πœ‘π‘œ πœ‘π‘œ ADP ejpam-5960 195 50 satisfying satisfy VERB ejpam-5960 195 51 int(𝑐𝑙 int(𝑐𝑙 NUM ejpam-5960 195 52 ( ( PUNCT ejpam-5960 195 53 1π‘ˆ 1π‘ˆ NOUN ejpam-5960 195 54 𝑗 𝑗 NOUN ejpam-5960 195 55 ) ) PUNCT ejpam-5960 195 56 ) ) PUNCT ejpam-5960 196 1 ( ( PUNCT ejpam-5960 196 2 π‘₯ π‘₯ X ejpam-5960 196 3 ) ) PUNCT ejpam-5960 196 4 β‰° β‰° PROPN ejpam-5960 196 5 𝛼. 𝛼. NOUN ejpam-5960 196 6 however however ADV ejpam-5960 196 7 int(𝑐𝑙 int(𝑐𝑙 PRON ejpam-5960 196 8 ( ( PUNCT ejpam-5960 196 9 1π‘ˆ 1π‘ˆ NOUN ejpam-5960 196 10 𝑗 𝑗 NOUN ejpam-5960 196 11 ) ) PUNCT ejpam-5960 196 12 ) ) PUNCT ejpam-5960 196 13 ( ( PUNCT ejpam-5960 196 14 π‘₯ π‘₯ X ejpam-5960 196 15 ) ) PUNCT ejpam-5960 196 16 = = SYM ejpam-5960 197 1 1int(𝑐𝑙 1int(𝑐𝑙 NUM ejpam-5960 197 2 ( ( PUNCT ejpam-5960 197 3 π‘ˆ π‘ˆ PROPN ejpam-5960 197 4 𝑗 𝑗 PROPN ejpam-5960 197 5 ) ) PUNCT ejpam-5960 197 6 ) ) PUNCT ejpam-5960 197 7 , , PUNCT ejpam-5960 197 8 and and CCONJ ejpam-5960 197 9 so so ADV ejpam-5960 197 10 π‘₯ π‘₯ PRON ejpam-5960 197 11 ∈ ∈ NOUN ejpam-5960 197 12 int(𝑐𝑙 int(𝑐𝑙 NOUN ejpam-5960 197 13 ( ( PUNCT ejpam-5960 197 14 π‘ˆ π‘ˆ PROPN ejpam-5960 197 15 𝑗 𝑗 PROPN ejpam-5960 197 16 ) ) PUNCT ejpam-5960 197 17 ) ) PUNCT ejpam-5960 197 18 . . PUNCT ejpam-5960 198 1 this this PRON ejpam-5960 198 2 implies imply VERB ejpam-5960 198 3 that that SCONJ ejpam-5960 198 4 ⋃𝑛 ⋃𝑛 NOUN ejpam-5960 198 5 𝑗=1 𝑗=1 PRON ejpam-5960 198 6 int(𝑐𝑙 int(𝑐𝑙 PRON ejpam-5960 198 7 ( ( PUNCT ejpam-5960 198 8 π‘ˆ π‘ˆ PROPN ejpam-5960 198 9 𝑗 𝑗 NOUN ejpam-5960 198 10 ) ) PUNCT ejpam-5960 198 11 ) ) PUNCT ejpam-5960 198 12 βŠƒ βŠƒ PROPN ejpam-5960 198 13 πœ‡πœ”π›Ό. πœ‡πœ”π›Ό. X ejpam-5960 198 14 hence hence ADV ejpam-5960 198 15 πœ‡πœ”π›Ό πœ‡πœ”π›Ό PROPN ejpam-5960 198 16 is be AUX ejpam-5960 198 17 a a PRON ejpam-5960 198 18 nearly nearly ADV ejpam-5960 198 19 bounded bounded ADJ ejpam-5960 198 20 in in ADP ejpam-5960 198 21 ( ( PUNCT ejpam-5960 198 22 𝑋,𝑇 𝑋,𝑇 NOUN ejpam-5960 198 23 ) ) PUNCT ejpam-5960 198 24 for for ADP ejpam-5960 198 25 any any DET ejpam-5960 198 26 𝛼 𝛼 PROPN ejpam-5960 198 27 ∈ ∈ PROPN ejpam-5960 198 28 𝑀 𝑀 PROPN ejpam-5960 198 29 ( ( PUNCT ejpam-5960 198 30 𝐿 𝐿 PROPN ejpam-5960 198 31 ) ) PUNCT ejpam-5960 198 32 . . PUNCT ejpam-5960 199 1 conversely conversely ADV ejpam-5960 199 2 , , PUNCT ejpam-5960 199 3 suppose suppose VERB ejpam-5960 199 4 that that SCONJ ejpam-5960 199 5 πœ‡πœ”π›Ό πœ‡πœ”π›Ό NOUN ejpam-5960 199 6 is be AUX ejpam-5960 199 7 a a DET ejpam-5960 199 8 nearly nearly ADV ejpam-5960 199 9 bounded bound VERB ejpam-5960 199 10 set set NOUN ejpam-5960 199 11 for for ADP ejpam-5960 199 12 any any DET ejpam-5960 199 13 𝛼 𝛼 PROPN ejpam-5960 199 14 ∈ ∈ PROPN ejpam-5960 199 15 𝑀 𝑀 PROPN ejpam-5960 199 16 ( ( PUNCT ejpam-5960 199 17 𝐿 𝐿 PROPN ejpam-5960 199 18 ) ) PUNCT ejpam-5960 199 19 and and CCONJ ejpam-5960 199 20 πœ‘ πœ‘ PROPN ejpam-5960 199 21 is be AUX ejpam-5960 199 22 a a DET ejpam-5960 199 23 π›Όβˆ’cover π›Όβˆ’cover NOUN ejpam-5960 199 24 of of ADP ejpam-5960 199 25 1𝑋 1𝑋 NOUN ejpam-5960 199 26 in in ADP ejpam-5960 199 27 ( ( PUNCT ejpam-5960 199 28 πΏπ‘‹π‘–πœ”πΏ πΏπ‘‹π‘–πœ”πΏ PROPN ejpam-5960 199 29 ( ( PUNCT ejpam-5960 199 30 𝑇 𝑇 PROPN ejpam-5960 199 31 ) ) PUNCT ejpam-5960 199 32 ) ) PUNCT ejpam-5960 199 33 . . PUNCT ejpam-5960 200 1 then then ADV ejpam-5960 200 2 for for ADP ejpam-5960 200 3 any any DET ejpam-5960 200 4 π‘₯ π‘₯ PRON ejpam-5960 200 5 ∈ ∈ PROPN ejpam-5960 200 6 𝑋 𝑋 PROPN ejpam-5960 200 7 , , PUNCT ejpam-5960 200 8 𝛼 𝛼 PROPN ejpam-5960 200 9 ∈ ∈ PROPN ejpam-5960 200 10 𝑀 𝑀 PROPN ejpam-5960 200 11 ( ( PUNCT ejpam-5960 200 12 𝐿 𝐿 PROPN ejpam-5960 200 13 ) ) PUNCT ejpam-5960 200 14 there there PRON ejpam-5960 200 15 exists exist VERB ejpam-5960 200 16 πœ‚π‘₯ πœ‚π‘₯ PROPN ejpam-5960 200 17 ∈ ∈ NOUN ejpam-5960 200 18 πœ‘ πœ‘ NOUN ejpam-5960 200 19 such such ADJ ejpam-5960 200 20 that that DET ejpam-5960 200 21 πœ‚π‘₯ πœ‚π‘₯ PROPN ejpam-5960 200 22 ( ( PUNCT ejpam-5960 200 23 π‘₯ π‘₯ NOUN ejpam-5960 200 24 ) ) PUNCT ejpam-5960 200 25 β‰° β‰° PROPN ejpam-5960 200 26 𝛼. 𝛼. NOUN ejpam-5960 200 27 put put NOUN ejpam-5960 200 28 ( ( PUNCT ejpam-5960 200 29 πœ‚π‘₯)𝑠𝛼 πœ‚π‘₯)𝑠𝛼 PUNCT ejpam-5960 200 30 = = SYM ejpam-5960 200 31 { { PUNCT ejpam-5960 200 32 𝑦 𝑦 NOUN ejpam-5960 200 33 ∈ ∈ PROPN ejpam-5960 200 34 𝑋 𝑋 NOUN ejpam-5960 200 35 : : PUNCT ejpam-5960 200 36 πœ‚π‘₯ πœ‚π‘₯ PROPN ejpam-5960 200 37 ( ( PUNCT ejpam-5960 200 38 𝑦 𝑦 NOUN ejpam-5960 200 39 ) ) PUNCT ejpam-5960 200 40 β‰° β‰° X ejpam-5960 200 41 𝛼 𝛼 PROPN ejpam-5960 200 42 } } PUNCT ejpam-5960 200 43 , , PUNCT ejpam-5960 200 44 then then ADV ejpam-5960 200 45 π‘₯ π‘₯ PROPN ejpam-5960 200 46 ∈ ∈ PROPN ejpam-5960 200 47 ( ( PUNCT ejpam-5960 200 48 πœ‚π‘₯)𝑠𝛼. πœ‚π‘₯)𝑠𝛼. ADJ ejpam-5960 200 49 since since SCONJ ejpam-5960 200 50 πœ‚π‘₯ πœ‚π‘₯ PROPN ejpam-5960 200 51 ∈ ∈ PROPN ejpam-5960 200 52 πœ”πΏ πœ”πΏ PROPN ejpam-5960 200 53 ( ( PUNCT ejpam-5960 200 54 𝑇 𝑇 PROPN ejpam-5960 200 55 ) ) PUNCT ejpam-5960 200 56 then then ADV ejpam-5960 200 57 ( ( PUNCT ejpam-5960 200 58 πœ‚π‘₯)𝑠𝛼 πœ‚π‘₯)𝑠𝛼 PROPN ejpam-5960 200 59 ∈ ∈ PROPN ejpam-5960 200 60 𝑇 𝑇 PROPN ejpam-5960 200 61 . . PUNCT ejpam-5960 201 1 one one PRON ejpam-5960 201 2 can can AUX ejpam-5960 201 3 see see VERB ejpam-5960 201 4 that that DET ejpam-5960 201 5 π‘ˆ π‘ˆ PROPN ejpam-5960 201 6 = = PRON ejpam-5960 201 7 { { PUNCT ejpam-5960 201 8 ( ( PUNCT ejpam-5960 201 9 πœ‚π‘₯)𝑠𝛼 πœ‚π‘₯)𝑠𝛼 PUNCT ejpam-5960 201 10 : : PUNCT ejpam-5960 201 11 π‘₯ π‘₯ PROPN ejpam-5960 201 12 ∈ ∈ PROPN ejpam-5960 201 13 𝑋 𝑋 PROPN ejpam-5960 201 14 } } PUNCT ejpam-5960 201 15 is be AUX ejpam-5960 201 16 an an DET ejpam-5960 201 17 open open ADJ ejpam-5960 201 18 cover cover NOUN ejpam-5960 201 19 of of ADP ejpam-5960 201 20 𝑋 𝑋 PROPN ejpam-5960 201 21 in in ADP ejpam-5960 201 22 ( ( PUNCT ejpam-5960 201 23 𝑋,𝑇 𝑋,𝑇 NOUN ejpam-5960 201 24 ) ) PUNCT ejpam-5960 201 25 . . PUNCT ejpam-5960 202 1 since since SCONJ ejpam-5960 202 2 πœ‡πœ”π›Ό πœ‡πœ”π›Ό NOUN ejpam-5960 202 3 is be AUX ejpam-5960 202 4 a a PRON ejpam-5960 202 5 nearly nearly ADV ejpam-5960 202 6 bounded bounded ADJ ejpam-5960 202 7 in in ADP ejpam-5960 202 8 ( ( PUNCT ejpam-5960 202 9 𝑋,𝑇 𝑋,𝑇 NOUN ejpam-5960 202 10 ) ) PUNCT ejpam-5960 202 11 , , PUNCT ejpam-5960 202 12 then then ADV ejpam-5960 202 13 there there PRON ejpam-5960 202 14 exists exist VERB ejpam-5960 202 15 π‘₯1 π‘₯1 NOUN ejpam-5960 202 16 , , PUNCT ejpam-5960 202 17 π‘₯2 π‘₯2 NOUN ejpam-5960 202 18 , , PUNCT ejpam-5960 202 19 ... ... PUNCT ejpam-5960 202 20 , , PUNCT ejpam-5960 202 21 π‘₯𝑛 π‘₯𝑛 PROPN ejpam-5960 202 22 ∈ ∈ PROPN ejpam-5960 202 23 πœ‡π‘ π›Ό πœ‡π‘ π›Ό NOUN ejpam-5960 202 24 such such ADJ ejpam-5960 202 25 that that SCONJ ejpam-5960 202 26 π‘ˆπ‘œ π‘ˆπ‘œ PROPN ejpam-5960 202 27 = = SYM ejpam-5960 202 28 { { PUNCT ejpam-5960 202 29 ( ( PUNCT ejpam-5960 202 30 πœ‚π‘₯𝑖 πœ‚π‘₯𝑖 PROPN ejpam-5960 202 31 ) ) PUNCT ejpam-5960 202 32 𝑠𝛼 𝑠𝛼 PROPN ejpam-5960 202 33 : : PUNCT ejpam-5960 203 1 𝑖 𝑖 SYM ejpam-5960 203 2 = = SYM ejpam-5960 203 3 1 1 NUM ejpam-5960 203 4 , , PUNCT ejpam-5960 203 5 2 2 NUM ejpam-5960 203 6 , , PUNCT ejpam-5960 203 7 ... ... PUNCT ejpam-5960 203 8 , , PUNCT ejpam-5960 203 9 𝑛 𝑛 X ejpam-5960 203 10 } } PUNCT ejpam-5960 203 11 is be AUX ejpam-5960 203 12 an an DET ejpam-5960 203 13 nearly nearly ADV ejpam-5960 203 14 open open ADJ ejpam-5960 203 15 cover cover NOUN ejpam-5960 203 16 of of ADP ejpam-5960 203 17 πœ‡πœ”π›Ό. πœ‡πœ”π›Ό. NOUN ejpam-5960 203 18 thus thus ADV ejpam-5960 203 19 there there PRON ejpam-5960 203 20 exists exist VERB ejpam-5960 203 21 𝑖 𝑖 SYM ejpam-5960 203 22 ≀ ≀ NUM ejpam-5960 203 23 𝑛 𝑛 PRON ejpam-5960 203 24 with with ADP ejpam-5960 203 25 π‘₯ π‘₯ PRON ejpam-5960 203 26 ∈ ∈ PROPN ejpam-5960 203 27 int(𝑐𝑙 int(𝑐𝑙 NOUN ejpam-5960 203 28 ( ( PUNCT ejpam-5960 203 29 ( ( PUNCT ejpam-5960 203 30 πœ‚π‘₯𝑖 πœ‚π‘₯𝑖 PROPN ejpam-5960 203 31 ) ) PUNCT ejpam-5960 203 32 𝑠𝛼 𝑠𝛼 PROPN ejpam-5960 203 33 ) ) PUNCT ejpam-5960 203 34 ) ) PUNCT ejpam-5960 203 35 for for ADP ejpam-5960 203 36 each each DET ejpam-5960 203 37 π‘₯ π‘₯ PROPN ejpam-5960 203 38 ∈ ∈ PROPN ejpam-5960 203 39 πœ‡πœ”π›Ό. πœ‡πœ”π›Ό. NOUN ejpam-5960 203 40 however however SCONJ ejpam-5960 203 41 int(𝑐𝑙 int(𝑐𝑙 PRON ejpam-5960 203 42 ( ( PUNCT ejpam-5960 203 43 ( ( PUNCT ejpam-5960 203 44 πœ‚π‘₯𝑖 πœ‚π‘₯𝑖 PROPN ejpam-5960 203 45 ) ) PUNCT ejpam-5960 203 46 𝑠𝛼 𝑠𝛼 PROPN ejpam-5960 203 47 ) ) PUNCT ejpam-5960 203 48 ) ) PUNCT ejpam-5960 204 1 βŠ‚ βŠ‚ PROPN ejpam-5960 204 2 int(𝑐𝑙 int(𝑐𝑙 NUM ejpam-5960 204 3 ( ( PUNCT ejpam-5960 204 4 ( ( PUNCT ejpam-5960 204 5 πœ‚π‘₯𝑖 πœ‚π‘₯𝑖 PROPN ejpam-5960 204 6 ) ) PUNCT ejpam-5960 204 7 ) ) PUNCT ejpam-5960 204 8 ) ) PUNCT ejpam-5960 205 1 𝑠𝛼 𝑠𝛼 PROPN ejpam-5960 205 2 and and CCONJ ejpam-5960 205 3 so so ADV ejpam-5960 205 4 π‘₯ π‘₯ PRON ejpam-5960 205 5 ∈ ∈ NOUN ejpam-5960 205 6 int(𝑐𝑙 int(𝑐𝑙 NOUN ejpam-5960 205 7 ( ( PUNCT ejpam-5960 205 8 ( ( PUNCT ejpam-5960 205 9 πœ‚π‘₯𝑖 πœ‚π‘₯𝑖 PROPN ejpam-5960 205 10 ) ) PUNCT ejpam-5960 205 11 ) ) PUNCT ejpam-5960 205 12 ) ) PUNCT ejpam-5960 206 1 𝑠𝛼 𝑠𝛼 NOUN ejpam-5960 206 2 , , PUNCT ejpam-5960 206 3 thus thus ADV ejpam-5960 206 4 int(𝑐𝑙 int(𝑐𝑙 NUM ejpam-5960 206 5 ( ( PUNCT ejpam-5960 206 6 πœ‚π‘₯𝑖 πœ‚π‘₯𝑖 PROPN ejpam-5960 206 7 ) ) PUNCT ejpam-5960 206 8 ) ) PUNCT ejpam-5960 206 9 ( ( PUNCT ejpam-5960 206 10 π‘₯ π‘₯ X ejpam-5960 206 11 ) ) PUNCT ejpam-5960 206 12 β‰° β‰° PROPN ejpam-5960 206 13 𝛼. 𝛼. NOUN ejpam-5960 206 14 hence hence ADV ejpam-5960 206 15 πœ‘π‘œ πœ‘π‘œ VERB ejpam-5960 207 1 = = PUNCT ejpam-5960 207 2 { { PUNCT ejpam-5960 207 3 πœ‚π‘₯𝑖 πœ‚π‘₯𝑖 NOUN ejpam-5960 207 4 : : PUNCT ejpam-5960 207 5 𝑖 𝑖 SYM ejpam-5960 207 6 = = SYM ejpam-5960 207 7 1 1 NUM ejpam-5960 207 8 , , PUNCT ejpam-5960 207 9 2 2 NUM ejpam-5960 207 10 , , PUNCT ejpam-5960 207 11 ... ... PUNCT ejpam-5960 207 12 , , PUNCT ejpam-5960 207 13 𝑛 𝑛 PROPN ejpam-5960 207 14 } } PUNCT ejpam-5960 207 15 ∈ ∈ NOUN ejpam-5960 207 16 2(πœ‘ 2(πœ‘ NUM ejpam-5960 207 17 ) ) PUNCT ejpam-5960 207 18 is be AUX ejpam-5960 207 19 a a DET ejpam-5960 207 20 nearly nearly ADV ejpam-5960 207 21 π›Όβˆ’cover π›Όβˆ’cover ADP ejpam-5960 207 22 of of ADP ejpam-5960 207 23 πœ‡. πœ‡. NOUN ejpam-5960 207 24 according accord VERB ejpam-5960 207 25 to to ADP ejpam-5960 207 26 theorem theorem ADJ ejpam-5960 207 27 3.3 3.3 NUM ejpam-5960 207 28 , , PUNCT ejpam-5960 207 29 πœ‡ πœ‡ ADP ejpam-5960 207 30 is be AUX ejpam-5960 207 31 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 207 32 in in ADP ejpam-5960 207 33 ( ( PUNCT ejpam-5960 207 34 𝐿𝑋𝑖 𝐿𝑋𝑖 NOUN ejpam-5960 207 35 , , PUNCT ejpam-5960 207 36 πœ”πΏ πœ”πΏ PROPN ejpam-5960 207 37 ( ( PUNCT ejpam-5960 207 38 𝑇 𝑇 PROPN ejpam-5960 207 39 ) ) PUNCT ejpam-5960 207 40 ) ) PUNCT ejpam-5960 207 41 . . PUNCT ejpam-5960 208 1 n. n. PROPN ejpam-5960 208 2 a. a. PROPN ejpam-5960 208 3 alsaedi alsaedi PROPN ejpam-5960 208 4 / / SYM ejpam-5960 208 5 eur eur PROPN ejpam-5960 208 6 . . PUNCT ejpam-5960 209 1 j. j. PROPN ejpam-5960 209 2 pure pure PROPN ejpam-5960 209 3 appl appl PROPN ejpam-5960 209 4 . . PROPN ejpam-5960 209 5 math math PROPN ejpam-5960 209 6 , , PUNCT ejpam-5960 209 7 18 18 NUM ejpam-5960 209 8 ( ( PUNCT ejpam-5960 209 9 4 4 NUM ejpam-5960 209 10 ) ) PUNCT ejpam-5960 209 11 ( ( PUNCT ejpam-5960 209 12 2025 2025 NUM ejpam-5960 209 13 ) ) PUNCT ejpam-5960 209 14 , , PUNCT ejpam-5960 209 15 5960 5960 NUM ejpam-5960 209 16 8 8 NUM ejpam-5960 209 17 of of ADP ejpam-5960 209 18 22 22 NUM ejpam-5960 209 19 theorem theorem VERB ejpam-5960 209 20 3.7 3.7 NUM ejpam-5960 209 21 . . PUNCT ejpam-5960 210 1 let let VERB ejpam-5960 210 2 ( ( PUNCT ejpam-5960 210 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 210 4 , , PUNCT ejpam-5960 210 5 𝜏 𝜏 NOUN ejpam-5960 210 6 ) ) PUNCT ejpam-5960 210 7 be be AUX ejpam-5960 210 8 a a DET ejpam-5960 210 9 πΏβˆ’ts πΏβˆ’t NOUN ejpam-5960 210 10 . . PUNCT ejpam-5960 211 1 and and CCONJ ejpam-5960 211 2 let let VERB ejpam-5960 211 3 πœ‡ πœ‡ ADP ejpam-5960 211 4 ∈ ∈ X ejpam-5960 211 5 𝐿𝑋. 𝐿𝑋. INTJ ejpam-5960 211 6 if if SCONJ ejpam-5960 211 7 πœ‚ πœ‚ NOUN ejpam-5960 211 8 is be AUX ejpam-5960 211 9 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 211 10 and and CCONJ ejpam-5960 211 11 πœ‡ πœ‡ ADP ejpam-5960 211 12 ≀ ≀ NOUN ejpam-5960 211 13 πœ‚ πœ‚ NOUN ejpam-5960 211 14 , , PUNCT ejpam-5960 211 15 then then ADV ejpam-5960 211 16 πœ‡ πœ‡ ADP ejpam-5960 211 17 is be AUX ejpam-5960 211 18 a a DET ejpam-5960 211 19 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 211 20 . . PUNCT ejpam-5960 212 1 proof proof NOUN ejpam-5960 212 2 . . PUNCT ejpam-5960 213 1 let let VERB ejpam-5960 213 2 πœ‚ πœ‚ NOUN ejpam-5960 213 3 be be AUX ejpam-5960 213 4 a a DET ejpam-5960 213 5 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 213 6 set set NOUN ejpam-5960 213 7 and and CCONJ ejpam-5960 213 8 πœ‡ πœ‡ ADP ejpam-5960 213 9 ≀ ≀ NUM ejpam-5960 213 10 πœ‚. πœ‚. NOUN ejpam-5960 213 11 let let VERB ejpam-5960 213 12 ψ ψ AUX ejpam-5960 213 13 βŠ‚ βŠ‚ X ejpam-5960 213 14 πœβ€² πœβ€² PROPN ejpam-5960 213 15 be be AUX ejpam-5960 213 16 an an DET ejpam-5960 213 17 π›Όβˆ’rf π›Όβˆ’rf NOUN ejpam-5960 213 18 of of ADP ejpam-5960 213 19 1𝑋. 1𝑋. NUM ejpam-5960 213 20 since since SCONJ ejpam-5960 213 21 πœ‚ πœ‚ PROPN ejpam-5960 213 22 is be AUX ejpam-5960 213 23 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 213 24 set set NOUN ejpam-5960 213 25 , , PUNCT ejpam-5960 213 26 then then ADV ejpam-5960 213 27 there there PRON ejpam-5960 213 28 exists exist VERB ejpam-5960 213 29 a a DET ejpam-5960 213 30 finite finite NOUN ejpam-5960 213 31 subfamily subfamily ADV ejpam-5960 213 32 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 213 33 ∈ ∈ PROPN ejpam-5960 213 34 2(ψ 2(ψ NUM ejpam-5960 213 35 ) ) PUNCT ejpam-5960 213 36 such such ADJ ejpam-5960 213 37 that that SCONJ ejpam-5960 213 38 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 213 39 is be AUX ejpam-5960 213 40 an an DET ejpam-5960 213 41 π›Όβˆ’rcrf π›Όβˆ’rcrf NOUN ejpam-5960 213 42 of of ADP ejpam-5960 213 43 πœ‚ πœ‚ NOUN ejpam-5960 213 44 , , PUNCT ejpam-5960 213 45 since since SCONJ ejpam-5960 213 46 πœ‡ πœ‡ ADP ejpam-5960 213 47 ≀ ≀ NOUN ejpam-5960 213 48 πœ‚ πœ‚ NOUN ejpam-5960 213 49 , , PUNCT ejpam-5960 213 50 then then ADV ejpam-5960 213 51 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 213 52 is be AUX ejpam-5960 213 53 an an DET ejpam-5960 213 54 π›Όβˆ’rcrf π›Όβˆ’rcrf NOUN ejpam-5960 213 55 of of ADP ejpam-5960 213 56 πœ‡ πœ‡ ADV ejpam-5960 213 57 and and CCONJ ejpam-5960 213 58 so so ADV ejpam-5960 213 59 πœ‡ πœ‡ PRON ejpam-5960 213 60 is be AUX ejpam-5960 213 61 a a DET ejpam-5960 213 62 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 213 63 set set NOUN ejpam-5960 213 64 . . PUNCT ejpam-5960 214 1 definition definition NOUN ejpam-5960 214 2 3.8 3.8 NUM ejpam-5960 214 3 . . PUNCT ejpam-5960 215 1 let let VERB ejpam-5960 215 2 ( ( PUNCT ejpam-5960 215 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 215 4 , , PUNCT ejpam-5960 215 5 𝜏 𝜏 NOUN ejpam-5960 215 6 ) ) PUNCT ejpam-5960 215 7 be be VERB ejpam-5960 215 8 an an DET ejpam-5960 215 9 πΏβˆ’ts πΏβˆ’ts PROPN ejpam-5960 215 10 and and CCONJ ejpam-5960 215 11 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 215 12 ∈ ∈ PROPN ejpam-5960 215 13 𝑀 𝑀 PROPN ejpam-5960 215 14 ( ( PUNCT ejpam-5960 215 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 215 16 ) ) PUNCT ejpam-5960 215 17 . . PUNCT ejpam-5960 216 1 if if SCONJ ejpam-5960 216 2 πœ‡ πœ‡ ADP ejpam-5960 216 3 ∈ ∈ PROPN ejpam-5960 216 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 216 5 is be AUX ejpam-5960 216 6 closed close VERB ejpam-5960 216 7 and and CCONJ ejpam-5960 216 8 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 216 9 set set NOUN ejpam-5960 216 10 , , PUNCT ejpam-5960 216 11 then then ADV ejpam-5960 216 12 πœ‡ πœ‡ ADP ejpam-5960 216 13 is be AUX ejpam-5960 216 14 called call VERB ejpam-5960 216 15 the the DET ejpam-5960 216 16 π‘π›Όπ΅βˆ’remoted π‘π›Όπ΅βˆ’remote VERB ejpam-5960 216 17 neighborhood neighborhood NOUN ejpam-5960 216 18 of of ADP ejpam-5960 216 19 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 216 20 ( ( PUNCT ejpam-5960 216 21 π‘π›Όπ΅π‘…βˆ’nbd π‘π›Όπ΅π‘…βˆ’nbd ADP ejpam-5960 216 22 , , PUNCT ejpam-5960 216 23 for for ADP ejpam-5960 216 24 short short ADJ ejpam-5960 216 25 ) ) PUNCT ejpam-5960 216 26 of of ADP ejpam-5960 216 27 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 216 28 if if SCONJ ejpam-5960 216 29 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 216 30 βˆ‰ βˆ‰ PROPN ejpam-5960 216 31 πœ‡. πœ‡. NOUN ejpam-5960 216 32 the the DET ejpam-5960 216 33 set set NOUN ejpam-5960 216 34 of of ADP ejpam-5960 216 35 all all DET ejpam-5960 216 36 π‘π›Όπ΅π‘…βˆ’nbds π‘π›Όπ΅π‘…βˆ’nbds NUM ejpam-5960 216 37 of of ADP ejpam-5960 216 38 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 216 39 is be AUX ejpam-5960 216 40 denoted denote VERB ejpam-5960 216 41 by by ADP ejpam-5960 216 42 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 216 43 . . PUNCT ejpam-5960 217 1 we we PRON ejpam-5960 217 2 note note VERB ejpam-5960 217 3 that that SCONJ ejpam-5960 217 4 [ [ X ejpam-5960 217 5 24 24 NUM ejpam-5960 217 6 ] ] X ejpam-5960 217 7 𝛼𝐡𝑅π‘₯𝛼 𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 217 8 βŠ† βŠ† NUM ejpam-5960 217 9 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 217 10 βŠ† βŠ† NUM ejpam-5960 217 11 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 217 12 , , PUNCT ejpam-5960 217 13 βˆ€ βˆ€ X ejpam-5960 217 14 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 217 15 ∈ ∈ PROPN ejpam-5960 217 16 𝑀 𝑀 PROPN ejpam-5960 217 17 ( ( PUNCT ejpam-5960 217 18 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 217 19 ) ) PUNCT ejpam-5960 217 20 . . PUNCT ejpam-5960 218 1 the the DET ejpam-5960 218 2 following follow VERB ejpam-5960 218 3 example example NOUN ejpam-5960 218 4 shows show VERB ejpam-5960 218 5 that that SCONJ ejpam-5960 218 6 the the DET ejpam-5960 218 7 converse converse NOUN ejpam-5960 218 8 is be AUX ejpam-5960 218 9 not not PART ejpam-5960 218 10 true true ADJ ejpam-5960 218 11 in in ADP ejpam-5960 218 12 general general ADJ ejpam-5960 218 13 . . PUNCT ejpam-5960 219 1 example example NOUN ejpam-5960 219 2 3.9 3.9 NUM ejpam-5960 219 3 let let VERB ejpam-5960 219 4 𝐿 𝐿 PROPN ejpam-5960 219 5 = = PUNCT ejpam-5960 220 1 [ [ X ejpam-5960 220 2 0 0 NUM ejpam-5960 220 3 , , PUNCT ejpam-5960 220 4 1 1 NUM ejpam-5960 220 5 ] ] PUNCT ejpam-5960 220 6 , , PUNCT ejpam-5960 220 7 𝑋 𝑋 NOUN ejpam-5960 220 8 = = PUNCT ejpam-5960 220 9 β–‘ β–‘ PUNCT ejpam-5960 220 10 and and CCONJ ejpam-5960 220 11 let let VERB ejpam-5960 220 12 𝜏 𝜏 VERB ejpam-5960 220 13 = = SYM ejpam-5960 220 14 { { PUNCT ejpam-5960 220 15 0𝑋 0𝑋 PROPN ejpam-5960 220 16 , , PUNCT ejpam-5960 220 17 π‘₯.5 π‘₯.5 ADV ejpam-5960 220 18 , , PUNCT ejpam-5960 220 19 1𝑋 1𝑋 PROPN ejpam-5960 220 20 } } PUNCT ejpam-5960 220 21 . . PUNCT ejpam-5960 221 1 then then ADV ejpam-5960 221 2 ( ( PUNCT ejpam-5960 221 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 221 4 , , PUNCT ejpam-5960 221 5 𝜏 𝜏 NOUN ejpam-5960 221 6 ) ) PUNCT ejpam-5960 221 7 is be AUX ejpam-5960 221 8 πΏβˆ’ts πΏβˆ’ts PROPN ejpam-5960 221 9 . . PUNCT ejpam-5960 222 1 firstly firstly ADV ejpam-5960 222 2 , , PUNCT ejpam-5960 222 3 we we PRON ejpam-5960 222 4 show show VERB ejpam-5960 222 5 that that SCONJ ejpam-5960 222 6 πœ‡ πœ‡ SCONJ ejpam-5960 222 7 = = X ejpam-5960 222 8 π‘₯.4 π‘₯.4 NOUN ejpam-5960 222 9 ∈ ∈ PROPN ejpam-5960 222 10 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 222 11 is be AUX ejpam-5960 222 12 not not PART ejpam-5960 222 13 a a DET ejpam-5960 222 14 π›Όβˆ’bounded π›Όβˆ’bounde VERB ejpam-5960 222 15 set set NOUN ejpam-5960 222 16 . . PUNCT ejpam-5960 223 1 in in ADP ejpam-5960 223 2 fact fact NOUN ejpam-5960 223 3 , , PUNCT ejpam-5960 223 4 we we PRON ejpam-5960 223 5 suppose suppose VERB ejpam-5960 223 6 that that SCONJ ejpam-5960 223 7 constant constant ADJ ejpam-5960 223 8 0.3βˆ’net 0.3βˆ’net NUM ejpam-5960 223 9 𝑆 𝑆 PROPN ejpam-5960 223 10 = = PRON ejpam-5960 223 11 { { PUNCT ejpam-5960 223 12 π‘₯.3 π‘₯.3 NOUN ejpam-5960 223 13 : : PUNCT ejpam-5960 223 14 π‘₯ π‘₯ PUNCT ejpam-5960 223 15 ∈ ∈ PROPN ejpam-5960 223 16 𝑋 𝑋 PROPN ejpam-5960 223 17 } } PUNCT ejpam-5960 223 18 in in ADP ejpam-5960 223 19 πœ‡. πœ‡. NOUN ejpam-5960 223 20 let let VERB ejpam-5960 223 21 𝑦.2 𝑦.2 X ejpam-5960 223 22 ∈ ∈ PROPN ejpam-5960 223 23 𝑀 𝑀 PROPN ejpam-5960 223 24 ( ( PUNCT ejpam-5960 223 25 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 223 26 ) ) PUNCT ejpam-5960 223 27 , , PUNCT ejpam-5960 223 28 π‘₯ π‘₯ PROPN ejpam-5960 223 29 β‰  β‰  PROPN ejpam-5960 223 30 𝑦 𝑦 NOUN ejpam-5960 223 31 , , PUNCT ejpam-5960 223 32 then then ADV ejpam-5960 223 33 𝑅𝑦.2 𝑅𝑦.2 ADJ ejpam-5960 223 34 = = SYM ejpam-5960 223 35 { { PUNCT ejpam-5960 223 36 0𝑋 0𝑋 PROPN ejpam-5960 223 37 , , PUNCT ejpam-5960 223 38 π‘₯.5 π‘₯.5 ADJ ejpam-5960 223 39 } } PUNCT ejpam-5960 223 40 . . PUNCT ejpam-5960 224 1 since since SCONJ ejpam-5960 224 2 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 224 3 ) ) PUNCT ejpam-5960 224 4 = = PUNCT ejpam-5960 224 5 π‘₯.3 π‘₯.3 X ejpam-5960 224 6 ≀ ≀ NUM ejpam-5960 224 7 π‘₯.5 π‘₯.5 X ejpam-5960 224 8 ∈ ∈ PROPN ejpam-5960 224 9 𝑅𝑦.2 𝑅𝑦.2 PROPN ejpam-5960 224 10 . . PUNCT ejpam-5960 225 1 so so ADV ejpam-5960 225 2 𝑦.2 𝑦.2 X ejpam-5960 225 3 is be AUX ejpam-5960 225 4 not not PART ejpam-5960 225 5 cluster cluster NOUN ejpam-5960 225 6 point point NOUN ejpam-5960 225 7 of of ADP ejpam-5960 225 8 𝑆. 𝑆. PROPN ejpam-5960 225 9 on on ADP ejpam-5960 225 10 account account NOUN ejpam-5960 225 11 of of ADP ejpam-5960 225 12 the the DET ejpam-5960 225 13 arbitrariness arbitrariness NOUN ejpam-5960 225 14 of of ADP ejpam-5960 225 15 𝑦 𝑦 PRON ejpam-5960 225 16 it it PRON ejpam-5960 225 17 follows follow VERB ejpam-5960 225 18 that that SCONJ ejpam-5960 225 19 the the DET ejpam-5960 225 20 𝑆 𝑆 PROPN ejpam-5960 225 21 has have VERB ejpam-5960 225 22 no no DET ejpam-5960 225 23 cluster cluster NOUN ejpam-5960 225 24 point point NOUN ejpam-5960 225 25 in in ADP ejpam-5960 225 26 1𝑋 1𝑋 NOUN ejpam-5960 225 27 with with ADP ejpam-5960 225 28 height height NOUN ejpam-5960 225 29 0.3 0.3 NUM ejpam-5960 225 30 . . PUNCT ejpam-5960 226 1 thus thus ADV ejpam-5960 226 2 πœ‡ πœ‡ PRON ejpam-5960 226 3 is be AUX ejpam-5960 226 4 not not PART ejpam-5960 226 5 π›Όβˆ’bounded π›Όβˆ’bounde VERB ejpam-5960 226 6 set set NOUN ejpam-5960 226 7 . . PUNCT ejpam-5960 227 1 now now ADV ejpam-5960 227 2 , , PUNCT ejpam-5960 227 3 we we PRON ejpam-5960 227 4 show show VERB ejpam-5960 227 5 that that SCONJ ejpam-5960 227 6 πœ‡ πœ‡ ADP ejpam-5960 227 7 is be AUX ejpam-5960 227 8 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 227 9 set set VERB ejpam-5960 227 10 . . PUNCT ejpam-5960 228 1 let let VERB ejpam-5960 228 2 𝑆 𝑆 PROPN ejpam-5960 228 3 = = PRON ejpam-5960 228 4 { { PUNCT ejpam-5960 228 5 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 228 6 : : PUNCT ejpam-5960 228 7 π‘₯ π‘₯ PROPN ejpam-5960 228 8 ∈ ∈ PROPN ejpam-5960 228 9 𝑋 𝑋 PROPN ejpam-5960 228 10 , , PUNCT ejpam-5960 228 11 𝛼 𝛼 NOUN ejpam-5960 228 12 ≀ ≀ NOUN ejpam-5960 228 13 .4 .4 NUM ejpam-5960 228 14 } } PUNCT ejpam-5960 228 15 is be AUX ejpam-5960 228 16 any any DET ejpam-5960 228 17 constant constant ADJ ejpam-5960 228 18 π›Όβˆ’net π›Όβˆ’net NOUN ejpam-5960 228 19 in in ADP ejpam-5960 228 20 πœ‡. πœ‡. NOUN ejpam-5960 228 21 if if SCONJ ejpam-5960 228 22 𝛼 𝛼 X ejpam-5960 228 23 < < X ejpam-5960 228 24 .4 .4 PUNCT ejpam-5960 228 25 then then ADV ejpam-5960 228 26 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 228 27 = = SYM ejpam-5960 228 28 { { PUNCT ejpam-5960 228 29 0𝑋 0𝑋 PROPN ejpam-5960 228 30 } } PUNCT ejpam-5960 228 31 , , PUNCT ejpam-5960 229 1 where where SCONJ ejpam-5960 229 2 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 229 3 ( ( PUNCT ejpam-5960 229 4 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 229 5 ( ( PUNCT ejpam-5960 229 6 0𝑋 0𝑋 PROPN ejpam-5960 229 7 ) ) PUNCT ejpam-5960 229 8 ) ) PUNCT ejpam-5960 230 1 = = PUNCT ejpam-5960 230 2 0𝑋. 0𝑋. NOUN ejpam-5960 230 3 since since SCONJ ejpam-5960 230 4 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 230 5 ) ) PUNCT ejpam-5960 230 6 = = PUNCT ejpam-5960 230 7 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 230 8 βˆ‰ βˆ‰ PROPN ejpam-5960 230 9 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 230 10 ( ( PUNCT ejpam-5960 230 11 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 230 12 ( ( PUNCT ejpam-5960 230 13 0𝑋 0𝑋 PROPN ejpam-5960 230 14 ) ) PUNCT ejpam-5960 230 15 ) ) PUNCT ejpam-5960 231 1 = = SYM ejpam-5960 231 2 0𝑋 0𝑋 NOUN ejpam-5960 231 3 βˆ€ βˆ€ X ejpam-5960 231 4 𝑛 𝑛 PRON ejpam-5960 231 5 ∈ ∈ PROPN ejpam-5960 231 6 𝑁 𝑁 PROPN ejpam-5960 231 7 , , PUNCT ejpam-5960 231 8 βˆ€π›Ό βˆ€π›Ό X ejpam-5960 231 9 < < X ejpam-5960 231 10 .4 .4 NUM ejpam-5960 231 11 . . PUNCT ejpam-5960 232 1 if if SCONJ ejpam-5960 232 2 𝛼 𝛼 X ejpam-5960 232 3 = = SYM ejpam-5960 232 4 .4 .4 NUM ejpam-5960 232 5 , , PUNCT ejpam-5960 232 6 then then ADV ejpam-5960 232 7 𝑅π‘₯.4 𝑅π‘₯.4 NOUN ejpam-5960 232 8 = = PUNCT ejpam-5960 232 9 { { PUNCT ejpam-5960 232 10 0𝑋 0𝑋 PROPN ejpam-5960 232 11 } } PUNCT ejpam-5960 232 12 and and CCONJ ejpam-5960 232 13 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 232 14 ) ) PUNCT ejpam-5960 232 15 = = PUNCT ejpam-5960 232 16 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 232 17 βˆ‰ βˆ‰ PROPN ejpam-5960 232 18 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 232 19 ( ( PUNCT ejpam-5960 232 20 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 232 21 ( ( PUNCT ejpam-5960 232 22 0𝑋 0𝑋 PROPN ejpam-5960 232 23 ) ) PUNCT ejpam-5960 232 24 ) ) PUNCT ejpam-5960 233 1 = = SYM ejpam-5960 233 2 0𝑋. 0𝑋. NOUN ejpam-5960 234 1 so so ADV ejpam-5960 234 2 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 234 3 is be AUX ejpam-5960 234 4 the the DET ejpam-5960 234 5 π›Ώβˆ’cluster π›Ώβˆ’cluster ADJ ejpam-5960 234 6 point point NOUN ejpam-5960 234 7 of of ADP ejpam-5960 234 8 𝑆 𝑆 PROPN ejpam-5960 234 9 in in ADP ejpam-5960 234 10 1𝑋 1𝑋 PROPN ejpam-5960 234 11 1𝑋. 1𝑋. NUM ejpam-5960 234 12 thus thus ADV ejpam-5960 234 13 πœ‡ πœ‡ ADV ejpam-5960 234 14 is be AUX ejpam-5960 234 15 a a DET ejpam-5960 234 16 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 234 17 set set NOUN ejpam-5960 234 18 . . PUNCT ejpam-5960 235 1 example example NOUN ejpam-5960 235 2 3.10 3.10 NUM ejpam-5960 235 3 let let VERB ejpam-5960 235 4 𝑋 𝑋 PROPN ejpam-5960 235 5 = = SYM ejpam-5960 235 6 { { PUNCT ejpam-5960 235 7 2 2 NUM ejpam-5960 235 8 , , PUNCT ejpam-5960 235 9 3 3 NUM ejpam-5960 235 10 , , PUNCT ejpam-5960 235 11 4 4 NUM ejpam-5960 235 12 , , PUNCT ejpam-5960 235 13 . . PUNCT ejpam-5960 235 14 . . PUNCT ejpam-5960 236 1 . . PUNCT ejpam-5960 237 1 } } PUNCT ejpam-5960 237 2 , , PUNCT ejpam-5960 237 3 𝐿 𝐿 PROPN ejpam-5960 237 4 = = SYM ejpam-5960 238 1 [ [ X ejpam-5960 238 2 0 0 NUM ejpam-5960 238 3 , , PUNCT ejpam-5960 238 4 1 1 NUM ejpam-5960 238 5 ] ] PUNCT ejpam-5960 238 6 , , PUNCT ejpam-5960 238 7 πœŒπ‘› πœŒπ‘› X ejpam-5960 238 8 ( ( PUNCT ejpam-5960 238 9 π‘₯ π‘₯ NOUN ejpam-5960 238 10 ) ) PUNCT ejpam-5960 238 11 = = PRON ejpam-5960 238 12 { { PUNCT ejpam-5960 238 13 0 0 NUM ejpam-5960 238 14 : : PUNCT ejpam-5960 239 1 π‘₯ π‘₯ X ejpam-5960 239 2 = = SYM ejpam-5960 239 3 𝑛 𝑛 DET ejpam-5960 239 4 1 1 NUM ejpam-5960 239 5 2 2 NUM ejpam-5960 239 6 + + SYM ejpam-5960 239 7 1 1 NUM ejpam-5960 239 8 𝑛 𝑛 NOUN ejpam-5960 239 9 : : PUNCT ejpam-5960 239 10 π‘₯ π‘₯ PROPN ejpam-5960 239 11 β‰  β‰  PROPN ejpam-5960 239 12 𝑛 𝑛 PRON ejpam-5960 239 13 𝑛 𝑛 PRON ejpam-5960 239 14 ∈ ∈ NOUN ejpam-5960 239 15 n n ADV ejpam-5960 239 16 πœ‚β€²2 πœ‚β€²2 NOUN ejpam-5960 239 17 = = PUNCT ejpam-5960 239 18 { { PUNCT ejpam-5960 239 19 1 1 NUM ejpam-5960 239 20 : : PUNCT ejpam-5960 239 21 π‘₯ π‘₯ NOUN ejpam-5960 239 22 = = SYM ejpam-5960 239 23 2 2 NUM ejpam-5960 239 24 1 1 NUM ejpam-5960 239 25 2 2 NUM ejpam-5960 239 26 : : PUNCT ejpam-5960 239 27 π‘₯ π‘₯ X ejpam-5960 239 28 > > SYM ejpam-5960 239 29 2 2 NUM ejpam-5960 239 30 πœŽπ‘› πœŽπ‘› PRON ejpam-5960 239 31 ( ( PUNCT ejpam-5960 239 32 π‘₯ π‘₯ NOUN ejpam-5960 239 33 ) ) PUNCT ejpam-5960 239 34 = = PUNCT ejpam-5960 239 35 πœ‚β€²π‘› πœ‚β€²π‘› X ejpam-5960 239 36 ( ( PUNCT ejpam-5960 239 37 π‘₯ π‘₯ X ejpam-5960 239 38 ) ) PUNCT ejpam-5960 239 39 = = NOUN ejpam-5960 239 40 { { PUNCT ejpam-5960 239 41 1 1 NUM ejpam-5960 239 42 : : PUNCT ejpam-5960 239 43 π‘₯ π‘₯ X ejpam-5960 239 44 = = SYM ejpam-5960 239 45 𝑛 𝑛 DET ejpam-5960 239 46 1 1 NUM ejpam-5960 239 47 2 2 NUM ejpam-5960 239 48 βˆ’ βˆ’ NUM ejpam-5960 239 49 1 1 NUM ejpam-5960 239 50 𝑛 𝑛 NOUN ejpam-5960 239 51 : : PUNCT ejpam-5960 239 52 π‘₯ π‘₯ PROPN ejpam-5960 239 53 β‰  β‰  PROPN ejpam-5960 239 54 𝑛 𝑛 PROPN ejpam-5960 239 55 for for ADP ejpam-5960 239 56 𝑛 𝑛 DET ejpam-5960 239 57 ∈ ∈ PROPN ejpam-5960 239 58 { { PUNCT ejpam-5960 239 59 3 3 NUM ejpam-5960 239 60 , , PUNCT ejpam-5960 239 61 4 4 NUM ejpam-5960 239 62 , , PUNCT ejpam-5960 239 63 . . PUNCT ejpam-5960 239 64 . . PUNCT ejpam-5960 240 1 . . PUNCT ejpam-5960 240 2 } } PUNCT ejpam-5960 241 1 then then ADV ejpam-5960 241 2 we we PRON ejpam-5960 241 3 have have VERB ejpam-5960 241 4 : : PUNCT ejpam-5960 241 5 ( ( PUNCT ejpam-5960 241 6 i i NOUN ejpam-5960 241 7 ) ) PUNCT ejpam-5960 241 8 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 242 1 ( ( PUNCT ejpam-5960 242 2 πœŒπ‘› πœŒπ‘› NOUN ejpam-5960 242 3 ) ) PUNCT ejpam-5960 242 4 = = SYM ejpam-5960 242 5 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 242 6 , , PUNCT ejpam-5960 242 7 βˆ€ βˆ€ VERB ejpam-5960 242 8 𝑛 𝑛 PRON ejpam-5960 242 9 ∈ ∈ PROPN ejpam-5960 242 10 𝑋. 𝑋. PROPN ejpam-5960 242 11 ( ( PUNCT ejpam-5960 242 12 ii ii PROPN ejpam-5960 242 13 ) ) PUNCT ejpam-5960 242 14 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 242 15 ( ( PUNCT ejpam-5960 242 16 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 242 17 ( ( PUNCT ejpam-5960 242 18 𝜌2 𝜌2 ADJ ejpam-5960 242 19 ) ) PUNCT ejpam-5960 242 20 ) ) PUNCT ejpam-5960 243 1 = = SYM ejpam-5960 243 2 𝜌2 𝜌2 ADJ ejpam-5960 243 3 ∨ ∨ NUM ejpam-5960 243 4 𝜎2 𝜎2 NOUN ejpam-5960 243 5 . . PUNCT ejpam-5960 244 1 ( ( PUNCT ejpam-5960 244 2 iii iii X ejpam-5960 244 3 ) ) PUNCT ejpam-5960 244 4 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 244 5 ( ( PUNCT ejpam-5960 244 6 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 244 7 ( ( PUNCT ejpam-5960 244 8 𝜌3 𝜌3 NOUN ejpam-5960 244 9 ) ) PUNCT ejpam-5960 244 10 ) ) PUNCT ejpam-5960 245 1 = = SYM ejpam-5960 245 2 𝜌3 𝜌3 VERB ejpam-5960 245 3 ∨ ∨ NUM ejpam-5960 245 4 𝜎2 𝜎2 PROPN ejpam-5960 245 5 , , PUNCT ejpam-5960 245 6 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 245 7 ( ( PUNCT ejpam-5960 245 8 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 245 9 ( ( PUNCT ejpam-5960 245 10 πœŒπ‘› πœŒπ‘› NOUN ejpam-5960 245 11 ) ) PUNCT ejpam-5960 245 12 ) ) PUNCT ejpam-5960 246 1 = = SYM ejpam-5960 246 2 πœŒπ‘› πœŒπ‘› NOUN ejpam-5960 246 3 , , PUNCT ejpam-5960 246 4 βˆ€ βˆ€ VERB ejpam-5960 246 5 𝑛 𝑛 PRON ejpam-5960 246 6 β‰₯ β‰₯ NOUN ejpam-5960 246 7 4 4 NUM ejpam-5960 246 8 . . PUNCT ejpam-5960 247 1 we we PRON ejpam-5960 247 2 show show VERB ejpam-5960 247 3 that that SCONJ ejpam-5960 247 4 𝜌5 𝜌5 PROPN ejpam-5960 247 5 ∈ ∈ PROPN ejpam-5960 247 6 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 247 7 is be AUX ejpam-5960 247 8 not not PART ejpam-5960 247 9 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 247 10 set set VERB ejpam-5960 247 11 . . PUNCT ejpam-5960 248 1 put put VERB ejpam-5960 248 2 ψ ψ X ejpam-5960 248 3 = = PUNCT ejpam-5960 248 4 { { PUNCT ejpam-5960 248 5 πœŒπ‘› πœŒπ‘› NOUN ejpam-5960 248 6 : : PUNCT ejpam-5960 248 7 𝑛 𝑛 DET ejpam-5960 248 8 ∈ ∈ PROPN ejpam-5960 248 9 𝑋 𝑋 PROPN ejpam-5960 248 10 } } PUNCT ejpam-5960 248 11 , , PUNCT ejpam-5960 248 12 then then ADV ejpam-5960 248 13 then then ADV ejpam-5960 248 14 ψ ψ NOUN ejpam-5960 248 15 is be AUX ejpam-5960 248 16 0.8 0.8 NUM ejpam-5960 248 17 – – PUNCT ejpam-5960 248 18 rf rf NOUN ejpam-5960 248 19 of of ADP ejpam-5960 248 20 1𝑋 1𝑋 NOUN ejpam-5960 248 21 where where SCONJ ejpam-5960 248 22 𝛼 𝛼 AUX ejpam-5960 248 23 = = PUNCT ejpam-5960 248 24 0.8 0.8 NUM ejpam-5960 248 25 ∈ ∈ PROPN ejpam-5960 248 26 𝑀 𝑀 PROPN ejpam-5960 248 27 ( ( PUNCT ejpam-5960 248 28 𝐿 𝐿 PROPN ejpam-5960 248 29 ) ) PUNCT ejpam-5960 248 30 = = SYM ejpam-5960 248 31 ( ( PUNCT ejpam-5960 248 32 0 0 NUM ejpam-5960 248 33 , , PUNCT ejpam-5960 248 34 1 1 NUM ejpam-5960 248 35 ] ] PUNCT ejpam-5960 248 36 ( ( PUNCT ejpam-5960 248 37 because because SCONJ ejpam-5960 248 38 βˆ€π‘₯ βˆ€π‘₯ PROPN ejpam-5960 248 39 ∈ ∈ PROPN ejpam-5960 248 40 𝑋 𝑋 PROPN ejpam-5960 248 41 βˆƒπœ† βˆƒπœ† PROPN ejpam-5960 249 1 = = PUNCT ejpam-5960 249 2 𝜌6 𝜌6 PROPN ejpam-5960 249 3 ∈ ∈ PROPN ejpam-5960 249 4 ψ ψ X ejpam-5960 249 5 βˆ‹ βˆ‹ X ejpam-5960 249 6 𝑐𝑙 𝑐𝑙 X ejpam-5960 249 7 ( ( PUNCT ejpam-5960 249 8 int(𝜌6 int(𝜌6 NOUN ejpam-5960 249 9 ) ) PUNCT ejpam-5960 249 10 ) ) PUNCT ejpam-5960 250 1 ∈ ∈ PROPN ejpam-5960 250 2 𝑅π‘₯0.8 𝑅π‘₯0.8 PROPN ejpam-5960 250 3 ) ) PUNCT ejpam-5960 250 4 , , PUNCT ejpam-5960 250 5 where where SCONJ ejpam-5960 250 6 𝜌6(π‘₯ 𝜌6(π‘₯ NOUN ejpam-5960 250 7 ) ) PUNCT ejpam-5960 250 8 = = SYM ejpam-5960 250 9 0 0 NUM ejpam-5960 251 1 at at ADP ejpam-5960 251 2 π‘₯ π‘₯ NOUN ejpam-5960 251 3 = = SYM ejpam-5960 251 4 6 6 NUM ejpam-5960 251 5 and and CCONJ ejpam-5960 251 6 𝜌6(π‘₯ 𝜌6(π‘₯ NUM ejpam-5960 251 7 ) ) PUNCT ejpam-5960 251 8 = = SYM ejpam-5960 251 9 0.6 0.6 NUM ejpam-5960 251 10 at at ADP ejpam-5960 251 11 π‘₯ π‘₯ DET ejpam-5960 251 12 β‰  β‰  PROPN ejpam-5960 251 13 6 6 NUM ejpam-5960 251 14 . . PUNCT ejpam-5960 251 15 n. n. PROPN ejpam-5960 251 16 a. a. PROPN ejpam-5960 251 17 alsaedi alsaedi PROPN ejpam-5960 251 18 / / SYM ejpam-5960 251 19 eur eur PROPN ejpam-5960 251 20 . . PUNCT ejpam-5960 252 1 j. j. PROPN ejpam-5960 252 2 pure pure PROPN ejpam-5960 252 3 appl appl PROPN ejpam-5960 252 4 . . PROPN ejpam-5960 252 5 math math PROPN ejpam-5960 252 6 , , PUNCT ejpam-5960 252 7 18 18 NUM ejpam-5960 252 8 ( ( PUNCT ejpam-5960 252 9 4 4 NUM ejpam-5960 252 10 ) ) PUNCT ejpam-5960 252 11 ( ( PUNCT ejpam-5960 252 12 2025 2025 NUM ejpam-5960 252 13 ) ) PUNCT ejpam-5960 252 14 , , PUNCT ejpam-5960 252 15 5960 5960 NUM ejpam-5960 252 16 9 9 NUM ejpam-5960 252 17 of of ADP ejpam-5960 252 18 22 22 NUM ejpam-5960 252 19 but but CCONJ ejpam-5960 252 20 the the DET ejpam-5960 252 21 family family NOUN ejpam-5960 252 22 { { PUNCT ejpam-5960 252 23 𝑐𝑙 𝑐𝑙 X ejpam-5960 252 24 ( ( PUNCT ejpam-5960 252 25 int(πœŒπ‘› int(πœŒπ‘› NOUN ejpam-5960 252 26 ) ) PUNCT ejpam-5960 252 27 ) ) PUNCT ejpam-5960 252 28 : : PUNCT ejpam-5960 252 29 𝑛 𝑛 DET ejpam-5960 252 30 ∈ ∈ PROPN ejpam-5960 252 31 𝑋 𝑋 PROPN ejpam-5960 252 32 } } PUNCT ejpam-5960 252 33 = = PUNCT ejpam-5960 252 34 { { PUNCT ejpam-5960 252 35 𝜌2 𝜌2 ADJ ejpam-5960 252 36 ∨ ∨ NUM ejpam-5960 252 37 𝜎2 𝜎2 NOUN ejpam-5960 252 38 , , PUNCT ejpam-5960 252 39 𝜌3 𝜌3 VERB ejpam-5960 252 40 ∨ ∨ NUM ejpam-5960 252 41 𝜎3 𝜎3 NOUN ejpam-5960 252 42 , , PUNCT ejpam-5960 252 43 πœŒπ‘› πœŒπ‘› ADP ejpam-5960 252 44 : : PUNCT ejpam-5960 252 45 𝑛 𝑛 PRON ejpam-5960 252 46 β‰₯ β‰₯ NOUN ejpam-5960 252 47 4 4 NUM ejpam-5960 252 48 } } PUNCT ejpam-5960 252 49 . . PUNCT ejpam-5960 253 1 then then ADV ejpam-5960 253 2 any any DET ejpam-5960 253 3 finite finite NOUN ejpam-5960 253 4 subfamily subfamily ADV ejpam-5960 253 5 ψ ψ X ejpam-5960 253 6 β—¦ β—¦ NOUN ejpam-5960 253 7 = = SYM ejpam-5960 253 8 { { PUNCT ejpam-5960 253 9 𝑐𝑙 𝑐𝑙 X ejpam-5960 253 10 ( ( PUNCT ejpam-5960 253 11 int(πœŒπ‘› int(πœŒπ‘› NOUN ejpam-5960 253 12 ) ) PUNCT ejpam-5960 253 13 ) ) PUNCT ejpam-5960 253 14 : : PUNCT ejpam-5960 254 1 𝑖 𝑖 X ejpam-5960 254 2 < < X ejpam-5960 254 3 𝑛 𝑛 PROPN ejpam-5960 254 4 } } PUNCT ejpam-5960 254 5 ∈ ∈ NOUN ejpam-5960 254 6 2(ψ 2(ψ NUM ejpam-5960 254 7 ) ) PUNCT ejpam-5960 254 8 is be AUX ejpam-5960 254 9 not not PART ejpam-5960 254 10 0.8 0.8 NUM ejpam-5960 254 11 – – PUNCT ejpam-5960 254 12 rf rf NOUN ejpam-5960 254 13 of of ADP ejpam-5960 254 14 𝜌5 𝜌5 PROPN ejpam-5960 254 15 ( ( PUNCT ejpam-5960 254 16 because because SCONJ ejpam-5960 254 17 βˆƒπ‘₯0.8 βˆƒπ‘₯0.8 PROPN ejpam-5960 254 18 ∈ ∈ PROPN ejpam-5960 254 19 𝜌5 𝜌5 PROPN ejpam-5960 254 20 and and CCONJ ejpam-5960 254 21 βˆ€πœ† βˆ€πœ† X ejpam-5960 254 22 = = SYM ejpam-5960 254 23 𝜌2 𝜌2 ADV ejpam-5960 254 24 we we PRON ejpam-5960 254 25 have have VERB ejpam-5960 254 26 𝜌2 𝜌2 ADJ ejpam-5960 254 27 βˆ‰ βˆ‰ PROPN ejpam-5960 254 28 𝑅π‘₯0.8 𝑅π‘₯0.8 NOUN ejpam-5960 254 29 ) ) PUNCT ejpam-5960 254 30 . . PUNCT ejpam-5960 255 1 where where SCONJ ejpam-5960 255 2 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 255 3 ( ( PUNCT ejpam-5960 255 4 int(𝜌2 int(𝜌2 NOUN ejpam-5960 255 5 ) ) PUNCT ejpam-5960 255 6 ) ) PUNCT ejpam-5960 256 1 ( ( PUNCT ejpam-5960 256 2 π‘₯ π‘₯ X ejpam-5960 256 3 ) ) PUNCT ejpam-5960 256 4 = = SYM ejpam-5960 256 5 𝜌2 𝜌2 ADJ ejpam-5960 256 6 ∨ ∨ NUM ejpam-5960 256 7 𝜎2(π‘₯ 𝜎2(π‘₯ NOUN ejpam-5960 256 8 ) ) PUNCT ejpam-5960 256 9 = = SYM ejpam-5960 256 10 1 1 NUM ejpam-5960 256 11 at at ADP ejpam-5960 256 12 π‘₯ π‘₯ NOUN ejpam-5960 256 13 = = SYM ejpam-5960 256 14 2 2 NUM ejpam-5960 256 15 and and CCONJ ejpam-5960 256 16 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 256 17 ( ( PUNCT ejpam-5960 256 18 int(𝜌2 int(𝜌2 NOUN ejpam-5960 256 19 ) ) PUNCT ejpam-5960 256 20 ) ) PUNCT ejpam-5960 257 1 ( ( PUNCT ejpam-5960 257 2 π‘₯ π‘₯ X ejpam-5960 257 3 ) ) PUNCT ejpam-5960 257 4 = = SYM ejpam-5960 257 5 𝜌2 𝜌2 ADJ ejpam-5960 257 6 ∨ ∨ NUM ejpam-5960 257 7 𝜎2(π‘₯ 𝜎2(π‘₯ NOUN ejpam-5960 257 8 ) ) PUNCT ejpam-5960 257 9 = = SYM ejpam-5960 257 10 1 1 NUM ejpam-5960 257 11 at at ADP ejpam-5960 257 12 π‘₯ π‘₯ PRON ejpam-5960 257 13 β‰  β‰  PROPN ejpam-5960 257 14 2 2 NUM ejpam-5960 257 15 . . PUNCT ejpam-5960 258 1 thus thus ADV ejpam-5960 258 2 𝜌5 𝜌5 VERB ejpam-5960 258 3 is be AUX ejpam-5960 258 4 not not PART ejpam-5960 258 5 a a DET ejpam-5960 258 6 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 258 7 set set NOUN ejpam-5960 258 8 , , PUNCT ejpam-5960 258 9 however however ADV ejpam-5960 258 10 𝜌5 𝜌5 PROPN ejpam-5960 258 11 ∈ ∈ PROPN ejpam-5960 258 12 𝑅π‘₯0.8 𝑅π‘₯0.8 PROPN ejpam-5960 258 13 . . PUNCT ejpam-5960 259 1 thus thus ADV ejpam-5960 259 2 𝑁𝛼𝐡𝑅π‘₯0.8 𝑁𝛼𝐡𝑅π‘₯0.8 VERB ejpam-5960 259 3 βŠ† βŠ† NUM ejpam-5960 259 4 𝑅π‘₯0.8 𝑅π‘₯0.8 PROPN ejpam-5960 259 5 . . PUNCT ejpam-5960 260 1 definition definition NOUN ejpam-5960 260 2 3.11 3.11 NUM ejpam-5960 260 3 . . PUNCT ejpam-5960 261 1 let let VERB ejpam-5960 261 2 ( ( PUNCT ejpam-5960 261 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 261 4 , , PUNCT ejpam-5960 261 5 𝜏 𝜏 NOUN ejpam-5960 261 6 ) ) PUNCT ejpam-5960 261 7 be be VERB ejpam-5960 261 8 an an DET ejpam-5960 261 9 𝐿 𝐿 PROPN ejpam-5960 261 10 βˆ’ βˆ’ PROPN ejpam-5960 261 11 𝑑𝑠 𝑑𝑠 ADJ ejpam-5960 261 12 and and CCONJ ejpam-5960 261 13 πœ‡ πœ‡ ADP ejpam-5960 261 14 ∈ ∈ NOUN ejpam-5960 261 15 𝐿𝑋. 𝐿𝑋. VERB ejpam-5960 261 16 then then ADV ejpam-5960 261 17 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 261 18 ∈ ∈ PROPN ejpam-5960 261 19 𝑀 𝑀 PROPN ejpam-5960 261 20 ( ( PUNCT ejpam-5960 261 21 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 261 22 ) ) PUNCT ejpam-5960 261 23 is be AUX ejpam-5960 261 24 called call VERB ejpam-5960 261 25 a a DET ejpam-5960 261 26 𝑁.𝛼 𝑁.𝛼 PROPN ejpam-5960 261 27 βˆ’ βˆ’ PROPN ejpam-5960 261 28 π‘π‘œπ‘’π‘›π‘‘π‘’π‘‘ π‘π‘œπ‘’π‘›π‘‘π‘’π‘‘ VERB ejpam-5960 261 29 adherent adherent ADJ ejpam-5960 261 30 point point NOUN ejpam-5960 261 31 of of ADP ejpam-5960 261 32 πœ‡ πœ‡ PRON ejpam-5960 261 33 and and CCONJ ejpam-5960 261 34 write write VERB ejpam-5960 261 35 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 261 36 ∈ ∈ PROPN ejpam-5960 261 37 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 261 38 ( ( PUNCT ejpam-5960 261 39 πœ‡ πœ‡ NOUN ejpam-5960 261 40 ) ) PUNCT ejpam-5960 261 41 iff iff NOUN ejpam-5960 261 42 πœ‡ πœ‡ ADP ejpam-5960 261 43 β‰° β‰° PROPN ejpam-5960 261 44 πœ† πœ† PRON ejpam-5960 261 45 for for ADP ejpam-5960 261 46 each each DET ejpam-5960 261 47 πœ† πœ† DET ejpam-5960 261 48 ∈ ∈ NOUN ejpam-5960 261 49 𝛼𝐡𝑅π‘₯𝛼 𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 261 50 . . PUNCT ejpam-5960 262 1 if if SCONJ ejpam-5960 262 2 πœ‡ πœ‡ ADP ejpam-5960 262 3 = = X ejpam-5960 262 4 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 262 5 ( ( PUNCT ejpam-5960 262 6 πœ‡ πœ‡ NOUN ejpam-5960 262 7 ) ) PUNCT ejpam-5960 262 8 , , PUNCT ejpam-5960 262 9 then then ADV ejpam-5960 262 10 πœ‡ πœ‡ ADP ejpam-5960 262 11 is be AUX ejpam-5960 262 12 called call VERB ejpam-5960 262 13 a a DET ejpam-5960 262 14 𝑁𝛼𝐡 𝑁𝛼𝐡 NOUN ejpam-5960 262 15 βˆ’ βˆ’ NOUN ejpam-5960 262 16 π‘π‘™π‘œπ‘ π‘’π‘‘ π‘π‘™π‘œπ‘ π‘’π‘‘ NOUN ejpam-5960 262 17 𝐿 𝐿 PROPN ejpam-5960 262 18 βˆ’ βˆ’ PROPN ejpam-5960 262 19 𝑠𝑒𝑏𝑠𝑒𝑑. 𝑠𝑒𝑏𝑠𝑒𝑑. NOUN ejpam-5960 262 20 the the DET ejpam-5960 262 21 family family NOUN ejpam-5960 262 22 of of ADP ejpam-5960 262 23 all all DET ejpam-5960 262 24 𝑁𝛼𝐡 𝑁𝛼𝐡 PROPN ejpam-5960 262 25 βˆ’ βˆ’ PROPN ejpam-5960 262 26 π‘π‘™π‘œπ‘ π‘’π‘‘ π‘π‘™π‘œπ‘ π‘’π‘‘ NOUN ejpam-5960 262 27 𝐿 𝐿 PROPN ejpam-5960 262 28 βˆ’ βˆ’ PROPN ejpam-5960 262 29 𝑠𝑒𝑏𝑠𝑒𝑑𝑠 𝑠𝑒𝑏𝑠𝑒𝑑𝑠 NOUN ejpam-5960 262 30 is be AUX ejpam-5960 262 31 denoted denote VERB ejpam-5960 262 32 by by ADP ejpam-5960 262 33 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 262 34 ( ( PUNCT ejpam-5960 262 35 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 262 36 , , PUNCT ejpam-5960 262 37 𝜏 𝜏 NOUN ejpam-5960 262 38 ) ) PUNCT ejpam-5960 262 39 and and CCONJ ejpam-5960 262 40 its its PRON ejpam-5960 262 41 complement complement NOUN ejpam-5960 262 42 is be AUX ejpam-5960 262 43 called call VERB ejpam-5960 262 44 the the DET ejpam-5960 262 45 family family NOUN ejpam-5960 262 46 of of ADP ejpam-5960 262 47 all all DET ejpam-5960 262 48 𝑁𝛼𝐡 𝑁𝛼𝐡 NOUN ejpam-5960 262 49 βˆ’ βˆ’ PROPN ejpam-5960 262 50 π‘œπ‘π‘’π‘› π‘œπ‘π‘’π‘› NOUN ejpam-5960 262 51 𝐿 𝐿 PROPN ejpam-5960 262 52 βˆ’ βˆ’ PROPN ejpam-5960 262 53 𝑠𝑒𝑏𝑠𝑒𝑑𝑠 𝑠𝑒𝑏𝑠𝑒𝑑𝑠 NOUN ejpam-5960 262 54 and and CCONJ ejpam-5960 262 55 denoted denote VERB ejpam-5960 262 56 by by ADP ejpam-5960 262 57 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 PROPN ejpam-5960 262 58 ( ( PUNCT ejpam-5960 262 59 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 262 60 , , PUNCT ejpam-5960 262 61 𝜏 𝜏 NOUN ejpam-5960 262 62 ) ) PUNCT ejpam-5960 262 63 . . PUNCT ejpam-5960 263 1 theorem theorem VERB ejpam-5960 263 2 3.12 3.12 NUM ejpam-5960 263 3 . . PUNCT ejpam-5960 264 1 let let VERB ejpam-5960 264 2 ( ( PUNCT ejpam-5960 264 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 264 4 , , PUNCT ejpam-5960 264 5 𝜏 𝜏 NOUN ejpam-5960 264 6 ) ) PUNCT ejpam-5960 264 7 be be VERB ejpam-5960 264 8 an an DET ejpam-5960 264 9 𝐿 𝐿 PROPN ejpam-5960 264 10 βˆ’ βˆ’ PROPN ejpam-5960 264 11 𝑑𝑠 𝑑𝑠 VERB ejpam-5960 264 12 and and CCONJ ejpam-5960 264 13 let let VERB ejpam-5960 264 14 πœ‡ πœ‡ ADP ejpam-5960 264 15 ∈ ∈ X ejpam-5960 264 16 𝐿𝑋. 𝐿𝑋. ADJ ejpam-5960 264 17 then then ADV ejpam-5960 264 18 the the DET ejpam-5960 264 19 following follow VERB ejpam-5960 264 20 statements statement NOUN ejpam-5960 264 21 are be AUX ejpam-5960 264 22 true true ADJ ejpam-5960 264 23 : : PUNCT ejpam-5960 264 24 ( ( PUNCT ejpam-5960 264 25 i i NOUN ejpam-5960 264 26 ) ) PUNCT ejpam-5960 264 27 πœ‡ πœ‡ ADP ejpam-5960 264 28 ≀ ≀ ADJ ejpam-5960 264 29 𝑐𝑙 𝑐𝑙 ADP ejpam-5960 264 30 ( ( PUNCT ejpam-5960 264 31 πœ‡ πœ‡ ADP ejpam-5960 264 32 ) ) PUNCT ejpam-5960 264 33 ≀ ≀ NOUN ejpam-5960 264 34 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 264 35 ( ( PUNCT ejpam-5960 264 36 πœ‡ πœ‡ NOUN ejpam-5960 264 37 ) ) PUNCT ejpam-5960 264 38 . . PUNCT ejpam-5960 265 1 moreover moreover ADV ejpam-5960 265 2 , , PUNCT ejpam-5960 265 3 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 265 4 ( ( PUNCT ejpam-5960 265 5 πœ‡ πœ‡ NOUN ejpam-5960 265 6 ) ) PUNCT ejpam-5960 265 7 ≀ ≀ NOUN ejpam-5960 265 8 𝛼𝐡.𝑐𝑙 𝛼𝐡.𝑐𝑙 ADV ejpam-5960 265 9 ( ( PUNCT ejpam-5960 265 10 πœ‡ πœ‡ ADP ejpam-5960 265 11 ) ) PUNCT ejpam-5960 265 12 [ [ X ejpam-5960 265 13 26 26 NUM ejpam-5960 265 14 ] ] PUNCT ejpam-5960 265 15 ( ( PUNCT ejpam-5960 265 16 ii ii NOUN ejpam-5960 265 17 ) ) PUNCT ejpam-5960 265 18 if if SCONJ ejpam-5960 265 19 πœ‚ πœ‚ PROPN ejpam-5960 265 20 ∈ ∈ PROPN ejpam-5960 265 21 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 265 22 and and CCONJ ejpam-5960 265 23 πœ‡ πœ‡ ADP ejpam-5960 265 24 ≀ ≀ NOUN ejpam-5960 265 25 πœ‚ πœ‚ NOUN ejpam-5960 265 26 then then ADV ejpam-5960 265 27 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 265 28 ( ( PUNCT ejpam-5960 265 29 πœ‡ πœ‡ ADP ejpam-5960 265 30 ) ) PUNCT ejpam-5960 265 31 ≀ ≀ NOUN ejpam-5960 265 32 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 265 33 ( ( PUNCT ejpam-5960 265 34 πœ‚ πœ‚ NOUN ejpam-5960 265 35 ) ) PUNCT ejpam-5960 265 36 . . PUNCT ejpam-5960 266 1 ( ( PUNCT ejpam-5960 266 2 iii iii X ejpam-5960 266 3 ) ) PUNCT ejpam-5960 266 4 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 266 5 ( ( PUNCT ejpam-5960 266 6 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 266 7 ( ( PUNCT ejpam-5960 266 8 πœ‡ πœ‡ NOUN ejpam-5960 266 9 ) ) PUNCT ejpam-5960 266 10 ) ) PUNCT ejpam-5960 267 1 = = SYM ejpam-5960 267 2 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 267 3 ( ( PUNCT ejpam-5960 267 4 πœ‡ πœ‡ NOUN ejpam-5960 267 5 ) ) PUNCT ejpam-5960 267 6 . . PUNCT ejpam-5960 268 1 ( ( PUNCT ejpam-5960 268 2 iv iv X ejpam-5960 268 3 ) ) PUNCT ejpam-5960 268 4 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 268 5 ( ( PUNCT ejpam-5960 268 6 πœ‡ πœ‡ NOUN ejpam-5960 268 7 ) ) PUNCT ejpam-5960 268 8 = = SYM ejpam-5960 268 9 ∧{πœ‚ ∧{πœ‚ PROPN ejpam-5960 268 10 ∈ ∈ NOUN ejpam-5960 268 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 268 12 : : PUNCT ejpam-5960 268 13 πœ‚ πœ‚ X ejpam-5960 268 14 ∈ ∈ PROPN ejpam-5960 268 15 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 268 16 ( ( PUNCT ejpam-5960 268 17 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 268 18 , , PUNCT ejpam-5960 268 19 𝜏 𝜏 NOUN ejpam-5960 268 20 ) ) PUNCT ejpam-5960 268 21 , , PUNCT ejpam-5960 268 22 πœ‡ πœ‡ ADP ejpam-5960 268 23 ≀ ≀ PROPN ejpam-5960 268 24 πœ‚ πœ‚ NOUN ejpam-5960 268 25 } } PUNCT ejpam-5960 268 26 . . PUNCT ejpam-5960 269 1 proof proof NOUN ejpam-5960 269 2 . . PUNCT ejpam-5960 270 1 ( ( PUNCT ejpam-5960 270 2 i i NOUN ejpam-5960 270 3 ) ) PUNCT ejpam-5960 270 4 let let VERB ejpam-5960 270 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 270 6 ∈ ∈ PROPN ejpam-5960 270 7 𝑀 𝑀 PROPN ejpam-5960 270 8 ( ( PUNCT ejpam-5960 270 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 270 10 ) ) PUNCT ejpam-5960 270 11 such such ADJ ejpam-5960 270 12 that that SCONJ ejpam-5960 270 13 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 270 14 βˆ‰ βˆ‰ PROPN ejpam-5960 270 15 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 270 16 ( ( PUNCT ejpam-5960 270 17 πœ‡ πœ‡ NOUN ejpam-5960 270 18 ) ) PUNCT ejpam-5960 270 19 , , PUNCT ejpam-5960 270 20 then then ADV ejpam-5960 270 21 there there PRON ejpam-5960 270 22 exists exist VERB ejpam-5960 270 23 πœ† πœ† DET ejpam-5960 270 24 ∈ ∈ PROPN ejpam-5960 270 25 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 270 26 such such ADJ ejpam-5960 270 27 that that SCONJ ejpam-5960 270 28 πœ‡ πœ‡ ADP ejpam-5960 270 29 ≀ ≀ NUM ejpam-5960 270 30 πœ†. πœ†. NOUN ejpam-5960 270 31 since since SCONJ ejpam-5960 270 32 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 270 33 βŠ† βŠ† NUM ejpam-5960 270 34 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 270 35 and and CCONJ ejpam-5960 270 36 so so ADV ejpam-5960 270 37 πœ† πœ† DET ejpam-5960 270 38 ∈ ∈ PROPN ejpam-5960 270 39 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 270 40 and and CCONJ ejpam-5960 270 41 hence hence ADV ejpam-5960 270 42 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 270 43 βˆ‰ βˆ‰ PROPN ejpam-5960 270 44 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 270 45 ( ( PUNCT ejpam-5960 270 46 πœ‡ πœ‡ NOUN ejpam-5960 270 47 ) ) PUNCT ejpam-5960 270 48 . . PUNCT ejpam-5960 271 1 thus thus ADV ejpam-5960 271 2 𝑐𝑙 𝑐𝑙 DET ejpam-5960 271 3 ( ( PUNCT ejpam-5960 271 4 πœ‡ πœ‡ ADP ejpam-5960 271 5 ) ) PUNCT ejpam-5960 271 6 ≀ ≀ NOUN ejpam-5960 271 7 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 271 8 ( ( PUNCT ejpam-5960 271 9 πœ‡ πœ‡ NOUN ejpam-5960 271 10 ) ) PUNCT ejpam-5960 271 11 . . PUNCT ejpam-5960 272 1 ( ( PUNCT ejpam-5960 272 2 ii ii NOUN ejpam-5960 272 3 ) ) PUNCT ejpam-5960 272 4 let let VERB ejpam-5960 272 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 272 6 ∈ ∈ PROPN ejpam-5960 272 7 𝑀 𝑀 PROPN ejpam-5960 272 8 ( ( PUNCT ejpam-5960 272 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 272 10 ) ) PUNCT ejpam-5960 272 11 such such ADJ ejpam-5960 272 12 that that SCONJ ejpam-5960 272 13 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 272 14 βˆ‰ βˆ‰ PROPN ejpam-5960 272 15 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 272 16 ( ( PUNCT ejpam-5960 272 17 πœ‚ πœ‚ NOUN ejpam-5960 272 18 ) ) PUNCT ejpam-5960 272 19 , , PUNCT ejpam-5960 272 20 then then ADV ejpam-5960 272 21 there there PRON ejpam-5960 272 22 exists exist VERB ejpam-5960 272 23 πœ† πœ† DET ejpam-5960 272 24 ∈ ∈ PROPN ejpam-5960 272 25 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 272 26 such such ADJ ejpam-5960 272 27 that that SCONJ ejpam-5960 272 28 πœ‚ πœ‚ NOUN ejpam-5960 272 29 ≀ ≀ NUM ejpam-5960 272 30 πœ†. πœ†. NOUN ejpam-5960 272 31 since since SCONJ ejpam-5960 272 32 πœ‡ πœ‡ ADP ejpam-5960 272 33 ≀ ≀ NUM ejpam-5960 272 34 πœ‚ πœ‚ NOUN ejpam-5960 272 35 , , PUNCT ejpam-5960 272 36 then then ADV ejpam-5960 272 37 πœ‡ πœ‡ ADP ejpam-5960 272 38 ≀ ≀ NUM ejpam-5960 272 39 πœ† πœ† PRON ejpam-5960 273 1 and and CCONJ ejpam-5960 273 2 so so ADV ejpam-5960 273 3 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 273 4 βˆ‰ βˆ‰ PROPN ejpam-5960 273 5 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 273 6 ( ( PUNCT ejpam-5960 273 7 πœ‡ πœ‡ NOUN ejpam-5960 273 8 ) ) PUNCT ejpam-5960 273 9 . . PUNCT ejpam-5960 274 1 thus thus ADV ejpam-5960 274 2 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 274 3 ( ( PUNCT ejpam-5960 274 4 πœ‡ πœ‡ NOUN ejpam-5960 274 5 ) ) PUNCT ejpam-5960 274 6 ≀ ≀ NOUN ejpam-5960 274 7 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 274 8 ( ( PUNCT ejpam-5960 274 9 πœ‚ πœ‚ NOUN ejpam-5960 274 10 ) ) PUNCT ejpam-5960 274 11 . . PUNCT ejpam-5960 275 1 ( ( PUNCT ejpam-5960 275 2 iii iii X ejpam-5960 275 3 ) ) PUNCT ejpam-5960 275 4 suppose suppose VERB ejpam-5960 275 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 275 6 ∈ ∈ PROPN ejpam-5960 275 7 𝑀 𝑀 PROPN ejpam-5960 275 8 ( ( PUNCT ejpam-5960 275 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 275 10 ) ) PUNCT ejpam-5960 275 11 such such ADJ ejpam-5960 275 12 that that SCONJ ejpam-5960 275 13 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 275 14 ∈ ∈ PROPN ejpam-5960 275 15 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 275 16 ( ( PUNCT ejpam-5960 275 17 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 275 18 ( ( PUNCT ejpam-5960 275 19 πœ‡ πœ‡ NOUN ejpam-5960 275 20 ) ) PUNCT ejpam-5960 275 21 ) ) PUNCT ejpam-5960 275 22 . . PUNCT ejpam-5960 276 1 according accord VERB ejpam-5960 276 2 to to ADP ejpam-5960 276 3 definition definition NOUN ejpam-5960 276 4 3.11 3.11 NUM ejpam-5960 276 5 , , PUNCT ejpam-5960 276 6 we we PRON ejpam-5960 276 7 have have VERB ejpam-5960 276 8 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 276 9 ( ( PUNCT ejpam-5960 276 10 πœ‡ πœ‡ NOUN ejpam-5960 276 11 ) ) PUNCT ejpam-5960 276 12 β‰° β‰° PROPN ejpam-5960 276 13 πœ† πœ† PRON ejpam-5960 276 14 for for ADP ejpam-5960 276 15 each each DET ejpam-5960 276 16 πœ† πœ† PRON ejpam-5960 276 17 ∈ ∈ PROPN ejpam-5960 276 18 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 276 19 . . PUNCT ejpam-5960 277 1 hence hence ADV ejpam-5960 277 2 , , PUNCT ejpam-5960 277 3 there there PRON ejpam-5960 277 4 exists exist VERB ejpam-5960 277 5 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 277 6 ∈ ∈ PROPN ejpam-5960 277 7 𝑀 𝑀 PROPN ejpam-5960 277 8 ( ( PUNCT ejpam-5960 277 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 277 10 ) ) PUNCT ejpam-5960 277 11 such such ADJ ejpam-5960 277 12 that that SCONJ ejpam-5960 277 13 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 277 14 ∈ ∈ PROPN ejpam-5960 277 15 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 277 16 ( ( PUNCT ejpam-5960 277 17 πœ‡ πœ‡ NOUN ejpam-5960 277 18 ) ) PUNCT ejpam-5960 277 19 with with ADP ejpam-5960 277 20 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 277 21 βˆ‰ βˆ‰ PROPN ejpam-5960 277 22 πœ† πœ† PROPN ejpam-5960 277 23 and and CCONJ ejpam-5960 277 24 so so ADV ejpam-5960 277 25 πœ‡ πœ‡ ADP ejpam-5960 277 26 β‰° β‰° PROPN ejpam-5960 277 27 πœ† πœ† SYM ejpam-5960 277 28 , , PUNCT ejpam-5960 277 29 that that ADV ejpam-5960 277 30 is is ADV ejpam-5960 277 31 , , PUNCT ejpam-5960 277 32 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 277 33 ∈ ∈ PROPN ejpam-5960 277 34 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 277 35 ( ( PUNCT ejpam-5960 277 36 πœ‡ πœ‡ NOUN ejpam-5960 277 37 ) ) PUNCT ejpam-5960 277 38 . . PUNCT ejpam-5960 278 1 this this PRON ejpam-5960 278 2 shows show VERB ejpam-5960 278 3 that that SCONJ ejpam-5960 278 4 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 278 5 ( ( PUNCT ejpam-5960 278 6 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 278 7 ( ( PUNCT ejpam-5960 278 8 πœ‡ πœ‡ NOUN ejpam-5960 278 9 ) ) PUNCT ejpam-5960 278 10 ) ) PUNCT ejpam-5960 278 11 ≀ ≀ PUNCT ejpam-5960 278 12 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 278 13 ( ( PUNCT ejpam-5960 278 14 πœ‡ πœ‡ NOUN ejpam-5960 278 15 ) ) PUNCT ejpam-5960 278 16 . . PUNCT ejpam-5960 279 1 on on ADP ejpam-5960 279 2 the the DET ejpam-5960 279 3 other other ADJ ejpam-5960 279 4 hand hand NOUN ejpam-5960 279 5 , , PUNCT ejpam-5960 279 6 πœ‡ πœ‡ ADP ejpam-5960 279 7 ≀ ≀ ADJ ejpam-5960 279 8 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 279 9 ( ( PUNCT ejpam-5960 279 10 πœ‡ πœ‡ NOUN ejpam-5960 279 11 ) ) PUNCT ejpam-5960 279 12 follows follow VERB ejpam-5960 279 13 from from ADP ejpam-5960 279 14 ( ( PUNCT ejpam-5960 279 15 i i NOUN ejpam-5960 279 16 ) ) PUNCT ejpam-5960 279 17 and and CCONJ ejpam-5960 279 18 so so ADV ejpam-5960 279 19 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 279 20 ( ( PUNCT ejpam-5960 279 21 πœ‡ πœ‡ NOUN ejpam-5960 279 22 ) ) PUNCT ejpam-5960 279 23 ≀ ≀ NOUN ejpam-5960 279 24 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 279 25 ( ( PUNCT ejpam-5960 279 26 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 279 27 ( ( PUNCT ejpam-5960 279 28 πœ‡ πœ‡ NOUN ejpam-5960 279 29 ) ) PUNCT ejpam-5960 279 30 ) ) PUNCT ejpam-5960 279 31 . . PUNCT ejpam-5960 280 1 therefore therefore ADV ejpam-5960 280 2 , , PUNCT ejpam-5960 280 3 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 280 4 ( ( PUNCT ejpam-5960 280 5 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 280 6 ( ( PUNCT ejpam-5960 280 7 πœ‡ πœ‡ NOUN ejpam-5960 280 8 ) ) PUNCT ejpam-5960 280 9 ) ) PUNCT ejpam-5960 281 1 = = SYM ejpam-5960 281 2 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 281 3 ( ( PUNCT ejpam-5960 281 4 πœ‡ πœ‡ NOUN ejpam-5960 281 5 ) ) PUNCT ejpam-5960 281 6 . . PUNCT ejpam-5960 282 1 ( ( PUNCT ejpam-5960 282 2 iv iv X ejpam-5960 282 3 ) ) PUNCT ejpam-5960 282 4 on on ADP ejpam-5960 282 5 account account NOUN ejpam-5960 282 6 of of ADP ejpam-5960 282 7 ( ( PUNCT ejpam-5960 282 8 i i NOUN ejpam-5960 282 9 ) ) PUNCT ejpam-5960 282 10 and and CCONJ ejpam-5960 282 11 ( ( PUNCT ejpam-5960 282 12 iii iii NOUN ejpam-5960 282 13 ) ) PUNCT ejpam-5960 282 14 , , PUNCT ejpam-5960 282 15 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 282 16 ( ( PUNCT ejpam-5960 282 17 πœ‡ πœ‡ NOUN ejpam-5960 282 18 ) ) PUNCT ejpam-5960 282 19 is be AUX ejpam-5960 282 20 a a DET ejpam-5960 282 21 𝑁𝛼𝐡 𝑁𝛼𝐡 NOUN ejpam-5960 282 22 βˆ’ βˆ’ NOUN ejpam-5960 282 23 π‘π‘™π‘œπ‘ π‘’π‘‘ π‘π‘™π‘œπ‘ π‘’π‘‘ NOUN ejpam-5960 282 24 set set NOUN ejpam-5960 282 25 containing contain VERB ejpam-5960 282 26 πœ‡ πœ‡ ADP ejpam-5960 282 27 , , PUNCT ejpam-5960 282 28 and and CCONJ ejpam-5960 282 29 so so ADV ejpam-5960 282 30 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 282 31 ( ( PUNCT ejpam-5960 282 32 πœ‡ πœ‡ NOUN ejpam-5960 282 33 ) ) PUNCT ejpam-5960 282 34 β‰₯ β‰₯ NOUN ejpam-5960 282 35 ∧{πœ‚ ∧{πœ‚ PROPN ejpam-5960 282 36 ∈ ∈ PROPN ejpam-5960 283 1 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 283 2 : : PUNCT ejpam-5960 283 3 πœ‚ πœ‚ X ejpam-5960 283 4 ∈ ∈ PROPN ejpam-5960 283 5 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 283 6 ( ( PUNCT ejpam-5960 283 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 283 8 , , PUNCT ejpam-5960 283 9 𝜏 𝜏 NOUN ejpam-5960 283 10 ) ) PUNCT ejpam-5960 283 11 , , PUNCT ejpam-5960 283 12 πœ‡ πœ‡ ADP ejpam-5960 283 13 ≀ ≀ PROPN ejpam-5960 283 14 πœ‚ πœ‚ NOUN ejpam-5960 283 15 } } PUNCT ejpam-5960 283 16 . . PUNCT ejpam-5960 284 1 conversely conversely ADV ejpam-5960 284 2 , , PUNCT ejpam-5960 284 3 in in ADP ejpam-5960 284 4 case case NOUN ejpam-5960 284 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 284 6 ∈ ∈ PROPN ejpam-5960 284 7 𝑀 𝑀 PROPN ejpam-5960 284 8 ( ( PUNCT ejpam-5960 284 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 284 10 ) ) PUNCT ejpam-5960 284 11 and and CCONJ ejpam-5960 284 12 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 284 13 ∈ ∈ PROPN ejpam-5960 284 14 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 284 15 ( ( PUNCT ejpam-5960 284 16 πœ‡ πœ‡ NOUN ejpam-5960 284 17 ) ) PUNCT ejpam-5960 284 18 , , PUNCT ejpam-5960 284 19 then then ADV ejpam-5960 284 20 πœ‡ πœ‡ ADP ejpam-5960 284 21 β‰° β‰° PROPN ejpam-5960 284 22 πœ† πœ† PRON ejpam-5960 284 23 for for ADP ejpam-5960 284 24 each each DET ejpam-5960 284 25 πœ† πœ† PRON ejpam-5960 284 26 ∈ ∈ PROPN ejpam-5960 284 27 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 284 28 . . PUNCT ejpam-5960 285 1 hence hence ADV ejpam-5960 285 2 , , PUNCT ejpam-5960 285 3 if if SCONJ ejpam-5960 285 4 πœ‚ πœ‚ NOUN ejpam-5960 285 5 is be AUX ejpam-5960 285 6 an an DET ejpam-5960 285 7 𝑁𝛼𝐡 𝑁𝛼𝐡 NOUN ejpam-5960 285 8 βˆ’ βˆ’ NOUN ejpam-5960 285 9 π‘π‘™π‘œπ‘ π‘’π‘‘ π‘π‘™π‘œπ‘ π‘’π‘‘ NOUN ejpam-5960 285 10 set set NOUN ejpam-5960 285 11 containing contain VERB ejpam-5960 285 12 πœ‡ πœ‡ ADP ejpam-5960 285 13 , , PUNCT ejpam-5960 285 14 then then ADV ejpam-5960 285 15 πœ‚ πœ‚ PROPN ejpam-5960 285 16 β‰° β‰° PROPN ejpam-5960 285 17 πœ† πœ† ADV ejpam-5960 285 18 , , PUNCT ejpam-5960 285 19 and and CCONJ ejpam-5960 285 20 then then ADV ejpam-5960 285 21 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 285 22 ∈ ∈ PROPN ejpam-5960 285 23 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 285 24 ( ( PUNCT ejpam-5960 285 25 πœ‚ πœ‚ NOUN ejpam-5960 285 26 ) ) PUNCT ejpam-5960 285 27 = = VERB ejpam-5960 286 1 πœ‚. πœ‚. NOUN ejpam-5960 286 2 this this PRON ejpam-5960 286 3 implies imply VERB ejpam-5960 286 4 that that SCONJ ejpam-5960 286 5 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 286 6 ( ( PUNCT ejpam-5960 286 7 πœ‡ πœ‡ NOUN ejpam-5960 286 8 ) ) PUNCT ejpam-5960 286 9 ≀ ≀ NOUN ejpam-5960 286 10 ∧{πœ‚ ∧{πœ‚ PROPN ejpam-5960 286 11 ∈ ∈ PROPN ejpam-5960 286 12 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 286 13 : : PUNCT ejpam-5960 286 14 πœ‚ πœ‚ X ejpam-5960 286 15 ∈ ∈ PROPN ejpam-5960 286 16 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 286 17 ( ( PUNCT ejpam-5960 286 18 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 286 19 , , PUNCT ejpam-5960 286 20 𝜏 𝜏 NOUN ejpam-5960 286 21 ) ) PUNCT ejpam-5960 286 22 , , PUNCT ejpam-5960 286 23 πœ‡ πœ‡ ADP ejpam-5960 286 24 ≀ ≀ PROPN ejpam-5960 286 25 πœ‚ πœ‚ NOUN ejpam-5960 286 26 } } PUNCT ejpam-5960 286 27 . . PUNCT ejpam-5960 287 1 hence hence ADV ejpam-5960 287 2 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 287 3 ( ( PUNCT ejpam-5960 287 4 πœ‡ πœ‡ NOUN ejpam-5960 287 5 ) ) PUNCT ejpam-5960 287 6 = = SYM ejpam-5960 287 7 ∧{πœ‚ ∧{πœ‚ PROPN ejpam-5960 287 8 ∈ ∈ NOUN ejpam-5960 287 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 287 10 : : PUNCT ejpam-5960 287 11 πœ‚ πœ‚ X ejpam-5960 287 12 ∈ ∈ PROPN ejpam-5960 287 13 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 287 14 ( ( PUNCT ejpam-5960 287 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 287 16 , , PUNCT ejpam-5960 287 17 𝜏 𝜏 NOUN ejpam-5960 287 18 ) ) PUNCT ejpam-5960 287 19 , , PUNCT ejpam-5960 287 20 πœ‡ πœ‡ ADP ejpam-5960 287 21 ≀ ≀ PROPN ejpam-5960 287 22 πœ‚ πœ‚ NOUN ejpam-5960 287 23 } } PUNCT ejpam-5960 287 24 from from ADP ejpam-5960 287 25 theorem theorem ADJ ejpam-5960 287 26 3.12 3.12 NUM ejpam-5960 287 27 , , PUNCT ejpam-5960 287 28 one one PRON ejpam-5960 287 29 can can AUX ejpam-5960 287 30 see see VERB ejpam-5960 287 31 that that SCONJ ejpam-5960 287 32 every every DET ejpam-5960 287 33 𝑁𝛼𝐡 𝑁𝛼𝐡 NOUN ejpam-5960 287 34 βˆ’ βˆ’ PROPN ejpam-5960 287 35 π‘π‘™π‘œπ‘ π‘’π‘‘ π‘π‘™π‘œπ‘ π‘’π‘‘ NOUN ejpam-5960 287 36 𝐿 𝐿 PROPN ejpam-5960 287 37 βˆ’ βˆ’ PROPN ejpam-5960 287 38 𝑠𝑒𝑏𝑠𝑒𝑑 𝑠𝑒𝑏𝑠𝑒𝑑 NOUN ejpam-5960 287 39 is be AUX ejpam-5960 287 40 a a DET ejpam-5960 287 41 closed closed ADJ ejpam-5960 287 42 πΏβˆ’ πΏβˆ’ NOUN ejpam-5960 287 43 𝑠𝑒𝑏𝑠𝑒𝑑 𝑠𝑒𝑏𝑠𝑒𝑑 NOUN ejpam-5960 287 44 , , PUNCT ejpam-5960 287 45 but but CCONJ ejpam-5960 287 46 the the DET ejpam-5960 287 47 inverse inverse NOUN ejpam-5960 287 48 is be AUX ejpam-5960 287 49 not not PART ejpam-5960 287 50 true true ADJ ejpam-5960 287 51 since since SCONJ ejpam-5960 287 52 every every DET ejpam-5960 287 53 closed close VERB ejpam-5960 287 54 πΏβˆ’ πΏβˆ’ NOUN ejpam-5960 287 55 𝑠𝑒𝑏𝑠𝑒𝑑 𝑠𝑒𝑏𝑠𝑒𝑑 NOUN ejpam-5960 287 56 is be AUX ejpam-5960 287 57 not not PART ejpam-5960 287 58 a a DET ejpam-5960 287 59 𝑁.π›Όβˆ’π‘π‘œπ‘’π‘›π‘‘π‘’π‘‘ 𝑁.π›Όβˆ’π‘π‘œπ‘’π‘›π‘‘π‘’π‘‘ NOUN ejpam-5960 287 60 set set VERB ejpam-5960 287 61 in in ADP ejpam-5960 287 62 general general ADJ ejpam-5960 287 63 , , PUNCT ejpam-5960 287 64 as as SCONJ ejpam-5960 287 65 the the DET ejpam-5960 287 66 following follow VERB ejpam-5960 287 67 example example NOUN ejpam-5960 287 68 shows show NOUN ejpam-5960 287 69 . . PUNCT ejpam-5960 288 1 example example NOUN ejpam-5960 288 2 3.13 3.13 NUM ejpam-5960 288 3 . . PUNCT ejpam-5960 289 1 by by ADP ejpam-5960 289 2 example example NOUN ejpam-5960 289 3 3.10 3.10 NUM ejpam-5960 289 4 , , PUNCT ejpam-5960 289 5 let let VERB ejpam-5960 289 6 𝜌 𝜌 PART ejpam-5960 289 7 ∈ ∈ PROPN ejpam-5960 289 8 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 289 9 be be VERB ejpam-5960 289 10 a a DET ejpam-5960 289 11 𝐿–subset 𝐿–subset NOUN ejpam-5960 289 12 , , PUNCT ejpam-5960 289 13 define define VERB ejpam-5960 289 14 as as SCONJ ejpam-5960 289 15 follows follow VERB ejpam-5960 289 16 : : PUNCT ejpam-5960 289 17 𝜌(π‘₯ 𝜌(π‘₯ NUM ejpam-5960 289 18 ) ) PUNCT ejpam-5960 289 19 = = PRON ejpam-5960 289 20 { { PUNCT ejpam-5960 289 21 1 1 X ejpam-5960 289 22 : : PUNCT ejpam-5960 289 23 π‘₯ π‘₯ X ejpam-5960 289 24 = = SYM ejpam-5960 289 25 3 3 NUM ejpam-5960 289 26 , , PUNCT ejpam-5960 289 27 4 4 NUM ejpam-5960 289 28 , , PUNCT ejpam-5960 289 29 5 5 NUM ejpam-5960 289 30 , , PUNCT ejpam-5960 289 31 ... ... PUNCT ejpam-5960 289 32 1 1 NUM ejpam-5960 289 33 6 6 NUM ejpam-5960 289 34 : : PUNCT ejpam-5960 289 35 π‘₯ π‘₯ NOUN ejpam-5960 289 36 = = SYM ejpam-5960 289 37 2 2 NUM ejpam-5960 289 38 we we PRON ejpam-5960 289 39 note note VERB ejpam-5960 289 40 that that SCONJ ejpam-5960 289 41 𝜌 𝜌 PRON ejpam-5960 289 42 is be AUX ejpam-5960 289 43 a a DET ejpam-5960 289 44 closed closed ADJ ejpam-5960 289 45 𝐿–subset 𝐿–subset NOUN ejpam-5960 289 46 because because SCONJ ejpam-5960 289 47 𝜌 𝜌 X ejpam-5960 289 48 ∈ ∈ NOUN ejpam-5960 289 49 πœβ€² πœβ€² NOUN ejpam-5960 289 50 where where SCONJ ejpam-5960 289 51 πœβ€² πœβ€² PROPN ejpam-5960 289 52 is be AUX ejpam-5960 289 53 a a DET ejpam-5960 289 54 𝐿–topology 𝐿–topology NOUN ejpam-5960 289 55 with with ADP ejpam-5960 289 56 a a DET ejpam-5960 289 57 subbase subbase NOUN ejpam-5960 289 58 { { PUNCT ejpam-5960 289 59 πœ‡β€²π‘› πœ‡β€²π‘› ADV ejpam-5960 289 60 , , PUNCT ejpam-5960 289 61 πœ‚β€²π‘› πœ‚β€²π‘› ADV ejpam-5960 289 62 : : PUNCT ejpam-5960 289 63 𝑛 𝑛 DET ejpam-5960 289 64 ∈ ∈ PROPN ejpam-5960 289 65 𝑋 𝑋 PROPN ejpam-5960 289 66 } } PUNCT ejpam-5960 289 67 , , PUNCT ejpam-5960 289 68 and and CCONJ ejpam-5960 289 69 we we PRON ejpam-5960 289 70 have have VERB ejpam-5960 289 71 : : PUNCT ejpam-5960 289 72 n. n. NOUN ejpam-5960 289 73 a. a. PROPN ejpam-5960 289 74 alsaedi alsaedi PROPN ejpam-5960 289 75 / / SYM ejpam-5960 289 76 eur eur PROPN ejpam-5960 289 77 . . PUNCT ejpam-5960 290 1 j. j. PROPN ejpam-5960 290 2 pure pure PROPN ejpam-5960 290 3 appl appl PROPN ejpam-5960 290 4 . . PROPN ejpam-5960 290 5 math math PROPN ejpam-5960 290 6 , , PUNCT ejpam-5960 290 7 18 18 NUM ejpam-5960 290 8 ( ( PUNCT ejpam-5960 290 9 4 4 NUM ejpam-5960 290 10 ) ) PUNCT ejpam-5960 290 11 ( ( PUNCT ejpam-5960 290 12 2025 2025 NUM ejpam-5960 290 13 ) ) PUNCT ejpam-5960 290 14 , , PUNCT ejpam-5960 290 15 5960 5960 NUM ejpam-5960 290 16 10 10 NUM ejpam-5960 290 17 of of ADP ejpam-5960 290 18 22 22 NUM ejpam-5960 290 19 πœ‚3(π‘₯ πœ‚3(π‘₯ NUM ejpam-5960 290 20 ) ) PUNCT ejpam-5960 290 21 = = PRON ejpam-5960 290 22 { { PUNCT ejpam-5960 290 23 0 0 NUM ejpam-5960 290 24 : : PUNCT ejpam-5960 290 25 π‘₯ π‘₯ PRON ejpam-5960 290 26 β‰₯ β‰₯ NUM ejpam-5960 290 27 3 3 NUM ejpam-5960 290 28 5 5 NUM ejpam-5960 290 29 6 6 NUM ejpam-5960 290 30 : : PUNCT ejpam-5960 290 31 π‘₯ π‘₯ X ejpam-5960 290 32 < < X ejpam-5960 290 33 3 3 NUM ejpam-5960 291 1 and and CCONJ ejpam-5960 291 2 so so ADV ejpam-5960 291 3 𝜌 𝜌 X ejpam-5960 291 4 = = SYM ejpam-5960 291 5 πœ‚β€²3(π‘₯ πœ‚β€²3(π‘₯ NOUN ejpam-5960 291 6 ) ) PUNCT ejpam-5960 292 1 = = PRON ejpam-5960 292 2 { { PUNCT ejpam-5960 292 3 1 1 NUM ejpam-5960 292 4 : : PUNCT ejpam-5960 292 5 π‘₯ π‘₯ PRON ejpam-5960 292 6 β‰₯ β‰₯ NUM ejpam-5960 292 7 3 3 NUM ejpam-5960 292 8 1 1 NUM ejpam-5960 292 9 6 6 NUM ejpam-5960 292 10 : : PUNCT ejpam-5960 292 11 π‘₯ π‘₯ X ejpam-5960 292 12 < < X ejpam-5960 292 13 3 3 NUM ejpam-5960 292 14 therefore therefore ADV ejpam-5960 292 15 𝜌 𝜌 X ejpam-5960 292 16 ∈ ∈ PROPN ejpam-5960 292 17 πœβ€². πœβ€². NOUN ejpam-5960 292 18 now now ADV ejpam-5960 292 19 , , PUNCT ejpam-5960 292 20 the the DET ejpam-5960 292 21 family family NOUN ejpam-5960 292 22 ψ ψ X ejpam-5960 292 23 = = X ejpam-5960 292 24 { { PUNCT ejpam-5960 292 25 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 292 26 : : PUNCT ejpam-5960 292 27 𝑛 𝑛 DET ejpam-5960 292 28 ∈ ∈ PROPN ejpam-5960 292 29 𝑋 𝑋 PROPN ejpam-5960 292 30 } } PUNCT ejpam-5960 292 31 is be AUX ejpam-5960 292 32 0βˆ’cover 0βˆ’cover NOUN ejpam-5960 292 33 of of ADP ejpam-5960 292 34 1𝑋. 1𝑋. NUM ejpam-5960 292 35 since since SCONJ ejpam-5960 292 36 𝛼 𝛼 NOUN ejpam-5960 292 37 = = SYM ejpam-5960 292 38 0 0 NUM ejpam-5960 292 39 ∈ ∈ PROPN ejpam-5960 292 40 π‘ƒπ‘Ÿ π‘ƒπ‘Ÿ PROPN ejpam-5960 292 41 ( ( PUNCT ejpam-5960 292 42 𝐿 𝐿 PROPN ejpam-5960 292 43 ) ) PUNCT ejpam-5960 292 44 = = PUNCT ejpam-5960 293 1 [ [ X ejpam-5960 293 2 0 0 NUM ejpam-5960 293 3 , , PUNCT ejpam-5960 293 4 1 1 NUM ejpam-5960 293 5 ) ) PUNCT ejpam-5960 293 6 ( ( PUNCT ejpam-5960 293 7 βˆ€π‘₯ βˆ€π‘₯ NOUN ejpam-5960 293 8 ∈ ∈ PROPN ejpam-5960 293 9 𝑋 𝑋 NOUN ejpam-5960 293 10 βˆƒπœ† βˆƒπœ† PROPN ejpam-5960 293 11 ∈ ∈ PROPN ejpam-5960 293 12 ψ ψ NOUN ejpam-5960 293 13 βˆ‹ βˆ‹ NOUN ejpam-5960 293 14 πœ†(π‘₯ πœ†(π‘₯ NOUN ejpam-5960 293 15 ) ) PUNCT ejpam-5960 293 16 > > X ejpam-5960 293 17 0 0 X ejpam-5960 293 18 ) ) PUNCT ejpam-5960 293 19 which which PRON ejpam-5960 293 20 has have VERB ejpam-5960 293 21 no no DET ejpam-5960 293 22 finite finite NOUN ejpam-5960 293 23 subfamily subfamily ADV ejpam-5960 293 24 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 293 25 of of ADP ejpam-5960 293 26 ψ ψ PRON ejpam-5960 293 27 such such ADJ ejpam-5960 293 28 that that SCONJ ejpam-5960 293 29 { { PUNCT ejpam-5960 293 30 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 293 31 ( ( PUNCT ejpam-5960 293 32 𝑐𝑙 𝑐𝑙 X ejpam-5960 293 33 ( ( PUNCT ejpam-5960 293 34 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 293 35 ) ) PUNCT ejpam-5960 293 36 ) ) PUNCT ejpam-5960 293 37 : : PUNCT ejpam-5960 293 38 𝑖 𝑖 X ejpam-5960 293 39 < < X ejpam-5960 293 40 𝑛 𝑛 X ejpam-5960 293 41 } } PUNCT ejpam-5960 293 42 is be AUX ejpam-5960 293 43 a a DET ejpam-5960 293 44 0βˆ’cover 0βˆ’cover NOUN ejpam-5960 293 45 of of ADP ejpam-5960 293 46 𝜌 𝜌 X ejpam-5960 293 47 ( ( PUNCT ejpam-5960 293 48 since since SCONJ ejpam-5960 293 49 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 293 50 = = PUNCT ejpam-5960 293 51 { { PUNCT ejpam-5960 293 52 πœ‡2 πœ‡2 PROPN ejpam-5960 293 53 ∨ ∨ NUM ejpam-5960 293 54 πœ‚2 πœ‚2 PROPN ejpam-5960 293 55 , , PUNCT ejpam-5960 293 56 πœ‡π‘› πœ‡π‘› INTJ ejpam-5960 293 57 : : PUNCT ejpam-5960 293 58 𝑛 𝑛 DET ejpam-5960 293 59 β‰₯ β‰₯ NOUN ejpam-5960 293 60 4 4 NUM ejpam-5960 293 61 } } PUNCT ejpam-5960 293 62 is be AUX ejpam-5960 293 63 not not PART ejpam-5960 293 64 a a DET ejpam-5960 293 65 0βˆ’cover 0βˆ’cover NOUN ejpam-5960 293 66 of of ADP ejpam-5960 293 67 𝜌 𝜌 NOUN ejpam-5960 293 68 ) ) PUNCT ejpam-5960 293 69 . . PUNCT ejpam-5960 294 1 hence hence ADV ejpam-5960 294 2 𝜌 𝜌 PRON ejpam-5960 294 3 is be AUX ejpam-5960 294 4 not not PART ejpam-5960 294 5 a a DET ejpam-5960 294 6 𝑁.π›Όβˆ’bounded 𝑁.π›Όβˆ’bounde VERB ejpam-5960 294 7 set set NOUN ejpam-5960 294 8 . . PUNCT ejpam-5960 295 1 theorem theorem VERB ejpam-5960 295 2 3.14 3.14 NUM ejpam-5960 295 3 . . PUNCT ejpam-5960 296 1 let let VERB ejpam-5960 296 2 ( ( PUNCT ejpam-5960 296 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 296 4 , , PUNCT ejpam-5960 296 5 𝜏 𝜏 NOUN ejpam-5960 296 6 ) ) PUNCT ejpam-5960 296 7 be be VERB ejpam-5960 296 8 an an DET ejpam-5960 296 9 πΏβˆ’ts πΏβˆ’ts PROPN ejpam-5960 296 10 . . PUNCT ejpam-5960 297 1 the the DET ejpam-5960 297 2 following follow VERB ejpam-5960 297 3 statements statement NOUN ejpam-5960 297 4 hold hold VERB ejpam-5960 297 5 : : PUNCT ejpam-5960 297 6 ( ( PUNCT ejpam-5960 297 7 i i NOUN ejpam-5960 297 8 ) ) PUNCT ejpam-5960 297 9 0𝑋 0𝑋 PROPN ejpam-5960 297 10 , , PUNCT ejpam-5960 297 11 1𝑋 1𝑋 PROPN ejpam-5960 297 12 ∈ ∈ PROPN ejpam-5960 297 13 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 297 14 ( ( PUNCT ejpam-5960 297 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 297 16 , , PUNCT ejpam-5960 297 17 𝜏 𝜏 NOUN ejpam-5960 297 18 ) ) PUNCT ejpam-5960 297 19 . . PUNCT ejpam-5960 298 1 ( ( PUNCT ejpam-5960 298 2 ii ii NOUN ejpam-5960 298 3 ) ) PUNCT ejpam-5960 298 4 if if SCONJ ejpam-5960 298 5 πœ‡1 πœ‡1 ADJ ejpam-5960 298 6 , , PUNCT ejpam-5960 298 7 πœ‡2 πœ‡2 PROPN ejpam-5960 298 8 , , PUNCT ejpam-5960 298 9 ... ... PUNCT ejpam-5960 298 10 , , PUNCT ejpam-5960 298 11 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 298 12 ∈ ∈ PROPN ejpam-5960 298 13 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 298 14 ( ( PUNCT ejpam-5960 298 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 298 16 , , PUNCT ejpam-5960 298 17 𝜏 𝜏 NOUN ejpam-5960 298 18 ) ) PUNCT ejpam-5960 298 19 , , PUNCT ejpam-5960 298 20 then then ADV ejpam-5960 298 21 βˆ¨π‘› βˆ¨π‘› VERB ejpam-5960 298 22 𝑖=1 𝑖=1 PROPN ejpam-5960 298 23 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 298 24 ∈ ∈ PROPN ejpam-5960 298 25 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 298 26 ( ( PUNCT ejpam-5960 298 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 298 28 , , PUNCT ejpam-5960 298 29 𝜏 𝜏 NOUN ejpam-5960 298 30 ) ) PUNCT ejpam-5960 298 31 . . PUNCT ejpam-5960 299 1 ( ( PUNCT ejpam-5960 299 2 iii iii X ejpam-5960 299 3 ) ) PUNCT ejpam-5960 299 4 if if SCONJ ejpam-5960 299 5 { { PUNCT ejpam-5960 299 6 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 299 7 : : PUNCT ejpam-5960 299 8 𝑖 𝑖 SYM ejpam-5960 299 9 ∈ ∈ PROPN ejpam-5960 299 10 𝐼 𝐼 PROPN ejpam-5960 299 11 } } PUNCT ejpam-5960 299 12 βŠ† βŠ† NUM ejpam-5960 299 13 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 299 14 ( ( PUNCT ejpam-5960 299 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 299 16 , , PUNCT ejpam-5960 299 17 𝜏 𝜏 NOUN ejpam-5960 299 18 ) ) PUNCT ejpam-5960 299 19 , , PUNCT ejpam-5960 299 20 then then ADV ejpam-5960 299 21 ∧ ∧ PROPN ejpam-5960 299 22 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 299 23 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 299 24 ∈ ∈ PROPN ejpam-5960 299 25 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 299 26 ( ( PUNCT ejpam-5960 299 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 299 28 , , PUNCT ejpam-5960 299 29 𝜏 𝜏 NOUN ejpam-5960 299 30 ) ) PUNCT ejpam-5960 299 31 . . PUNCT ejpam-5960 300 1 ( ( PUNCT ejpam-5960 300 2 iv iv X ejpam-5960 300 3 ) ) PUNCT ejpam-5960 300 4 every every PRON ejpam-5960 300 5 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 300 6 and and CCONJ ejpam-5960 300 7 closed closed ADJ ejpam-5960 300 8 set set NOUN ejpam-5960 300 9 is be AUX ejpam-5960 300 10 π‘π›Όπ΅βˆ’closed π‘π›Όπ΅βˆ’close VERB ejpam-5960 300 11 . . PUNCT ejpam-5960 301 1 ( ( PUNCT ejpam-5960 301 2 v v NOUN ejpam-5960 301 3 ) ) PUNCT ejpam-5960 301 4 πœ‡ πœ‡ ADP ejpam-5960 301 5 ∈ ∈ PROPN ejpam-5960 301 6 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 301 7 is be AUX ejpam-5960 301 8 π‘π›Όπ΅βˆ’closed π‘π›Όπ΅βˆ’close VERB ejpam-5960 301 9 iff iff NOUN ejpam-5960 301 10 there there PRON ejpam-5960 301 11 exists exist VERB ejpam-5960 301 12 πœ† πœ† DET ejpam-5960 301 13 ∈ ∈ PROPN ejpam-5960 301 14 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 301 15 such such ADJ ejpam-5960 301 16 that that SCONJ ejpam-5960 301 17 πœ‡ πœ‡ ADP ejpam-5960 301 18 ≀ ≀ NOUN ejpam-5960 301 19 πœ† πœ† PRON ejpam-5960 301 20 for for ADP ejpam-5960 301 21 each each DET ejpam-5960 301 22 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 301 23 ∈ ∈ PROPN ejpam-5960 301 24 𝑀 𝑀 PROPN ejpam-5960 301 25 ( ( PUNCT ejpam-5960 301 26 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 301 27 ) ) PUNCT ejpam-5960 301 28 with with ADP ejpam-5960 301 29 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 301 30 βˆ‰ βˆ‰ PROPN ejpam-5960 301 31 πœ‡. πœ‡. NOUN ejpam-5960 301 32 proof proof NOUN ejpam-5960 301 33 . . PUNCT ejpam-5960 302 1 ( ( PUNCT ejpam-5960 302 2 i i NOUN ejpam-5960 302 3 ) ) PUNCT ejpam-5960 302 4 obvious obvious ADJ ejpam-5960 302 5 . . PUNCT ejpam-5960 303 1 ( ( PUNCT ejpam-5960 303 2 ii ii NOUN ejpam-5960 303 3 ) ) PUNCT ejpam-5960 303 4 let let VERB ejpam-5960 303 5 πœ‡1 πœ‡1 ADJ ejpam-5960 303 6 , , PUNCT ejpam-5960 303 7 πœ‡2 πœ‡2 PROPN ejpam-5960 303 8 , , PUNCT ejpam-5960 303 9 ... ... PUNCT ejpam-5960 303 10 , , PUNCT ejpam-5960 303 11 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 303 12 ∈ ∈ PROPN ejpam-5960 303 13 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 303 14 ( ( PUNCT ejpam-5960 303 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 303 16 , , PUNCT ejpam-5960 303 17 𝜏 𝜏 NOUN ejpam-5960 303 18 ) ) PUNCT ejpam-5960 303 19 and and CCONJ ejpam-5960 303 20 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 303 21 ∈ ∈ PROPN ejpam-5960 303 22 𝑀 𝑀 PROPN ejpam-5960 303 23 ( ( PUNCT ejpam-5960 303 24 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 303 25 ) ) PUNCT ejpam-5960 303 26 such such ADJ ejpam-5960 303 27 that that SCONJ ejpam-5960 303 28 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 303 29 ∈ ∈ PROPN ejpam-5960 303 30 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 303 31 ( ( PUNCT ejpam-5960 303 32 βˆ¨π‘› βˆ¨π‘› ADJ ejpam-5960 303 33 𝑖=1 𝑖=1 PROPN ejpam-5960 303 34 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 35 ) ) PUNCT ejpam-5960 303 36 , , PUNCT ejpam-5960 303 37 then then ADV ejpam-5960 303 38 for for ADP ejpam-5960 303 39 each each DET ejpam-5960 303 40 πœ† πœ† PRON ejpam-5960 303 41 ∈ ∈ PROPN ejpam-5960 303 42 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 303 43 we we PRON ejpam-5960 303 44 have have AUX ejpam-5960 303 45 βˆ¨π‘› βˆ¨π‘› VERB ejpam-5960 303 46 𝑖=1 𝑖=1 PROPN ejpam-5960 303 47 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 48 β‰° β‰° PROPN ejpam-5960 303 49 πœ† πœ† PRON ejpam-5960 303 50 and and CCONJ ejpam-5960 303 51 so so ADV ejpam-5960 303 52 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 53 β‰° β‰° PROPN ejpam-5960 303 54 πœ† πœ† PRON ejpam-5960 303 55 for for ADP ejpam-5960 303 56 some some PRON ejpam-5960 303 57 𝑖 𝑖 NOUN ejpam-5960 303 58 = = SYM ejpam-5960 303 59 1 1 NUM ejpam-5960 303 60 , , PUNCT ejpam-5960 303 61 2 2 NUM ejpam-5960 303 62 , , PUNCT ejpam-5960 303 63 ... ... PUNCT ejpam-5960 303 64 , , PUNCT ejpam-5960 303 65 𝑛. 𝑛. NOUN ejpam-5960 303 66 hence hence ADV ejpam-5960 303 67 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 303 68 ∈ ∈ PROPN ejpam-5960 303 69 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 303 70 ( ( PUNCT ejpam-5960 303 71 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 72 ) ) PUNCT ejpam-5960 303 73 for for ADP ejpam-5960 303 74 some some PRON ejpam-5960 303 75 𝑖 𝑖 NOUN ejpam-5960 303 76 = = SYM ejpam-5960 303 77 1 1 NUM ejpam-5960 303 78 , , PUNCT ejpam-5960 303 79 2 2 NUM ejpam-5960 303 80 , , PUNCT ejpam-5960 303 81 ... ... PUNCT ejpam-5960 303 82 , , PUNCT ejpam-5960 303 83 𝑛. 𝑛. NOUN ejpam-5960 303 84 since since SCONJ ejpam-5960 303 85 πœ‡π‘– πœ‡π‘– ADV ejpam-5960 303 86 is be AUX ejpam-5960 303 87 a a DET ejpam-5960 303 88 𝑁𝛼𝐡–closed 𝑁𝛼𝐡–closed PROPN ejpam-5960 303 89 set set NOUN ejpam-5960 303 90 , , PUNCT ejpam-5960 303 91 then then ADV ejpam-5960 303 92 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 303 93 ( ( PUNCT ejpam-5960 303 94 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 95 ) ) PUNCT ejpam-5960 303 96 ≀ ≀ NOUN ejpam-5960 303 97 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 98 for for ADP ejpam-5960 303 99 some some PRON ejpam-5960 303 100 𝑖 𝑖 NOUN ejpam-5960 303 101 = = SYM ejpam-5960 303 102 1 1 NUM ejpam-5960 303 103 , , PUNCT ejpam-5960 303 104 2 2 NUM ejpam-5960 303 105 , , PUNCT ejpam-5960 303 106 ... ... PUNCT ejpam-5960 303 107 , , PUNCT ejpam-5960 303 108 𝑛 𝑛 PROPN ejpam-5960 303 109 and and CCONJ ejpam-5960 303 110 so so ADV ejpam-5960 303 111 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 303 112 ∈ ∈ PROPN ejpam-5960 303 113 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 114 for for ADP ejpam-5960 303 115 some some PRON ejpam-5960 303 116 𝑖 𝑖 NOUN ejpam-5960 303 117 = = SYM ejpam-5960 303 118 1 1 NUM ejpam-5960 303 119 , , PUNCT ejpam-5960 303 120 2 2 NUM ejpam-5960 303 121 , , PUNCT ejpam-5960 303 122 ... ... PUNCT ejpam-5960 303 123 , , PUNCT ejpam-5960 303 124 𝑛 𝑛 PROPN ejpam-5960 303 125 and and CCONJ ejpam-5960 303 126 hence hence ADV ejpam-5960 303 127 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 303 128 ∈ ∈ PROPN ejpam-5960 303 129 βˆ¨π‘› βˆ¨π‘› ADJ ejpam-5960 303 130 𝑖=1 𝑖=1 PROPN ejpam-5960 303 131 πœ‡π‘–. πœ‡π‘–. AUX ejpam-5960 303 132 thus thus ADV ejpam-5960 303 133 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 VERB ejpam-5960 303 134 ( ( PUNCT ejpam-5960 303 135 βˆ¨π‘› βˆ¨π‘› ADJ ejpam-5960 303 136 𝑖=1 𝑖=1 PROPN ejpam-5960 303 137 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 138 ) ) PUNCT ejpam-5960 303 139 ≀ ≀ NOUN ejpam-5960 303 140 βˆ¨π‘› βˆ¨π‘› VERB ejpam-5960 303 141 𝑖=1 𝑖=1 PROPN ejpam-5960 303 142 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 303 143 . . PUNCT ejpam-5960 303 144 . . PUNCT ejpam-5960 303 145 . . PUNCT ejpam-5960 304 1 ( ( PUNCT ejpam-5960 304 2 βˆ— βˆ— NOUN ejpam-5960 304 3 ) ) PUNCT ejpam-5960 304 4 conversely conversely ADV ejpam-5960 304 5 , , PUNCT ejpam-5960 304 6 since since SCONJ ejpam-5960 304 7 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 304 8 ≀ ≀ NOUN ejpam-5960 304 9 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 304 10 ( ( PUNCT ejpam-5960 304 11 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 304 12 ) ) PUNCT ejpam-5960 304 13 then then ADV ejpam-5960 304 14 βˆ¨π‘› βˆ¨π‘› VERB ejpam-5960 304 15 𝑖=1 𝑖=1 PROPN ejpam-5960 304 16 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 304 17 ≀ ≀ NOUN ejpam-5960 304 18 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 304 19 ( ( PUNCT ejpam-5960 304 20 βˆ¨π‘› βˆ¨π‘› ADJ ejpam-5960 304 21 𝑖=1 𝑖=1 PROPN ejpam-5960 304 22 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 304 23 ) ) PUNCT ejpam-5960 304 24 . . PUNCT ejpam-5960 304 25 . . PUNCT ejpam-5960 304 26 . . PUNCT ejpam-5960 305 1 ( ( PUNCT ejpam-5960 305 2 βˆ—βˆ— βˆ—βˆ— NOUN ejpam-5960 305 3 ) ) PUNCT ejpam-5960 305 4 . . PUNCT ejpam-5960 306 1 hence hence ADV ejpam-5960 306 2 from from ADP ejpam-5960 306 3 ( ( PUNCT ejpam-5960 306 4 βˆ— βˆ— NOUN ejpam-5960 306 5 ) ) PUNCT ejpam-5960 306 6 and and CCONJ ejpam-5960 306 7 ( ( PUNCT ejpam-5960 306 8 βˆ—βˆ— βˆ—βˆ— X ejpam-5960 306 9 ) ) PUNCT ejpam-5960 306 10 we we PRON ejpam-5960 306 11 have have AUX ejpam-5960 306 12 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 VERB ejpam-5960 306 13 ( ( PUNCT ejpam-5960 306 14 βˆ¨π‘› βˆ¨π‘› ADJ ejpam-5960 306 15 𝑖=1 𝑖=1 PROPN ejpam-5960 306 16 πœ‡π‘– πœ‡π‘– ADV ejpam-5960 306 17 ) ) PUNCT ejpam-5960 306 18 = = VERB ejpam-5960 307 1 βˆ¨π‘› βˆ¨π‘› ADJ ejpam-5960 307 2 𝑖=1 𝑖=1 PROPN ejpam-5960 307 3 πœ‡π‘–. πœ‡π‘–. AUX ejpam-5960 307 4 thus thus ADV ejpam-5960 307 5 βˆ¨π‘› βˆ¨π‘› VERB ejpam-5960 307 6 𝑖=1 𝑖=1 PROPN ejpam-5960 307 7 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 307 8 ∈ ∈ PROPN ejpam-5960 307 9 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 307 10 ( ( PUNCT ejpam-5960 307 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 307 12 , , PUNCT ejpam-5960 307 13 𝜏 𝜏 NOUN ejpam-5960 307 14 ) ) PUNCT ejpam-5960 307 15 . . PUNCT ejpam-5960 308 1 ( ( PUNCT ejpam-5960 308 2 iii iii X ejpam-5960 308 3 ) ) PUNCT ejpam-5960 308 4 let let VERB ejpam-5960 308 5 πœ‡1 πœ‡1 ADJ ejpam-5960 308 6 , , PUNCT ejpam-5960 308 7 πœ‡2 πœ‡2 PROPN ejpam-5960 308 8 , , PUNCT ejpam-5960 308 9 ... ... PUNCT ejpam-5960 308 10 , , PUNCT ejpam-5960 308 11 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 308 12 ∈ ∈ PROPN ejpam-5960 308 13 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 308 14 ( ( PUNCT ejpam-5960 308 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 308 16 , , PUNCT ejpam-5960 308 17 𝜏 𝜏 NOUN ejpam-5960 308 18 ) ) PUNCT ejpam-5960 308 19 and and CCONJ ejpam-5960 308 20 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 308 21 ∈ ∈ PROPN ejpam-5960 308 22 𝑀 𝑀 PROPN ejpam-5960 308 23 ( ( PUNCT ejpam-5960 308 24 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 308 25 ) ) PUNCT ejpam-5960 308 26 such such ADJ ejpam-5960 308 27 that that SCONJ ejpam-5960 308 28 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 308 29 ∈ ∈ PROPN ejpam-5960 308 30 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 308 31 ( ( PUNCT ejpam-5960 308 32 βˆ§π‘–βˆˆπΌ βˆ§π‘–βˆˆπΌ NOUN ejpam-5960 308 33 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 34 ) ) PUNCT ejpam-5960 308 35 , , PUNCT ejpam-5960 308 36 then then ADV ejpam-5960 308 37 for for ADP ejpam-5960 308 38 each each DET ejpam-5960 308 39 πœ† πœ† PRON ejpam-5960 308 40 ∈ ∈ PROPN ejpam-5960 308 41 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 308 42 we we PRON ejpam-5960 308 43 have have VERB ejpam-5960 308 44 ∧ ∧ PROPN ejpam-5960 308 45 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 308 46 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 47 β‰° β‰° PROPN ejpam-5960 308 48 πœ† πœ† PRON ejpam-5960 308 49 and and CCONJ ejpam-5960 308 50 so so ADV ejpam-5960 308 51 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 52 β‰° β‰° PROPN ejpam-5960 308 53 πœ† πœ† ADV ejpam-5960 308 54 for for ADP ejpam-5960 308 55 each each DET ejpam-5960 308 56 𝑖 𝑖 SYM ejpam-5960 308 57 ∈ ∈ PROPN ejpam-5960 308 58 𝐼. 𝐼. PROPN ejpam-5960 308 59 hence hence ADV ejpam-5960 308 60 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 308 61 ∈ ∈ PROPN ejpam-5960 308 62 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 308 63 ( ( PUNCT ejpam-5960 308 64 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 65 ) ) PUNCT ejpam-5960 308 66 for for ADP ejpam-5960 308 67 each each DET ejpam-5960 308 68 𝑖 𝑖 SYM ejpam-5960 308 69 ∈ ∈ PROPN ejpam-5960 308 70 𝐼. 𝐼. PROPN ejpam-5960 308 71 since since SCONJ ejpam-5960 308 72 πœ‡π‘– πœ‡π‘– ADV ejpam-5960 308 73 is be AUX ejpam-5960 308 74 a a DET ejpam-5960 308 75 𝑁𝛼𝐡–closed 𝑁𝛼𝐡–closed PROPN ejpam-5960 308 76 set set NOUN ejpam-5960 308 77 , , PUNCT ejpam-5960 308 78 then then ADV ejpam-5960 308 79 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 308 80 ( ( PUNCT ejpam-5960 308 81 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 82 ) ) PUNCT ejpam-5960 308 83 ≀ ≀ NOUN ejpam-5960 308 84 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 85 for for ADP ejpam-5960 308 86 each each DET ejpam-5960 308 87 𝑖 𝑖 SYM ejpam-5960 308 88 ∈ ∈ PROPN ejpam-5960 308 89 𝐼 𝐼 PROPN ejpam-5960 308 90 and and CCONJ ejpam-5960 308 91 so so ADV ejpam-5960 308 92 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 308 93 ∈ ∈ PROPN ejpam-5960 308 94 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 95 for for ADP ejpam-5960 308 96 each each DET ejpam-5960 308 97 𝑖 𝑖 SYM ejpam-5960 308 98 ∈ ∈ PROPN ejpam-5960 308 99 𝐼 𝐼 PROPN ejpam-5960 308 100 and and CCONJ ejpam-5960 308 101 hence hence ADV ejpam-5960 308 102 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 308 103 ∈ ∈ PROPN ejpam-5960 308 104 ∧ ∧ PROPN ejpam-5960 308 105 π‘–βˆˆπΌ π‘–βˆˆπΌ X ejpam-5960 308 106 πœ‡π‘–. πœ‡π‘–. VERB ejpam-5960 308 107 thus thus ADV ejpam-5960 308 108 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 VERB ejpam-5960 308 109 ( ( PUNCT ejpam-5960 308 110 βˆ§π‘–βˆˆπΌ βˆ§π‘–βˆˆπΌ NOUN ejpam-5960 308 111 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 112 ) ) PUNCT ejpam-5960 308 113 β‰€βˆ§ β‰€βˆ§ X ejpam-5960 308 114 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 308 115 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 308 116 . . PUNCT ejpam-5960 308 117 . . PUNCT ejpam-5960 308 118 . . PUNCT ejpam-5960 309 1 ( ( PUNCT ejpam-5960 309 2 βˆ— βˆ— NOUN ejpam-5960 309 3 ) ) PUNCT ejpam-5960 309 4 . . PUNCT ejpam-5960 310 1 conversely conversely ADV ejpam-5960 310 2 , , PUNCT ejpam-5960 310 3 since since SCONJ ejpam-5960 310 4 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 310 5 ≀ ≀ NOUN ejpam-5960 310 6 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 310 7 ( ( PUNCT ejpam-5960 310 8 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 310 9 ) ) PUNCT ejpam-5960 310 10 then then ADV ejpam-5960 310 11 ∧ ∧ PROPN ejpam-5960 310 12 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 310 13 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 310 14 ≀ ≀ NOUN ejpam-5960 310 15 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 310 16 ( ( PUNCT ejpam-5960 310 17 βˆ§π‘–βˆˆπΌ βˆ§π‘–βˆˆπΌ NOUN ejpam-5960 310 18 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 310 19 ) ) PUNCT ejpam-5960 310 20 . . PUNCT ejpam-5960 310 21 . . PUNCT ejpam-5960 310 22 . . PUNCT ejpam-5960 311 1 ( ( PUNCT ejpam-5960 311 2 βˆ—βˆ— βˆ—βˆ— NOUN ejpam-5960 311 3 ) ) PUNCT ejpam-5960 311 4 . . PUNCT ejpam-5960 312 1 hence hence ADV ejpam-5960 312 2 from from ADP ejpam-5960 312 3 ( ( PUNCT ejpam-5960 312 4 βˆ— βˆ— NOUN ejpam-5960 312 5 ) ) PUNCT ejpam-5960 312 6 and and CCONJ ejpam-5960 312 7 ( ( PUNCT ejpam-5960 312 8 βˆ—βˆ— βˆ—βˆ— X ejpam-5960 312 9 ) ) PUNCT ejpam-5960 312 10 we we PRON ejpam-5960 312 11 have have AUX ejpam-5960 312 12 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 VERB ejpam-5960 312 13 ( ( PUNCT ejpam-5960 312 14 βˆ§π‘–βˆˆπΌ βˆ§π‘–βˆˆπΌ NOUN ejpam-5960 312 15 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 312 16 ) ) PUNCT ejpam-5960 312 17 = = SYM ejpam-5960 313 1 ∧ ∧ PROPN ejpam-5960 313 2 π‘–βˆˆπΌ π‘–βˆˆπΌ X ejpam-5960 313 3 πœ‡π‘–. πœ‡π‘–. VERB ejpam-5960 313 4 thus thus ADV ejpam-5960 313 5 ∧ ∧ PROPN ejpam-5960 313 6 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 313 7 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 313 8 ∈ ∈ PROPN ejpam-5960 313 9 𝑁𝛼𝐡𝐢 𝑁𝛼𝐡𝐢 PROPN ejpam-5960 313 10 ( ( PUNCT ejpam-5960 313 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 313 12 , , PUNCT ejpam-5960 313 13 𝜏 𝜏 NOUN ejpam-5960 313 14 ) ) PUNCT ejpam-5960 313 15 . . PUNCT ejpam-5960 314 1 ( ( PUNCT ejpam-5960 314 2 iv iv X ejpam-5960 314 3 ) ) PUNCT ejpam-5960 314 4 let let VERB ejpam-5960 314 5 πœ‡ πœ‡ SCONJ ejpam-5960 314 6 ∈ ∈ NOUN ejpam-5960 314 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 314 8 be be AUX ejpam-5960 314 9 a a DET ejpam-5960 314 10 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 314 11 and and CCONJ ejpam-5960 314 12 closed close VERB ejpam-5960 314 13 set set VERB ejpam-5960 314 14 and and CCONJ ejpam-5960 314 15 let let VERB ejpam-5960 314 16 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 314 17 ∈ ∈ PROPN ejpam-5960 314 18 𝑀 𝑀 PROPN ejpam-5960 314 19 ( ( PUNCT ejpam-5960 314 20 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 314 21 ) ) PUNCT ejpam-5960 314 22 such such ADJ ejpam-5960 314 23 that that SCONJ ejpam-5960 314 24 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 314 25 βˆ‰ βˆ‰ PROPN ejpam-5960 314 26 πœ‡ πœ‡ X ejpam-5960 314 27 , , PUNCT ejpam-5960 314 28 since since SCONJ ejpam-5960 314 29 πœ‡ πœ‡ ADV ejpam-5960 314 30 is be AUX ejpam-5960 314 31 a a DET ejpam-5960 314 32 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 314 33 and and CCONJ ejpam-5960 314 34 closed close VERB ejpam-5960 314 35 set set NOUN ejpam-5960 314 36 , , PUNCT ejpam-5960 314 37 then then ADV ejpam-5960 314 38 πœ‡ πœ‡ ADP ejpam-5960 314 39 ∈ ∈ PROPN ejpam-5960 314 40 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 314 41 . . PUNCT ejpam-5960 315 1 since since SCONJ ejpam-5960 315 2 πœ‡ πœ‡ ADP ejpam-5960 315 3 ≀ ≀ NUM ejpam-5960 315 4 πœ‡ πœ‡ ADP ejpam-5960 315 5 , , PUNCT ejpam-5960 315 6 then then ADV ejpam-5960 315 7 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 315 8 βˆ‰ βˆ‰ PROPN ejpam-5960 315 9 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 315 10 ( ( PUNCT ejpam-5960 315 11 πœ‡ πœ‡ NOUN ejpam-5960 315 12 ) ) PUNCT ejpam-5960 315 13 and and CCONJ ejpam-5960 315 14 so so ADV ejpam-5960 315 15 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 315 16 ( ( PUNCT ejpam-5960 315 17 πœ‡ πœ‡ NOUN ejpam-5960 315 18 ) ) PUNCT ejpam-5960 315 19 ≀ ≀ NUM ejpam-5960 315 20 πœ‡. πœ‡. NOUN ejpam-5960 315 21 therefore therefore ADV ejpam-5960 315 22 πœ‡ πœ‡ ADV ejpam-5960 315 23 is be AUX ejpam-5960 315 24 π‘π›Όπ΅βˆ’closed π‘π›Όπ΅βˆ’close VERB ejpam-5960 315 25 set set VERB ejpam-5960 315 26 . . PUNCT ejpam-5960 316 1 ( ( PUNCT ejpam-5960 316 2 v v NOUN ejpam-5960 316 3 ) ) PUNCT ejpam-5960 316 4 suppose suppose VERB ejpam-5960 316 5 that that SCONJ ejpam-5960 316 6 πœ‡ πœ‡ ADP ejpam-5960 316 7 is be AUX ejpam-5960 316 8 a a DET ejpam-5960 316 9 π‘π›Όπ΅βˆ’closed π‘π›Όπ΅βˆ’close VERB ejpam-5960 316 10 set set NOUN ejpam-5960 316 11 , , PUNCT ejpam-5960 316 12 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 316 13 ∈ ∈ PROPN ejpam-5960 316 14 𝑀 𝑀 PROPN ejpam-5960 316 15 ( ( PUNCT ejpam-5960 316 16 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 316 17 ) ) PUNCT ejpam-5960 316 18 and and CCONJ ejpam-5960 316 19 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 316 20 βˆ‰ βˆ‰ PROPN ejpam-5960 316 21 πœ‡. πœ‡. NOUN ejpam-5960 316 22 by by ADP ejpam-5960 316 23 definition definition NOUN ejpam-5960 316 24 3.11 3.11 NUM ejpam-5960 316 25 , , PUNCT ejpam-5960 316 26 there there PRON ejpam-5960 316 27 exists exist VERB ejpam-5960 316 28 πœ† πœ† DET ejpam-5960 316 29 ∈ ∈ PROPN ejpam-5960 316 30 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 316 31 with with ADP ejpam-5960 316 32 πœ‡ πœ‡ ADP ejpam-5960 316 33 ≀ ≀ NUM ejpam-5960 316 34 πœ†. πœ†. NOUN ejpam-5960 316 35 conversely conversely ADV ejpam-5960 316 36 , , PUNCT ejpam-5960 316 37 provided provide VERB ejpam-5960 316 38 that that SCONJ ejpam-5960 316 39 the the DET ejpam-5960 316 40 condition condition NOUN ejpam-5960 316 41 is be AUX ejpam-5960 316 42 satisfied satisfied ADJ ejpam-5960 316 43 . . PUNCT ejpam-5960 317 1 if if SCONJ ejpam-5960 317 2 πœ‡ πœ‡ ADV ejpam-5960 317 3 is be AUX ejpam-5960 317 4 not not PART ejpam-5960 317 5 a a DET ejpam-5960 317 6 π‘π›Όπ΅βˆ’closed π‘π›Όπ΅βˆ’close VERB ejpam-5960 317 7 set set NOUN ejpam-5960 317 8 , , PUNCT ejpam-5960 317 9 then then ADV ejpam-5960 317 10 there there PRON ejpam-5960 317 11 exists exist VERB ejpam-5960 317 12 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 317 13 ∈ ∈ PROPN ejpam-5960 317 14 𝑀 𝑀 PROPN ejpam-5960 317 15 ( ( PUNCT ejpam-5960 317 16 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 317 17 ) ) PUNCT ejpam-5960 317 18 such such ADJ ejpam-5960 317 19 that that SCONJ ejpam-5960 317 20 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 317 21 ∈ ∈ PROPN ejpam-5960 317 22 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 317 23 ( ( PUNCT ejpam-5960 317 24 πœ‡ πœ‡ NOUN ejpam-5960 317 25 ) ) PUNCT ejpam-5960 317 26 and and CCONJ ejpam-5960 317 27 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 317 28 βˆ‰ βˆ‰ PROPN ejpam-5960 317 29 πœ‡. πœ‡. NOUN ejpam-5960 317 30 hence hence ADV ejpam-5960 317 31 πœ‡ πœ‡ ADP ejpam-5960 317 32 β‰° β‰° PROPN ejpam-5960 317 33 πœ† πœ† PRON ejpam-5960 317 34 for for ADP ejpam-5960 317 35 each each DET ejpam-5960 317 36 πœ† πœ† PRON ejpam-5960 317 37 ∈ ∈ PROPN ejpam-5960 317 38 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 317 39 . . PUNCT ejpam-5960 318 1 it it PRON ejpam-5960 318 2 conflicts conflict VERB ejpam-5960 318 3 with with ADP ejpam-5960 318 4 the the DET ejpam-5960 318 5 hypothesis hypothesis NOUN ejpam-5960 318 6 , , PUNCT ejpam-5960 318 7 and and CCONJ ejpam-5960 318 8 so so ADV ejpam-5960 318 9 πœ‡ πœ‡ PRON ejpam-5960 318 10 is be AUX ejpam-5960 318 11 a a DET ejpam-5960 318 12 π‘π›Όπ΅βˆ’closed π‘π›Όπ΅βˆ’close VERB ejpam-5960 318 13 set set NOUN ejpam-5960 318 14 . . PUNCT ejpam-5960 319 1 n. n. PROPN ejpam-5960 319 2 a. a. PROPN ejpam-5960 319 3 alsaedi alsaedi PROPN ejpam-5960 319 4 / / SYM ejpam-5960 319 5 eur eur PROPN ejpam-5960 319 6 . . PUNCT ejpam-5960 320 1 j. j. PROPN ejpam-5960 320 2 pure pure PROPN ejpam-5960 320 3 appl appl PROPN ejpam-5960 320 4 . . PROPN ejpam-5960 320 5 math math PROPN ejpam-5960 320 6 , , PUNCT ejpam-5960 320 7 18 18 NUM ejpam-5960 320 8 ( ( PUNCT ejpam-5960 320 9 4 4 NUM ejpam-5960 320 10 ) ) PUNCT ejpam-5960 320 11 ( ( PUNCT ejpam-5960 320 12 2025 2025 NUM ejpam-5960 320 13 ) ) PUNCT ejpam-5960 320 14 , , PUNCT ejpam-5960 320 15 5960 5960 NUM ejpam-5960 320 16 11 11 NUM ejpam-5960 320 17 of of ADP ejpam-5960 320 18 22 22 NUM ejpam-5960 320 19 theorem theorem VERB ejpam-5960 320 20 3.15 3.15 NUM ejpam-5960 320 21 . . PUNCT ejpam-5960 321 1 let let VERB ejpam-5960 321 2 ( ( PUNCT ejpam-5960 321 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 321 4 , , PUNCT ejpam-5960 321 5 𝜏 𝜏 NOUN ejpam-5960 321 6 ) ) PUNCT ejpam-5960 321 7 be be VERB ejpam-5960 321 8 an an DET ejpam-5960 321 9 𝐿 𝐿 PROPN ejpam-5960 321 10 βˆ’ βˆ’ PROPN ejpam-5960 321 11 𝑑𝑠 𝑑𝑠 ADJ ejpam-5960 321 12 and and CCONJ ejpam-5960 321 13 πœ‡ πœ‡ ADP ejpam-5960 321 14 ∈ ∈ NOUN ejpam-5960 321 15 𝐿𝑋. 𝐿𝑋. X ejpam-5960 321 16 then then ADV ejpam-5960 321 17 the the DET ejpam-5960 321 18 mapping mapping NOUN ejpam-5960 321 19 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 321 20 : : PUNCT ejpam-5960 321 21 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 321 22 β†’ β†’ SYM ejpam-5960 321 23 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 321 24 is be AUX ejpam-5960 321 25 called call VERB ejpam-5960 321 26 a a DET ejpam-5960 321 27 closure closure NOUN ejpam-5960 321 28 operator operator NOUN ejpam-5960 321 29 of of ADP ejpam-5960 321 30 π‘π›Όβˆ’boundedness π‘π›Όβˆ’boundedness CCONJ ejpam-5960 321 31 iff iff VERB ejpam-5960 321 32 it it PRON ejpam-5960 321 33 satisfies satisfy VERB ejpam-5960 321 34 : : PUNCT ejpam-5960 321 35 ( ( PUNCT ejpam-5960 321 36 i i NOUN ejpam-5960 321 37 ) ) PUNCT ejpam-5960 321 38 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 321 39 ( ( PUNCT ejpam-5960 321 40 0𝑋 0𝑋 PROPN ejpam-5960 321 41 ) ) PUNCT ejpam-5960 321 42 = = SYM ejpam-5960 321 43 0𝑋. 0𝑋. NUM ejpam-5960 321 44 ( ( PUNCT ejpam-5960 321 45 ii ii NOUN ejpam-5960 321 46 ) ) PUNCT ejpam-5960 321 47 πœ‡ πœ‡ ADP ejpam-5960 321 48 ≀ ≀ ADJ ejpam-5960 321 49 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 321 50 ( ( PUNCT ejpam-5960 321 51 πœ‡ πœ‡ NOUN ejpam-5960 321 52 ) ) PUNCT ejpam-5960 321 53 . . PUNCT ejpam-5960 322 1 ( ( PUNCT ejpam-5960 322 2 iii iii X ejpam-5960 322 3 ) ) PUNCT ejpam-5960 322 4 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 322 5 ( ( PUNCT ejpam-5960 322 6 πœ‡ πœ‡ ADP ejpam-5960 322 7 ∨ ∨ X ejpam-5960 322 8 πœ‚ πœ‚ NOUN ejpam-5960 322 9 ) ) PUNCT ejpam-5960 322 10 = = SYM ejpam-5960 322 11 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 322 12 ( ( PUNCT ejpam-5960 322 13 πœ‡ πœ‡ NOUN ejpam-5960 322 14 ) ) PUNCT ejpam-5960 322 15 ∨ ∨ NOUN ejpam-5960 322 16 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 322 17 ( ( PUNCT ejpam-5960 322 18 πœ‚ πœ‚ NOUN ejpam-5960 322 19 ) ) PUNCT ejpam-5960 322 20 . . PUNCT ejpam-5960 323 1 ( ( PUNCT ejpam-5960 323 2 iv iv X ejpam-5960 323 3 ) ) PUNCT ejpam-5960 323 4 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 323 5 ( ( PUNCT ejpam-5960 323 6 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 323 7 ( ( PUNCT ejpam-5960 323 8 πœ‡ πœ‡ NOUN ejpam-5960 323 9 ) ) PUNCT ejpam-5960 323 10 ) ) PUNCT ejpam-5960 324 1 = = SYM ejpam-5960 324 2 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 324 3 ( ( PUNCT ejpam-5960 324 4 πœ‡ πœ‡ NOUN ejpam-5960 324 5 ) ) PUNCT ejpam-5960 324 6 . . PUNCT ejpam-5960 325 1 a a DET ejpam-5960 325 2 closure closure NOUN ejpam-5960 325 3 operator operator NOUN ejpam-5960 325 4 of of ADP ejpam-5960 325 5 π‘π›Όβˆ’boundedness π‘π›Όβˆ’boundedness CCONJ ejpam-5960 325 6 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 325 7 generates generate VERB ejpam-5960 325 8 πΏβˆ’topology πΏβˆ’topology NOUN ejpam-5960 325 9 πœπ‘π›Όπ΅π‘π‘™ πœπ‘π›Όπ΅π‘π‘™ NOUN ejpam-5960 325 10 on on ADP ejpam-5960 325 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 325 12 as as ADP ejpam-5960 325 13 : : PUNCT ejpam-5960 325 14 πœπ‘π›Όπ΅π‘π‘™ πœπ‘π›Όπ΅π‘π‘™ NOUN ejpam-5960 325 15 = = SYM ejpam-5960 325 16 { { PUNCT ejpam-5960 325 17 πœ‡ πœ‡ X ejpam-5960 325 18 ∈ ∈ NOUN ejpam-5960 325 19 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 325 20 : : PUNCT ejpam-5960 325 21 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 325 22 ( ( PUNCT ejpam-5960 325 23 πœ‡β€² πœ‡β€² NUM ejpam-5960 325 24 ) ) PUNCT ejpam-5960 325 25 = = SYM ejpam-5960 325 26 πœ‡β€² πœ‡β€² NOUN ejpam-5960 325 27 } } PUNCT ejpam-5960 325 28 . . PUNCT ejpam-5960 326 1 proof proof NOUN ejpam-5960 326 2 . . PUNCT ejpam-5960 327 1 it it PRON ejpam-5960 327 2 follows follow VERB ejpam-5960 327 3 directly directly ADV ejpam-5960 327 4 from from ADP ejpam-5960 327 5 theorems theorem NOUN ejpam-5960 327 6 3.12 3.12 NUM ejpam-5960 327 7 and and CCONJ ejpam-5960 327 8 3.14 3.14 NUM ejpam-5960 327 9 . . PUNCT ejpam-5960 328 1 theorem theorem VERB ejpam-5960 328 2 3.16 3.16 NUM ejpam-5960 328 3 . . PUNCT ejpam-5960 329 1 let let VERB ejpam-5960 329 2 ( ( PUNCT ejpam-5960 329 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 329 4 , , PUNCT ejpam-5960 329 5 𝜏 𝜏 NOUN ejpam-5960 329 6 ) ) PUNCT ejpam-5960 329 7 be be VERB ejpam-5960 329 8 an an DET ejpam-5960 329 9 𝐿 𝐿 PROPN ejpam-5960 329 10 βˆ’ βˆ’ PROPN ejpam-5960 329 11 𝑑𝑠. 𝑑𝑠. PROPN ejpam-5960 329 12 then then ADV ejpam-5960 329 13 : : PUNCT ejpam-5960 329 14 ( ( PUNCT ejpam-5960 329 15 i i NOUN ejpam-5960 329 16 ) ) PUNCT ejpam-5960 329 17 πœπ›Όπ΅ πœπ›Όπ΅ PROPN ejpam-5960 329 18 ≀ ≀ PROPN ejpam-5960 329 19 πœπ‘π›Όπ΅ πœπ‘π›Όπ΅ VERB ejpam-5960 329 20 ≀ ≀ NUM ejpam-5960 329 21 𝜏. 𝜏. NOUN ejpam-5960 329 22 ( ( PUNCT ejpam-5960 329 23 ii ii PROPN ejpam-5960 329 24 ) ) PUNCT ejpam-5960 329 25 if if SCONJ ejpam-5960 329 26 ( ( PUNCT ejpam-5960 329 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 329 28 , , PUNCT ejpam-5960 329 29 𝜏 𝜏 NOUN ejpam-5960 329 30 ) ) PUNCT ejpam-5960 329 31 is be AUX ejpam-5960 329 32 π›Όβˆ’bounded π›Όβˆ’bounde VERB ejpam-5960 329 33 ( ( PUNCT ejpam-5960 329 34 resp resp NOUN ejpam-5960 329 35 . . PUNCT ejpam-5960 329 36 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 329 37 space space NOUN ejpam-5960 329 38 ) ) PUNCT ejpam-5960 329 39 , , PUNCT ejpam-5960 329 40 then then ADV ejpam-5960 329 41 𝜏 𝜏 X ejpam-5960 329 42 = = SYM ejpam-5960 329 43 πœπ›Όπ΅ πœπ›Όπ΅ PROPN ejpam-5960 329 44 ( ( PUNCT ejpam-5960 329 45 resp resp NOUN ejpam-5960 329 46 . . PUNCT ejpam-5960 330 1 𝜏 𝜏 X ejpam-5960 330 2 = = SYM ejpam-5960 330 3 πœπ‘π›Όπ΅ πœπ‘π›Όπ΅ ADJ ejpam-5960 330 4 ) ) PUNCT ejpam-5960 330 5 . . PUNCT ejpam-5960 331 1 ( ( PUNCT ejpam-5960 331 2 iii iii X ejpam-5960 331 3 ) ) PUNCT ejpam-5960 331 4 if if SCONJ ejpam-5960 331 5 ( ( PUNCT ejpam-5960 331 6 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 331 7 , , PUNCT ejpam-5960 331 8 𝜏 𝜏 NOUN ejpam-5960 331 9 ) ) PUNCT ejpam-5960 331 10 is be AUX ejpam-5960 331 11 𝐿𝑅2βˆ’space 𝐿𝑅2βˆ’space NOUN ejpam-5960 331 12 , , PUNCT ejpam-5960 331 13 then then ADV ejpam-5960 331 14 πœπ›Όπ΅ πœπ›Όπ΅ PROPN ejpam-5960 331 15 = = PUNCT ejpam-5960 331 16 πœπ‘π›Όπ΅. πœπ‘π›Όπ΅. NOUN ejpam-5960 331 17 proof proof NOUN ejpam-5960 331 18 . . PUNCT ejpam-5960 332 1 ( ( PUNCT ejpam-5960 332 2 i i NOUN ejpam-5960 332 3 ) ) PUNCT ejpam-5960 332 4 let let VERB ejpam-5960 332 5 πœ‡ πœ‡ ADP ejpam-5960 332 6 ∈ ∈ PROPN ejpam-5960 332 7 πœπ›Όπ΅ πœπ›Όπ΅ PROPN ejpam-5960 332 8 , , PUNCT ejpam-5960 332 9 then then ADV ejpam-5960 332 10 𝛼𝐡𝑐𝑙 𝛼𝐡𝑐𝑙 PROPN ejpam-5960 332 11 ( ( PUNCT ejpam-5960 332 12 πœ‡β€² πœ‡β€² NOUN ejpam-5960 332 13 ) ) PUNCT ejpam-5960 332 14 ≀ ≀ NOUN ejpam-5960 332 15 πœ‡β€². πœ‡β€². NOUN ejpam-5960 333 1 since since SCONJ ejpam-5960 333 2 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 333 3 ( ( PUNCT ejpam-5960 333 4 πœ‡β€² πœ‡β€² NUM ejpam-5960 333 5 ) ) PUNCT ejpam-5960 333 6 ≀ ≀ NOUN ejpam-5960 333 7 𝛼𝐡𝑐𝑙 𝛼𝐡𝑐𝑙 PROPN ejpam-5960 333 8 ( ( PUNCT ejpam-5960 333 9 πœ‡β€² πœ‡β€² NUM ejpam-5960 333 10 ) ) PUNCT ejpam-5960 333 11 hence hence ADV ejpam-5960 333 12 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 333 13 ( ( PUNCT ejpam-5960 333 14 πœ‡β€² πœ‡β€² NUM ejpam-5960 333 15 ) ) PUNCT ejpam-5960 333 16 ≀ ≀ NOUN ejpam-5960 333 17 πœ‡β€² πœ‡β€² ADV ejpam-5960 333 18 and and CCONJ ejpam-5960 333 19 so so ADV ejpam-5960 333 20 πœ‡ πœ‡ ADP ejpam-5960 333 21 ∈ ∈ NOUN ejpam-5960 333 22 πœπ‘π›Όπ΅. πœπ‘π›Όπ΅. NOUN ejpam-5960 333 23 if if SCONJ ejpam-5960 333 24 πœ‡ πœ‡ SCONJ ejpam-5960 333 25 ∈ ∈ PROPN ejpam-5960 333 26 πœπ‘π›Όπ΅ πœπ‘π›Όπ΅ ADJ ejpam-5960 333 27 , , PUNCT ejpam-5960 333 28 then then ADV ejpam-5960 333 29 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 333 30 ( ( PUNCT ejpam-5960 333 31 πœ‡β€² πœ‡β€² NUM ejpam-5960 333 32 ) ) PUNCT ejpam-5960 333 33 ≀ ≀ NOUN ejpam-5960 333 34 πœ‡β€² πœ‡β€² ADV ejpam-5960 333 35 and and CCONJ ejpam-5960 333 36 so so ADV ejpam-5960 333 37 πœ‡ πœ‡ ADP ejpam-5960 333 38 ∈ ∈ PROPN ejpam-5960 333 39 𝜏. 𝜏. NOUN ejpam-5960 333 40 also also ADV ejpam-5960 333 41 if if SCONJ ejpam-5960 333 42 πœ‡ πœ‡ SCONJ ejpam-5960 333 43 ∈ ∈ PROPN ejpam-5960 333 44 πœπ‘π›Όπ΅ πœπ‘π›Όπ΅ ADJ ejpam-5960 333 45 , , PUNCT ejpam-5960 333 46 then then ADV ejpam-5960 333 47 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 333 48 ( ( PUNCT ejpam-5960 333 49 πœ‡β€² πœ‡β€² NUM ejpam-5960 333 50 ) ) PUNCT ejpam-5960 333 51 ≀ ≀ NOUN ejpam-5960 333 52 πœ‡β€². πœ‡β€². NOUN ejpam-5960 333 53 since since SCONJ ejpam-5960 333 54 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 333 55 ( ( PUNCT ejpam-5960 333 56 πœ‡β€² πœ‡β€² NOUN ejpam-5960 333 57 ) ) PUNCT ejpam-5960 333 58 ≀ ≀ NOUN ejpam-5960 333 59 𝑁𝐡𝑐𝑙 𝑁𝐡𝑐𝑙 PROPN ejpam-5960 333 60 ( ( PUNCT ejpam-5960 333 61 πœ‡β€² πœ‡β€² NUM ejpam-5960 333 62 ) ) PUNCT ejpam-5960 333 63 , , PUNCT ejpam-5960 333 64 hence hence ADV ejpam-5960 333 65 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 333 66 ( ( PUNCT ejpam-5960 333 67 πœ‡β€² πœ‡β€² NOUN ejpam-5960 333 68 ) ) PUNCT ejpam-5960 333 69 ≀ ≀ NOUN ejpam-5960 333 70 πœ‡β€² πœ‡β€² ADV ejpam-5960 333 71 and and CCONJ ejpam-5960 333 72 so so ADV ejpam-5960 333 73 πœ‡ πœ‡ ADP ejpam-5960 333 74 ∈ ∈ PROPN ejpam-5960 333 75 𝜏. 𝜏. NOUN ejpam-5960 333 76 ( ( PUNCT ejpam-5960 333 77 ii ii NOUN ejpam-5960 333 78 ) ) PUNCT ejpam-5960 333 79 we we PRON ejpam-5960 333 80 note note VERB ejpam-5960 333 81 that that SCONJ ejpam-5960 333 82 πœπ›Όπ΅ πœπ›Όπ΅ PROPN ejpam-5960 333 83 ≀ ≀ PROPN ejpam-5960 333 84 𝜏 𝜏 VERB ejpam-5960 333 85 from from ADP ejpam-5960 333 86 ( ( PUNCT ejpam-5960 333 87 i i NOUN ejpam-5960 333 88 ) ) PUNCT ejpam-5960 333 89 . . PUNCT ejpam-5960 334 1 now now ADV ejpam-5960 334 2 , , PUNCT ejpam-5960 334 3 let let VERB ejpam-5960 334 4 πœ‡ πœ‡ ADP ejpam-5960 334 5 ∈ ∈ VERB ejpam-5960 334 6 𝜏 𝜏 NOUN ejpam-5960 334 7 then then ADV ejpam-5960 334 8 πœ‡β€² πœ‡β€² ADJ ejpam-5960 334 9 ∈ ∈ PROPN ejpam-5960 334 10 πœβ€². πœβ€². NOUN ejpam-5960 334 11 since since SCONJ ejpam-5960 334 12 1𝑋 1𝑋 PROPN ejpam-5960 334 13 is be AUX ejpam-5960 334 14 a a DET ejpam-5960 334 15 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 334 16 and and CCONJ ejpam-5960 334 17 πœ‡β€² πœ‡β€² ADJ ejpam-5960 334 18 ≀ ≀ NUM ejpam-5960 334 19 1𝑋 1𝑋 NOUN ejpam-5960 334 20 , , PUNCT ejpam-5960 334 21 πœ‡ πœ‡ SCONJ ejpam-5960 334 22 β€² β€² NOUN ejpam-5960 334 23 is be AUX ejpam-5960 334 24 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 334 25 . . PUNCT ejpam-5960 335 1 by by ADP ejpam-5960 335 2 theorem theorem NOUN ejpam-5960 335 3 3.7 3.7 NUM ejpam-5960 335 4 and and CCONJ ejpam-5960 335 5 by by ADP ejpam-5960 335 6 theorem theorem ADJ ejpam-5960 335 7 3.14 3.14 NUM ejpam-5960 335 8 ( ( PUNCT ejpam-5960 335 9 iv iv NUM ejpam-5960 335 10 ) ) PUNCT ejpam-5960 335 11 , , PUNCT ejpam-5960 335 12 we we PRON ejpam-5960 335 13 have have VERB ejpam-5960 335 14 πœ‡β€² πœ‡β€² NOUN ejpam-5960 335 15 that that PRON ejpam-5960 335 16 is be AUX ejpam-5960 335 17 a a DET ejpam-5960 335 18 π‘π›Όπ΅βˆ’closed π‘π›Όπ΅βˆ’close VERB ejpam-5960 335 19 set set NOUN ejpam-5960 335 20 and and CCONJ ejpam-5960 335 21 so so ADV ejpam-5960 335 22 πœ‡β€² πœ‡β€² ADJ ejpam-5960 335 23 ∈ ∈ NOUN ejpam-5960 335 24 πœπ‘π›Όπ΅. πœπ‘π›Όπ΅. NOUN ejpam-5960 335 25 thus thus ADV ejpam-5960 335 26 𝜏 𝜏 NOUN ejpam-5960 335 27 = = SYM ejpam-5960 335 28 πœπ‘π›Όπ΅. πœπ‘π›Όπ΅. NOUN ejpam-5960 335 29 definition definition NOUN ejpam-5960 335 30 3.17 3.17 NUM ejpam-5960 335 31 . . PUNCT ejpam-5960 336 1 let let VERB ejpam-5960 336 2 ( ( PUNCT ejpam-5960 336 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 336 4 , , PUNCT ejpam-5960 336 5 𝜏 𝜏 NOUN ejpam-5960 336 6 ) ) PUNCT ejpam-5960 336 7 be be VERB ejpam-5960 336 8 an an DET ejpam-5960 336 9 𝐿 𝐿 PROPN ejpam-5960 336 10 βˆ’ βˆ’ PROPN ejpam-5960 336 11 𝑑𝑠 𝑑𝑠 NOUN ejpam-5960 336 12 , , PUNCT ejpam-5960 336 13 πœ‡ πœ‡ X ejpam-5960 336 14 ∈ ∈ ADP ejpam-5960 336 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 336 16 and and CCONJ ejpam-5960 336 17 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 336 18 ( ( PUNCT ejpam-5960 336 19 πœ‡ πœ‡ NOUN ejpam-5960 336 20 ) ) PUNCT ejpam-5960 336 21 = = PUNCT ejpam-5960 336 22 ∨{𝜌 ∨{𝜌 PROPN ejpam-5960 336 23 ∈ ∈ NUM ejpam-5960 336 24 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 336 25 : : PUNCT ejpam-5960 336 26 𝜌 𝜌 X ejpam-5960 336 27 ∈ ∈ NOUN ejpam-5960 336 28 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 NOUN ejpam-5960 336 29 ( ( PUNCT ejpam-5960 336 30 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 336 31 , , PUNCT ejpam-5960 336 32 𝜏 𝜏 NOUN ejpam-5960 336 33 ) ) PUNCT ejpam-5960 336 34 , , PUNCT ejpam-5960 336 35 𝜌 𝜌 ADP ejpam-5960 336 36 ≀ ≀ NUM ejpam-5960 336 37 πœ‡ πœ‡ ADP ejpam-5960 336 38 } } PUNCT ejpam-5960 336 39 . . PUNCT ejpam-5960 337 1 we we PRON ejpam-5960 337 2 say say VERB ejpam-5960 337 3 that that SCONJ ejpam-5960 337 4 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 337 5 ( ( PUNCT ejpam-5960 337 6 πœ‡ πœ‡ NOUN ejpam-5960 337 7 ) ) PUNCT ejpam-5960 337 8 is be AUX ejpam-5960 337 9 the the DET ejpam-5960 337 10 π‘π›Όπ΅βˆ’interior π‘π›Όπ΅βˆ’interior NOUN ejpam-5960 337 11 of of ADP ejpam-5960 337 12 πœ‡. πœ‡. NOUN ejpam-5960 337 13 the the DET ejpam-5960 337 14 following follow VERB ejpam-5960 337 15 theorem theorem NOUN ejpam-5960 337 16 shows show VERB ejpam-5960 337 17 the the DET ejpam-5960 337 18 relationships relationship NOUN ejpam-5960 337 19 between between ADP ejpam-5960 337 20 π‘π›Όπ΅βˆ’closure π‘π›Όπ΅βˆ’closure NOUN ejpam-5960 337 21 operator operator NOUN ejpam-5960 337 22 and and CCONJ ejpam-5960 337 23 π‘π›Όπ΅βˆ’interior π‘π›Όπ΅βˆ’interior ADJ ejpam-5960 337 24 operator operator NOUN ejpam-5960 337 25 . . PUNCT ejpam-5960 338 1 theorem theorem VERB ejpam-5960 338 2 3.18 3.18 NUM ejpam-5960 338 3 . . PUNCT ejpam-5960 339 1 let let AUX ejpam-5960 339 2 ( ( PUNCT ejpam-5960 339 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 339 4 , , PUNCT ejpam-5960 339 5 𝜏 𝜏 NOUN ejpam-5960 339 6 ) ) PUNCT ejpam-5960 339 7 be be VERB ejpam-5960 339 8 an an DET ejpam-5960 339 9 𝐿 𝐿 PROPN ejpam-5960 339 10 βˆ’ βˆ’ PROPN ejpam-5960 339 11 𝑑𝑠 𝑑𝑠 ADJ ejpam-5960 339 12 and and CCONJ ejpam-5960 339 13 πœ‡ πœ‡ ADP ejpam-5960 339 14 ∈ ∈ NOUN ejpam-5960 339 15 𝐿𝑋. 𝐿𝑋. X ejpam-5960 339 16 then then ADV ejpam-5960 339 17 the the DET ejpam-5960 339 18 following follow VERB ejpam-5960 339 19 are be AUX ejpam-5960 339 20 true true ADJ ejpam-5960 339 21 : : PUNCT ejpam-5960 339 22 ( ( PUNCT ejpam-5960 339 23 i i NOUN ejpam-5960 339 24 ) ) PUNCT ejpam-5960 339 25 πœ‡ πœ‡ SCONJ ejpam-5960 339 26 is be AUX ejpam-5960 339 27 π‘π›Όπ΅βˆ’open π‘π›Όπ΅βˆ’open ADJ ejpam-5960 339 28 iff iff PROPN ejpam-5960 339 29 πœ‡ πœ‡ ADP ejpam-5960 339 30 = = X ejpam-5960 339 31 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 339 32 ( ( PUNCT ejpam-5960 339 33 πœ‡ πœ‡ NOUN ejpam-5960 339 34 ) ) PUNCT ejpam-5960 339 35 . . PUNCT ejpam-5960 340 1 ( ( PUNCT ejpam-5960 340 2 ii ii NOUN ejpam-5960 340 3 ) ) PUNCT ejpam-5960 340 4 ( ( PUNCT ejpam-5960 340 5 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 340 6 ( ( PUNCT ejpam-5960 340 7 πœ‡))β€² πœ‡))β€² PROPN ejpam-5960 340 8 = = PUNCT ejpam-5960 340 9 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 340 10 ( ( PUNCT ejpam-5960 340 11 πœ‡β€² πœ‡β€² NUM ejpam-5960 340 12 ) ) PUNCT ejpam-5960 340 13 and and CCONJ ejpam-5960 340 14 ( ( PUNCT ejpam-5960 340 15 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 340 16 ( ( PUNCT ejpam-5960 340 17 πœ‡))β€² πœ‡))β€² PROPN ejpam-5960 340 18 = = PUNCT ejpam-5960 340 19 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 340 20 ( ( PUNCT ejpam-5960 340 21 πœ‡β€² πœ‡β€² NUM ejpam-5960 340 22 ) ) PUNCT ejpam-5960 340 23 . . PUNCT ejpam-5960 341 1 ( ( PUNCT ejpam-5960 341 2 iii iii X ejpam-5960 341 3 ) ) PUNCT ejpam-5960 341 4 𝑁𝛼𝐡𝑐𝑙 𝑁𝛼𝐡𝑐𝑙 PROPN ejpam-5960 341 5 ( ( PUNCT ejpam-5960 341 6 πœ‡ πœ‡ NOUN ejpam-5960 341 7 ) ) PUNCT ejpam-5960 341 8 = = SYM ejpam-5960 341 9 ( ( PUNCT ejpam-5960 341 10 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 341 11 ( ( PUNCT ejpam-5960 341 12 πœ‡β€²))β€² πœ‡β€²))β€² PROPN ejpam-5960 341 13 and and CCONJ ejpam-5960 341 14 𝑁𝛼𝑖𝑛𝑑 𝑁𝛼𝑖𝑛𝑑 PROPN ejpam-5960 341 15 ( ( PUNCT ejpam-5960 341 16 πœ‡ πœ‡ NOUN ejpam-5960 341 17 ) ) PUNCT ejpam-5960 341 18 = = SYM ejpam-5960 341 19 ( ( PUNCT ejpam-5960 341 20 𝑁𝛼𝑐𝑙 𝑁𝛼𝑐𝑙 PROPN ejpam-5960 341 21 ( ( PUNCT ejpam-5960 341 22 πœ‡β€²))β€². πœ‡β€²))β€². PROPN ejpam-5960 341 23 ( ( PUNCT ejpam-5960 341 24 iv iv X ejpam-5960 341 25 ) ) PUNCT ejpam-5960 341 26 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 341 27 ( ( PUNCT ejpam-5960 341 28 πœ‡ πœ‡ NOUN ejpam-5960 341 29 ) ) PUNCT ejpam-5960 341 30 ≀ ≀ NUM ejpam-5960 341 31 𝛼𝐡.𝑖𝑛𝑑 𝛼𝐡.𝑖𝑛𝑑 ADV ejpam-5960 341 32 ( ( PUNCT ejpam-5960 341 33 πœ‡ πœ‡ NOUN ejpam-5960 341 34 ) ) PUNCT ejpam-5960 341 35 ≀ ≀ NOUN ejpam-5960 341 36 𝑖𝑛𝑑 𝑖𝑛𝑑 NOUN ejpam-5960 341 37 ( ( PUNCT ejpam-5960 341 38 πœ‡ πœ‡ NOUN ejpam-5960 341 39 ) ) PUNCT ejpam-5960 341 40 ≀ ≀ NUM ejpam-5960 341 41 πœ‡. πœ‡. NOUN ejpam-5960 341 42 ( ( PUNCT ejpam-5960 341 43 v v NOUN ejpam-5960 341 44 ) ) PUNCT ejpam-5960 341 45 if if SCONJ ejpam-5960 341 46 πœ‚ πœ‚ PROPN ejpam-5960 341 47 ∈ ∈ PROPN ejpam-5960 341 48 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 341 49 and and CCONJ ejpam-5960 341 50 πœ‡ πœ‡ ADP ejpam-5960 341 51 ≀ ≀ NOUN ejpam-5960 341 52 πœ‚ πœ‚ NOUN ejpam-5960 341 53 then then ADV ejpam-5960 341 54 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 341 55 ( ( PUNCT ejpam-5960 341 56 πœ‡ πœ‡ NOUN ejpam-5960 341 57 ) ) PUNCT ejpam-5960 341 58 ≀ ≀ NUM ejpam-5960 341 59 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 341 60 ( ( PUNCT ejpam-5960 341 61 πœ‚ πœ‚ NOUN ejpam-5960 341 62 ) ) PUNCT ejpam-5960 341 63 . . PUNCT ejpam-5960 342 1 ( ( PUNCT ejpam-5960 342 2 vi vi NOUN ejpam-5960 342 3 ) ) PUNCT ejpam-5960 342 4 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 NOUN ejpam-5960 342 5 ( ( PUNCT ejpam-5960 342 6 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 342 7 ( ( PUNCT ejpam-5960 342 8 πœ‡ πœ‡ NOUN ejpam-5960 342 9 ) ) PUNCT ejpam-5960 342 10 ) ) PUNCT ejpam-5960 343 1 = = SYM ejpam-5960 343 2 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 343 3 ( ( PUNCT ejpam-5960 343 4 πœ‡ πœ‡ NOUN ejpam-5960 343 5 ) ) PUNCT ejpam-5960 343 6 . . PUNCT ejpam-5960 344 1 proof proof NOUN ejpam-5960 344 2 . . PUNCT ejpam-5960 345 1 ( ( PUNCT ejpam-5960 345 2 i i NOUN ejpam-5960 345 3 ) ) PUNCT ejpam-5960 345 4 let let VERB ejpam-5960 345 5 πœ‡ πœ‡ SCONJ ejpam-5960 345 6 ∈ ∈ NOUN ejpam-5960 345 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 345 8 be be VERB ejpam-5960 345 9 an an DET ejpam-5960 345 10 π‘π›Όπ΅βˆ’ π‘π›Όπ΅βˆ’ ADJ ejpam-5960 345 11 open open ADJ ejpam-5960 345 12 set set NOUN ejpam-5960 345 13 , , PUNCT ejpam-5960 345 14 then then ADV ejpam-5960 345 15 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 345 16 ( ( PUNCT ejpam-5960 345 17 πœ‡ πœ‡ NOUN ejpam-5960 345 18 ) ) PUNCT ejpam-5960 345 19 = = PUNCT ejpam-5960 345 20 ∨{𝜌 ∨{𝜌 PROPN ejpam-5960 345 21 ∈ ∈ NUM ejpam-5960 345 22 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 345 23 : : PUNCT ejpam-5960 345 24 𝜌 𝜌 X ejpam-5960 345 25 ∈ ∈ NOUN ejpam-5960 345 26 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 NOUN ejpam-5960 345 27 ( ( PUNCT ejpam-5960 345 28 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 345 29 , , PUNCT ejpam-5960 345 30 𝜏 𝜏 NOUN ejpam-5960 345 31 ) ) PUNCT ejpam-5960 345 32 , , PUNCT ejpam-5960 345 33 𝜌 𝜌 ADP ejpam-5960 345 34 ≀ ≀ NUM ejpam-5960 345 35 πœ‡ πœ‡ ADP ejpam-5960 345 36 } } PUNCT ejpam-5960 345 37 = = PUNCT ejpam-5960 345 38 πœ‡ πœ‡ ADP ejpam-5960 345 39 and and CCONJ ejpam-5960 345 40 so so ADV ejpam-5960 345 41 πœ‡ πœ‡ ADP ejpam-5960 345 42 = = X ejpam-5960 345 43 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 345 44 ( ( PUNCT ejpam-5960 345 45 πœ‡ πœ‡ NOUN ejpam-5960 345 46 ) ) PUNCT ejpam-5960 345 47 . . PUNCT ejpam-5960 346 1 conversely conversely ADV ejpam-5960 346 2 , , PUNCT ejpam-5960 346 3 let let VERB ejpam-5960 346 4 πœ‡ πœ‡ ADP ejpam-5960 346 5 = = X ejpam-5960 346 6 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 346 7 ( ( PUNCT ejpam-5960 346 8 πœ‡ πœ‡ NOUN ejpam-5960 346 9 ) ) PUNCT ejpam-5960 346 10 , , PUNCT ejpam-5960 346 11 since since SCONJ ejpam-5960 346 12 𝑁𝛼𝐡.𝑖𝑛𝑑 𝑁𝛼𝐡.𝑖𝑛𝑑 ADJ ejpam-5960 346 13 ( ( PUNCT ejpam-5960 346 14 πœ‡ πœ‡ NOUN ejpam-5960 346 15 ) ) PUNCT ejpam-5960 346 16 = = PUNCT ejpam-5960 346 17 ∨{𝜌 ∨{𝜌 PROPN ejpam-5960 346 18 ∈ ∈ NUM ejpam-5960 346 19 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 346 20 : : PUNCT ejpam-5960 346 21 𝜌 𝜌 X ejpam-5960 346 22 ∈ ∈ NOUN ejpam-5960 346 23 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 NOUN ejpam-5960 346 24 ( ( PUNCT ejpam-5960 346 25 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 346 26 , , PUNCT ejpam-5960 346 27 𝜏 𝜏 NOUN ejpam-5960 346 28 ) ) PUNCT ejpam-5960 346 29 , , PUNCT ejpam-5960 346 30 𝜌 𝜌 ADP ejpam-5960 346 31 ≀ ≀ NUM ejpam-5960 346 32 πœ‡ πœ‡ ADP ejpam-5960 346 33 } } PUNCT ejpam-5960 346 34 = = SYM ejpam-5960 346 35 πœ‡. πœ‡. NOUN ejpam-5960 346 36 therefore therefore ADV ejpam-5960 346 37 πœ‡ πœ‡ ADV ejpam-5960 346 38 is be AUX ejpam-5960 346 39 π‘π›Όπ΅βˆ’open π‘π›Όπ΅βˆ’open ADJ ejpam-5960 346 40 set set VERB ejpam-5960 346 41 . . PUNCT ejpam-5960 347 1 ( ( PUNCT ejpam-5960 347 2 ii ii X ejpam-5960 347 3 ) ) PUNCT ejpam-5960 347 4 it it PRON ejpam-5960 347 5 follows follow VERB ejpam-5960 347 6 directly directly ADV ejpam-5960 347 7 from from ADP ejpam-5960 347 8 theorem theorem ADJ ejpam-5960 347 9 3.12 3.12 NUM ejpam-5960 347 10 ( ( PUNCT ejpam-5960 347 11 iv iv NOUN ejpam-5960 347 12 ) ) PUNCT ejpam-5960 347 13 and and CCONJ ejpam-5960 347 14 definition definition NOUN ejpam-5960 347 15 3.17 3.17 NUM ejpam-5960 347 16 . . PUNCT ejpam-5960 348 1 ( ( PUNCT ejpam-5960 348 2 iii iii X ejpam-5960 348 3 ) ) PUNCT ejpam-5960 348 4 it it PRON ejpam-5960 348 5 follows follow VERB ejpam-5960 348 6 directly directly ADV ejpam-5960 348 7 from from ADP ejpam-5960 348 8 ( ( PUNCT ejpam-5960 348 9 ii ii NOUN ejpam-5960 348 10 ) ) PUNCT ejpam-5960 348 11 . . PUNCT ejpam-5960 349 1 ( ( PUNCT ejpam-5960 349 2 iv iv X ejpam-5960 349 3 ) ) PUNCT ejpam-5960 349 4 it it PRON ejpam-5960 349 5 follows follow VERB ejpam-5960 349 6 directly directly ADV ejpam-5960 349 7 from from ADP ejpam-5960 349 8 ( ( PUNCT ejpam-5960 349 9 ii ii NOUN ejpam-5960 349 10 ) ) PUNCT ejpam-5960 349 11 and and CCONJ ejpam-5960 349 12 theorems theorem NOUN ejpam-5960 349 13 3.12 3.12 NUM ejpam-5960 349 14 ( ( PUNCT ejpam-5960 349 15 i i NOUN ejpam-5960 349 16 ) ) PUNCT ejpam-5960 349 17 . . PUNCT ejpam-5960 350 1 ( ( PUNCT ejpam-5960 350 2 v v X ejpam-5960 350 3 ) ) PUNCT ejpam-5960 350 4 it it PRON ejpam-5960 350 5 follows follow VERB ejpam-5960 350 6 directly directly ADV ejpam-5960 350 7 from from ADP ejpam-5960 350 8 ( ( PUNCT ejpam-5960 350 9 ii ii NOUN ejpam-5960 350 10 ) ) PUNCT ejpam-5960 350 11 and and CCONJ ejpam-5960 350 12 theorem theorem VERB ejpam-5960 350 13 3.12 3.12 NUM ejpam-5960 350 14 ( ( PUNCT ejpam-5960 350 15 ii ii NOUN ejpam-5960 350 16 ) ) PUNCT ejpam-5960 350 17 . . PUNCT ejpam-5960 351 1 ( ( PUNCT ejpam-5960 351 2 vi vi X ejpam-5960 351 3 ) ) PUNCT ejpam-5960 351 4 it it PRON ejpam-5960 351 5 follows follow VERB ejpam-5960 351 6 directly directly ADV ejpam-5960 351 7 from from ADP ejpam-5960 351 8 ( ( PUNCT ejpam-5960 351 9 ii ii NOUN ejpam-5960 351 10 ) ) PUNCT ejpam-5960 351 11 and and CCONJ ejpam-5960 351 12 theorem theorem VERB ejpam-5960 351 13 3.12 3.12 NUM ejpam-5960 351 14 ( ( PUNCT ejpam-5960 351 15 iii iii NOUN ejpam-5960 351 16 ) ) PUNCT ejpam-5960 351 17 . . PUNCT ejpam-5960 352 1 theorem theorem VERB ejpam-5960 352 2 3.19 3.19 NUM ejpam-5960 352 3 . . PUNCT ejpam-5960 353 1 let let AUX ejpam-5960 353 2 ( ( PUNCT ejpam-5960 353 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 353 4 , , PUNCT ejpam-5960 353 5 𝜏 𝜏 NOUN ejpam-5960 353 6 ) ) PUNCT ejpam-5960 353 7 be be VERB ejpam-5960 353 8 an an DET ejpam-5960 353 9 𝐿 𝐿 PROPN ejpam-5960 353 10 βˆ’ βˆ’ PROPN ejpam-5960 353 11 𝑑𝑠. 𝑑𝑠. NOUN ejpam-5960 353 12 the the DET ejpam-5960 353 13 following follow VERB ejpam-5960 353 14 statements statement NOUN ejpam-5960 353 15 hold hold VERB ejpam-5960 353 16 : : PUNCT ejpam-5960 353 17 ( ( PUNCT ejpam-5960 353 18 i i NOUN ejpam-5960 353 19 ) ) PUNCT ejpam-5960 353 20 0𝑋 0𝑋 PROPN ejpam-5960 353 21 , , PUNCT ejpam-5960 353 22 1𝑋 1𝑋 PROPN ejpam-5960 353 23 ∈ ∈ PROPN ejpam-5960 353 24 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 PROPN ejpam-5960 353 25 ( ( PUNCT ejpam-5960 353 26 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 353 27 , , PUNCT ejpam-5960 353 28 𝜏 𝜏 NOUN ejpam-5960 353 29 ) ) PUNCT ejpam-5960 353 30 . . PUNCT ejpam-5960 354 1 ( ( PUNCT ejpam-5960 354 2 ii ii NOUN ejpam-5960 354 3 ) ) PUNCT ejpam-5960 354 4 if if SCONJ ejpam-5960 354 5 πœ‡1 πœ‡1 ADJ ejpam-5960 354 6 , , PUNCT ejpam-5960 354 7 πœ‡2 πœ‡2 PROPN ejpam-5960 354 8 , , PUNCT ejpam-5960 354 9 ... ... PUNCT ejpam-5960 354 10 , , PUNCT ejpam-5960 354 11 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 354 12 ∈ ∈ NOUN ejpam-5960 354 13 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 PROPN ejpam-5960 354 14 ( ( PUNCT ejpam-5960 354 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 354 16 , , PUNCT ejpam-5960 354 17 𝜏 𝜏 NOUN ejpam-5960 354 18 ) ) PUNCT ejpam-5960 354 19 , , PUNCT ejpam-5960 354 20 then then ADV ejpam-5960 354 21 βˆ§π‘› βˆ§π‘› ADV ejpam-5960 354 22 𝑖=1 𝑖=1 PUNCT ejpam-5960 354 23 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 354 24 ∈ ∈ NOUN ejpam-5960 354 25 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 NOUN ejpam-5960 354 26 ( ( PUNCT ejpam-5960 354 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 354 28 , , PUNCT ejpam-5960 354 29 𝜏 𝜏 NOUN ejpam-5960 354 30 ) ) PUNCT ejpam-5960 354 31 . . PUNCT ejpam-5960 355 1 n. n. PROPN ejpam-5960 355 2 a. a. PROPN ejpam-5960 355 3 alsaedi alsaedi PROPN ejpam-5960 355 4 / / SYM ejpam-5960 355 5 eur eur PROPN ejpam-5960 355 6 . . PUNCT ejpam-5960 356 1 j. j. PROPN ejpam-5960 356 2 pure pure PROPN ejpam-5960 356 3 appl appl PROPN ejpam-5960 356 4 . . PROPN ejpam-5960 356 5 math math PROPN ejpam-5960 356 6 , , PUNCT ejpam-5960 356 7 18 18 NUM ejpam-5960 356 8 ( ( PUNCT ejpam-5960 356 9 4 4 NUM ejpam-5960 356 10 ) ) PUNCT ejpam-5960 356 11 ( ( PUNCT ejpam-5960 356 12 2025 2025 NUM ejpam-5960 356 13 ) ) PUNCT ejpam-5960 356 14 , , PUNCT ejpam-5960 356 15 5960 5960 NUM ejpam-5960 356 16 12 12 NUM ejpam-5960 356 17 of of ADP ejpam-5960 356 18 22 22 NUM ejpam-5960 356 19 ( ( PUNCT ejpam-5960 356 20 iii iii NOUN ejpam-5960 356 21 ) ) PUNCT ejpam-5960 356 22 if if SCONJ ejpam-5960 356 23 { { PUNCT ejpam-5960 356 24 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 356 25 : : PUNCT ejpam-5960 356 26 𝑖 𝑖 SYM ejpam-5960 356 27 ∈ ∈ PROPN ejpam-5960 356 28 𝐼 𝐼 PROPN ejpam-5960 356 29 } } PUNCT ejpam-5960 356 30 βŠ† βŠ† NUM ejpam-5960 356 31 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 NOUN ejpam-5960 356 32 ( ( PUNCT ejpam-5960 356 33 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 356 34 , , PUNCT ejpam-5960 356 35 𝜏 𝜏 NOUN ejpam-5960 356 36 ) ) PUNCT ejpam-5960 356 37 , , PUNCT ejpam-5960 356 38 then then ADV ejpam-5960 356 39 ∨ ∨ NUM ejpam-5960 356 40 π‘–βˆˆπΌ π‘–βˆˆπΌ X ejpam-5960 356 41 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 356 42 ∈ ∈ NOUN ejpam-5960 356 43 𝑁𝛼𝐡𝑂 𝑁𝛼𝐡𝑂 NOUN ejpam-5960 356 44 ( ( PUNCT ejpam-5960 356 45 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 356 46 , , PUNCT ejpam-5960 356 47 𝜏 𝜏 NOUN ejpam-5960 356 48 ) ) PUNCT ejpam-5960 356 49 . . PUNCT ejpam-5960 357 1 proof proof NOUN ejpam-5960 357 2 . . PUNCT ejpam-5960 358 1 it it PRON ejpam-5960 358 2 is be AUX ejpam-5960 358 3 similar similar ADJ ejpam-5960 358 4 to to ADP ejpam-5960 358 5 the the DET ejpam-5960 358 6 proof proof NOUN ejpam-5960 358 7 of of ADP ejpam-5960 358 8 theorem theorem ADJ ejpam-5960 358 9 3.14 3.14 NUM ejpam-5960 358 10 . . PUNCT ejpam-5960 359 1 definition definition NOUN ejpam-5960 359 2 3.20 3.20 NUM ejpam-5960 359 3 . . PUNCT ejpam-5960 360 1 let let VERB ejpam-5960 360 2 ( ( PUNCT ejpam-5960 360 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 360 4 , , PUNCT ejpam-5960 360 5 𝜏 𝜏 NOUN ejpam-5960 360 6 ) ) PUNCT ejpam-5960 360 7 be be VERB ejpam-5960 360 8 an an DET ejpam-5960 360 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 360 10 and and CCONJ ejpam-5960 360 11 𝑆 𝑆 PROPN ejpam-5960 360 12 be be VERB ejpam-5960 360 13 a a DET ejpam-5960 360 14 molecular molecular ADJ ejpam-5960 360 15 net net NOUN ejpam-5960 360 16 in in ADP ejpam-5960 360 17 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 360 18 then then ADV ejpam-5960 360 19 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 360 20 ∈ ∈ PROPN ejpam-5960 360 21 𝑀 𝑀 PROPN ejpam-5960 360 22 ( ( PUNCT ejpam-5960 360 23 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 360 24 ) ) PUNCT ejpam-5960 360 25 is be AUX ejpam-5960 360 26 called call VERB ejpam-5960 360 27 a a DET ejpam-5960 360 28 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 360 29 limit limit NOUN ejpam-5960 360 30 point point NOUN ejpam-5960 360 31 of of ADP ejpam-5960 360 32 𝑆 𝑆 PROPN ejpam-5960 360 33 , , PUNCT ejpam-5960 360 34 ( ( PUNCT ejpam-5960 360 35 or or CCONJ ejpam-5960 360 36 𝑆 𝑆 PROPN ejpam-5960 360 37 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 360 38 to to ADP ejpam-5960 360 39 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 360 40 ) ) PUNCT ejpam-5960 360 41 in in ADP ejpam-5960 360 42 symbol symbol NOUN ejpam-5960 360 43 𝑆 𝑆 PROPN ejpam-5960 360 44 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ X ejpam-5960 360 45 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 360 46 if if SCONJ ejpam-5960 360 47 for for ADP ejpam-5960 360 48 every every DET ejpam-5960 360 49 πœ‡ πœ‡ ADP ejpam-5960 360 50 ∈ ∈ NOUN ejpam-5960 360 51 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 360 52 and and CCONJ ejpam-5960 360 53 there there PRON ejpam-5960 360 54 is be VERB ejpam-5960 360 55 𝑛 𝑛 DET ejpam-5960 360 56 ∈ ∈ PROPN ejpam-5960 360 57 𝐷 𝐷 NOUN ejpam-5960 360 58 such such ADJ ejpam-5960 360 59 that that SCONJ ejpam-5960 360 60 π‘š π‘š PROPN ejpam-5960 360 61 ∈ ∈ PROPN ejpam-5960 360 62 𝐷 𝐷 NOUN ejpam-5960 360 63 and and CCONJ ejpam-5960 360 64 π‘š π‘š ADP ejpam-5960 360 65 β‰₯ β‰₯ NOUN ejpam-5960 360 66 𝑛 𝑛 ADP ejpam-5960 360 67 we we PRON ejpam-5960 360 68 have have VERB ejpam-5960 360 69 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 360 70 ) ) PUNCT ejpam-5960 360 71 βˆ‰ βˆ‰ PROPN ejpam-5960 360 72 πœ‡. πœ‡. NOUN ejpam-5960 361 1 the the DET ejpam-5960 361 2 union union NOUN ejpam-5960 361 3 of of ADP ejpam-5960 361 4 all all DET ejpam-5960 361 5 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 361 6 limit limit NOUN ejpam-5960 361 7 points point NOUN ejpam-5960 361 8 of of ADP ejpam-5960 361 9 𝑆 𝑆 PROPN ejpam-5960 361 10 is be AUX ejpam-5960 361 11 denoted denote VERB ejpam-5960 361 12 by by ADP ejpam-5960 361 13 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 361 14 lim(𝑆 lim(𝑆 PROPN ejpam-5960 361 15 ) ) PUNCT ejpam-5960 361 16 . . PUNCT ejpam-5960 362 1 definition definition NOUN ejpam-5960 362 2 3.21 3.21 NUM ejpam-5960 362 3 . . PUNCT ejpam-5960 363 1 let let VERB ejpam-5960 363 2 ( ( PUNCT ejpam-5960 363 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 363 4 , , PUNCT ejpam-5960 363 5 𝜏 𝜏 NOUN ejpam-5960 363 6 ) ) PUNCT ejpam-5960 363 7 be be VERB ejpam-5960 363 8 an an DET ejpam-5960 363 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 363 10 and and CCONJ ejpam-5960 363 11 𝑆 𝑆 PROPN ejpam-5960 363 12 be be VERB ejpam-5960 363 13 a a DET ejpam-5960 363 14 molecular molecular ADJ ejpam-5960 363 15 net net NOUN ejpam-5960 363 16 in in ADP ejpam-5960 363 17 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 363 18 then then ADV ejpam-5960 363 19 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 363 20 ∈ ∈ PROPN ejpam-5960 363 21 𝑀 𝑀 PROPN ejpam-5960 363 22 ( ( PUNCT ejpam-5960 363 23 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 363 24 ) ) PUNCT ejpam-5960 363 25 is be AUX ejpam-5960 363 26 called call VERB ejpam-5960 363 27 a a DET ejpam-5960 363 28 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 363 29 cluster cluster NOUN ejpam-5960 363 30 point point NOUN ejpam-5960 363 31 of of ADP ejpam-5960 363 32 𝑆 𝑆 PROPN ejpam-5960 363 33 , , PUNCT ejpam-5960 363 34 in in ADP ejpam-5960 363 35 symbol symbol NOUN ejpam-5960 363 36 𝑆 𝑆 PROPN ejpam-5960 364 1 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ DET ejpam-5960 364 2 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 364 3 , , PUNCT ejpam-5960 364 4 if if SCONJ ejpam-5960 364 5 for for ADP ejpam-5960 364 6 every every DET ejpam-5960 364 7 πœ‡ πœ‡ ADP ejpam-5960 364 8 ∈ ∈ NOUN ejpam-5960 364 9 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 364 10 and and CCONJ ejpam-5960 364 11 every every DET ejpam-5960 364 12 𝑛 𝑛 PROPN ejpam-5960 364 13 ∈ ∈ PROPN ejpam-5960 364 14 𝐷 𝐷 NOUN ejpam-5960 364 15 there there PRON ejpam-5960 364 16 is be VERB ejpam-5960 364 17 π‘š π‘š PRON ejpam-5960 364 18 ∈ ∈ PROPN ejpam-5960 364 19 𝐷 𝐷 NOUN ejpam-5960 364 20 such such ADJ ejpam-5960 364 21 that that SCONJ ejpam-5960 364 22 π‘š π‘š PROPN ejpam-5960 364 23 β‰₯ β‰₯ PRON ejpam-5960 364 24 𝑛 𝑛 PROPN ejpam-5960 364 25 and and CCONJ ejpam-5960 364 26 𝑆(π‘š 𝑆(π‘š SYM ejpam-5960 364 27 ) ) PUNCT ejpam-5960 364 28 βˆ‰ βˆ‰ PROPN ejpam-5960 364 29 πœ‡. πœ‡. NOUN ejpam-5960 365 1 the the DET ejpam-5960 365 2 union union NOUN ejpam-5960 365 3 of of ADP ejpam-5960 365 4 all all DET ejpam-5960 365 5 𝑁𝛼–bounded 𝑁𝛼–bounde VERB ejpam-5960 365 6 cluster cluster NOUN ejpam-5960 365 7 points point NOUN ejpam-5960 365 8 of of ADP ejpam-5960 365 9 𝑆 𝑆 PROPN ejpam-5960 365 10 is be AUX ejpam-5960 365 11 denoted denote VERB ejpam-5960 365 12 by by ADP ejpam-5960 365 13 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝑆 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝑆 NOUN ejpam-5960 365 14 ) ) PUNCT ejpam-5960 365 15 . . PUNCT ejpam-5960 366 1 theorem theorem VERB ejpam-5960 366 2 3.22 3.22 NUM ejpam-5960 366 3 . . PUNCT ejpam-5960 367 1 suppose suppose VERB ejpam-5960 367 2 that that SCONJ ejpam-5960 367 3 𝑆 𝑆 PROPN ejpam-5960 367 4 is be AUX ejpam-5960 367 5 a a DET ejpam-5960 367 6 molecular molecular ADJ ejpam-5960 367 7 net net NOUN ejpam-5960 367 8 in in ADP ejpam-5960 367 9 ( ( PUNCT ejpam-5960 367 10 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 367 11 , , PUNCT ejpam-5960 367 12 𝜏 𝜏 NOUN ejpam-5960 367 13 ) ) PUNCT ejpam-5960 367 14 , , PUNCT ejpam-5960 367 15 πœ‡ πœ‡ ADP ejpam-5960 367 16 ∈ ∈ PROPN ejpam-5960 367 17 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 367 18 and and CCONJ ejpam-5960 367 19 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 367 20 ∈ ∈ PROPN ejpam-5960 367 21 𝑀 𝑀 PROPN ejpam-5960 367 22 ( ( PUNCT ejpam-5960 367 23 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 367 24 ) ) PUNCT ejpam-5960 367 25 . . PUNCT ejpam-5960 368 1 then then ADV ejpam-5960 368 2 the the DET ejpam-5960 368 3 following following ADJ ejpam-5960 368 4 statements statement NOUN ejpam-5960 368 5 hold hold VERB ejpam-5960 368 6 : : PUNCT ejpam-5960 368 7 ( ( PUNCT ejpam-5960 368 8 i i NOUN ejpam-5960 368 9 ) ) PUNCT ejpam-5960 368 10 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 368 11 ∈ ∈ PROPN ejpam-5960 368 12 𝑁𝛼𝐡. 𝑁𝛼𝐡. PROPN ejpam-5960 368 13 lim(𝑆 lim(𝑆 PROPN ejpam-5960 368 14 ) ) PUNCT ejpam-5960 368 15 iff iff PROPN ejpam-5960 368 16 𝑆 𝑆 PROPN ejpam-5960 368 17 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 368 18 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 368 19 ( ( PUNCT ejpam-5960 368 20 ii ii NOUN ejpam-5960 368 21 ) ) PUNCT ejpam-5960 368 22 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 368 23 ∈ ∈ PROPN ejpam-5960 368 24 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝑆 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝑆 NUM ejpam-5960 368 25 ) ) PUNCT ejpam-5960 369 1 iff iff PROPN ejpam-5960 369 2 𝑆 𝑆 PROPN ejpam-5960 369 3 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 369 4 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 369 5 ( ( PUNCT ejpam-5960 369 6 iii iii NOUN ejpam-5960 369 7 ) ) PUNCT ejpam-5960 369 8 lim(𝑆 lim(𝑆 PROPN ejpam-5960 369 9 ) ) PUNCT ejpam-5960 369 10 ≀ ≀ PUNCT ejpam-5960 369 11 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 369 12 lim(𝑆 lim(𝑆 PROPN ejpam-5960 369 13 ) ) PUNCT ejpam-5960 369 14 . . PUNCT ejpam-5960 370 1 ( ( PUNCT ejpam-5960 370 2 iv iv X ejpam-5960 370 3 ) ) PUNCT ejpam-5960 370 4 π‘Žπ‘‘β„Ž(𝑆 π‘Žπ‘‘β„Ž(𝑆 NOUN ejpam-5960 370 5 ) ) PUNCT ejpam-5960 370 6 ≀ ≀ NOUN ejpam-5960 370 7 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝑆 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝑆 NUM ejpam-5960 370 8 ) ) PUNCT ejpam-5960 370 9 . . PUNCT ejpam-5960 371 1 ( ( PUNCT ejpam-5960 371 2 v v NOUN ejpam-5960 371 3 ) ) PUNCT ejpam-5960 371 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 371 5 lim(𝑆 lim(𝑆 PROPN ejpam-5960 371 6 ) ) PUNCT ejpam-5960 371 7 and and CCONJ ejpam-5960 371 8 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝑆 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝑆 NUM ejpam-5960 371 9 ) ) PUNCT ejpam-5960 371 10 are be AUX ejpam-5960 371 11 𝑁𝛼𝐡–closed 𝑁𝛼𝐡–close VERB ejpam-5960 371 12 in in ADP ejpam-5960 371 13 𝐿𝑋. 𝐿𝑋. X ejpam-5960 371 14 ( ( PUNCT ejpam-5960 371 15 vi vi NOUN ejpam-5960 371 16 ) ) PUNCT ejpam-5960 371 17 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 371 18 ∈ ∈ PROPN ejpam-5960 371 19 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 371 20 ( ( PUNCT ejpam-5960 371 21 πœ‡ πœ‡ NOUN ejpam-5960 371 22 ) ) PUNCT ejpam-5960 371 23 ( ( PUNCT ejpam-5960 371 24 resp resp NOUN ejpam-5960 371 25 . . PUNCT ejpam-5960 372 1 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 372 2 ∈ ∈ PROPN ejpam-5960 372 3 𝛿𝑐𝑙 𝛿𝑐𝑙 X ejpam-5960 372 4 ( ( PUNCT ejpam-5960 372 5 πœ‡ πœ‡ NOUN ejpam-5960 372 6 ) ) PUNCT ejpam-5960 372 7 [ [ X ejpam-5960 372 8 9 9 NUM ejpam-5960 372 9 ] ] NUM ejpam-5960 372 10 ) ) PUNCT ejpam-5960 372 11 , , PUNCT ejpam-5960 372 12 iff iff PROPN ejpam-5960 372 13 there there PRON ejpam-5960 372 14 exists exist VERB ejpam-5960 372 15 a a DET ejpam-5960 372 16 molecular molecular ADJ ejpam-5960 372 17 net net ADJ ejpam-5960 372 18 𝑆 𝑆 PROPN ejpam-5960 372 19 in in ADP ejpam-5960 372 20 πœ‡ πœ‡ ADP ejpam-5960 372 21 such such ADJ ejpam-5960 372 22 that that SCONJ ejpam-5960 372 23 𝑆 𝑆 PROPN ejpam-5960 372 24 is be AUX ejpam-5960 372 25 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 372 26 ( ( PUNCT ejpam-5960 372 27 resp resp NOUN ejpam-5960 372 28 . . PUNCT ejpam-5960 373 1 𝛿–converges 𝛿–converge NOUN ejpam-5960 373 2 ) ) PUNCT ejpam-5960 373 3 to to PART ejpam-5960 373 4 π‘₯𝛼. π‘₯𝛼. VERB ejpam-5960 373 5 proof proof NOUN ejpam-5960 373 6 . . PUNCT ejpam-5960 374 1 ( ( PUNCT ejpam-5960 374 2 i i NOUN ejpam-5960 374 3 ) ) PUNCT ejpam-5960 374 4 let let VERB ejpam-5960 374 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 374 6 ∈ ∈ PROPN ejpam-5960 374 7 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 374 8 lim(𝑆 lim(𝑆 PROPN ejpam-5960 374 9 ) ) PUNCT ejpam-5960 374 10 and and CCONJ ejpam-5960 374 11 let let VERB ejpam-5960 374 12 πœ† πœ† DET ejpam-5960 374 13 ∈ ∈ PROPN ejpam-5960 374 14 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 374 15 . . PUNCT ejpam-5960 375 1 since since SCONJ ejpam-5960 375 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 375 3 βˆ‰ βˆ‰ PROPN ejpam-5960 375 4 πœ† πœ† NOUN ejpam-5960 375 5 , , PUNCT ejpam-5960 375 6 then then ADV ejpam-5960 375 7 𝑁𝛼𝐡. 𝑁𝛼𝐡. PROPN ejpam-5960 375 8 lim(𝑆 lim(𝑆 PROPN ejpam-5960 375 9 ) ) PUNCT ejpam-5960 375 10 βˆ‰ βˆ‰ PROPN ejpam-5960 375 11 πœ†. πœ†. PROPN ejpam-5960 375 12 therefore therefore ADV ejpam-5960 375 13 there there PRON ejpam-5960 375 14 exists exist VERB ejpam-5960 375 15 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 375 16 ∈ ∈ PROPN ejpam-5960 375 17 𝑀 𝑀 PROPN ejpam-5960 375 18 ( ( PUNCT ejpam-5960 375 19 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 375 20 ) ) PUNCT ejpam-5960 376 1 such such ADJ ejpam-5960 376 2 that that SCONJ ejpam-5960 376 3 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 376 4 ∈ ∈ PROPN ejpam-5960 376 5 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 376 6 lim(𝑆 lim(𝑆 PROPN ejpam-5960 376 7 ) ) PUNCT ejpam-5960 376 8 and and CCONJ ejpam-5960 376 9 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 376 10 βˆ‰ βˆ‰ PROPN ejpam-5960 376 11 πœ†. πœ†. PROPN ejpam-5960 376 12 then then ADV ejpam-5960 376 13 πœ† πœ† PROPN ejpam-5960 376 14 ∈ ∈ PROPN ejpam-5960 376 15 𝑁𝛼𝐡𝑅𝑦𝛾 𝑁𝛼𝐡𝑅𝑦𝛾 X ejpam-5960 376 16 , , PUNCT ejpam-5960 376 17 and and CCONJ ejpam-5960 376 18 so so ADV ejpam-5960 376 19 there there PRON ejpam-5960 376 20 is be VERB ejpam-5960 376 21 𝑛 𝑛 DET ejpam-5960 376 22 ∈ ∈ PROPN ejpam-5960 376 23 𝐷 𝐷 NOUN ejpam-5960 376 24 such such ADJ ejpam-5960 376 25 that that PRON ejpam-5960 376 26 for for ADP ejpam-5960 376 27 each each DET ejpam-5960 376 28 π‘š π‘š PROPN ejpam-5960 376 29 ∈ ∈ PROPN ejpam-5960 376 30 𝐷 𝐷 NOUN ejpam-5960 376 31 and and CCONJ ejpam-5960 376 32 π‘š π‘š ADP ejpam-5960 376 33 β‰₯ β‰₯ NOUN ejpam-5960 376 34 𝑛 𝑛 ADP ejpam-5960 376 35 we we PRON ejpam-5960 376 36 have have VERB ejpam-5960 376 37 𝑆(π‘š 𝑆(π‘š NOUN ejpam-5960 376 38 ) ) PUNCT ejpam-5960 376 39 βˆ‰ βˆ‰ PROPN ejpam-5960 376 40 πœ† πœ† NOUN ejpam-5960 376 41 , , PUNCT ejpam-5960 376 42 but but CCONJ ejpam-5960 376 43 since since SCONJ ejpam-5960 376 44 πœ† πœ† PRON ejpam-5960 376 45 ∈ ∈ PROPN ejpam-5960 376 46 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 376 47 so so ADV ejpam-5960 376 48 𝑆 𝑆 PROPN ejpam-5960 376 49 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ X ejpam-5960 376 50 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 376 51 conversely conversely ADV ejpam-5960 376 52 , , PUNCT ejpam-5960 376 53 let let VERB ejpam-5960 376 54 𝑆 𝑆 PROPN ejpam-5960 376 55 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ VERB ejpam-5960 376 56 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 376 57 , , PUNCT ejpam-5960 376 58 then then ADV ejpam-5960 376 59 by by ADP ejpam-5960 376 60 definition definition NOUN ejpam-5960 376 61 3.20 3.20 NUM ejpam-5960 376 62 , , PUNCT ejpam-5960 376 63 we we PRON ejpam-5960 376 64 have have VERB ejpam-5960 376 65 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 376 66 ∈ ∈ PROPN ejpam-5960 376 67 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 376 68 lim(𝑆 lim(𝑆 PROPN ejpam-5960 376 69 ) ) PUNCT ejpam-5960 376 70 . . PUNCT ejpam-5960 377 1 ( ( PUNCT ejpam-5960 377 2 ii ii X ejpam-5960 377 3 ) ) PUNCT ejpam-5960 377 4 let let VERB ejpam-5960 377 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 377 6 ∈ ∈ PROPN ejpam-5960 377 7 𝑁𝛼𝐡.(𝑆 𝑁𝛼𝐡.(𝑆 NOUN ejpam-5960 377 8 ) ) PUNCT ejpam-5960 377 9 and and CCONJ ejpam-5960 377 10 let let VERB ejpam-5960 377 11 πœ† πœ† DET ejpam-5960 377 12 ∈ ∈ PROPN ejpam-5960 377 13 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 377 14 . . PUNCT ejpam-5960 378 1 since since SCONJ ejpam-5960 378 2 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 378 3 ∈ ∈ PROPN ejpam-5960 378 4 𝑁𝛼𝐡.(𝑆 𝑁𝛼𝐡.(𝑆 NOUN ejpam-5960 378 5 ) ) PUNCT ejpam-5960 378 6 , , PUNCT ejpam-5960 378 7 then then ADV ejpam-5960 378 8 every every DET ejpam-5960 378 9 𝑛 𝑛 PROPN ejpam-5960 378 10 ∈ ∈ PROPN ejpam-5960 378 11 𝐷 𝐷 NOUN ejpam-5960 378 12 there there PRON ejpam-5960 378 13 is be VERB ejpam-5960 378 14 π‘š π‘š PRON ejpam-5960 378 15 ∈ ∈ PROPN ejpam-5960 378 16 𝐷 𝐷 NOUN ejpam-5960 378 17 such such ADJ ejpam-5960 378 18 that that SCONJ ejpam-5960 378 19 π‘š π‘š PROPN ejpam-5960 378 20 β‰₯ β‰₯ PRON ejpam-5960 378 21 𝑛 𝑛 PROPN ejpam-5960 378 22 and and CCONJ ejpam-5960 378 23 𝑆(π‘š 𝑆(π‘š SYM ejpam-5960 378 24 ) ) PUNCT ejpam-5960 378 25 βˆ‰ βˆ‰ PROPN ejpam-5960 378 26 πœ† πœ† NOUN ejpam-5960 378 27 , , PUNCT ejpam-5960 378 28 hence hence ADV ejpam-5960 378 29 𝑆 𝑆 PROPN ejpam-5960 378 30 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 378 31 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 378 32 conversely conversely ADV ejpam-5960 378 33 , , PUNCT ejpam-5960 378 34 let let VERB ejpam-5960 378 35 𝑆 𝑆 PROPN ejpam-5960 378 36 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 378 37 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 378 38 , , PUNCT ejpam-5960 378 39 then then ADV ejpam-5960 378 40 by by ADP ejpam-5960 378 41 definition definition NOUN ejpam-5960 378 42 3.21 3.21 NUM ejpam-5960 378 43 we we PRON ejpam-5960 378 44 have have VERB ejpam-5960 378 45 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 378 46 ∈ ∈ PROPN ejpam-5960 378 47 𝑁𝛼𝐡.(𝑆 𝑁𝛼𝐡.(𝑆 NOUN ejpam-5960 378 48 ) ) PUNCT ejpam-5960 378 49 . . PUNCT ejpam-5960 379 1 ( ( PUNCT ejpam-5960 379 2 iii iii X ejpam-5960 379 3 ) ) PUNCT ejpam-5960 379 4 let let VERB ejpam-5960 379 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 379 6 ∈ ∈ PROPN ejpam-5960 379 7 lim(𝑆 lim(𝑆 PROPN ejpam-5960 379 8 ) ) PUNCT ejpam-5960 379 9 and and CCONJ ejpam-5960 379 10 let let VERB ejpam-5960 379 11 πœ‚ πœ‚ X ejpam-5960 379 12 ∈ ∈ PROPN ejpam-5960 379 13 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 379 14 . . PUNCT ejpam-5960 380 1 since since SCONJ ejpam-5960 380 2 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 380 3 βŠ† βŠ† NUM ejpam-5960 380 4 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 380 5 , , PUNCT ejpam-5960 380 6 then then ADV ejpam-5960 380 7 πœ‚ πœ‚ PROPN ejpam-5960 380 8 ∈ ∈ PROPN ejpam-5960 380 9 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 380 10 . . PUNCT ejpam-5960 381 1 and and CCONJ ejpam-5960 381 2 since since SCONJ ejpam-5960 381 3 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 381 4 ∈ ∈ PROPN ejpam-5960 381 5 lim(𝑆 lim(𝑆 PROPN ejpam-5960 381 6 ) ) PUNCT ejpam-5960 381 7 , , PUNCT ejpam-5960 381 8 then then ADV ejpam-5960 381 9 , , PUNCT ejpam-5960 381 10 for for ADP ejpam-5960 381 11 each each DET ejpam-5960 381 12 πœ† πœ† DET ejpam-5960 381 13 ∈ ∈ PROPN ejpam-5960 381 14 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 381 15 there there PRON ejpam-5960 381 16 is be VERB ejpam-5960 381 17 𝑛 𝑛 DET ejpam-5960 381 18 ∈ ∈ PROPN ejpam-5960 381 19 𝐷 𝐷 NOUN ejpam-5960 381 20 such such ADJ ejpam-5960 381 21 that that PRON ejpam-5960 381 22 for for ADP ejpam-5960 381 23 each each DET ejpam-5960 381 24 π‘š π‘š PROPN ejpam-5960 381 25 ∈ ∈ PROPN ejpam-5960 381 26 𝐷 𝐷 NOUN ejpam-5960 381 27 and and CCONJ ejpam-5960 381 28 π‘š π‘š ADP ejpam-5960 381 29 β‰₯ β‰₯ NUM ejpam-5960 381 30 𝑛 𝑛 NOUN ejpam-5960 381 31 , , PUNCT ejpam-5960 381 32 we we PRON ejpam-5960 381 33 have have VERB ejpam-5960 381 34 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 381 35 ) ) PUNCT ejpam-5960 381 36 βˆ‰ βˆ‰ PROPN ejpam-5960 381 37 πœ† πœ† NOUN ejpam-5960 382 1 and and CCONJ ejpam-5960 382 2 so so ADV ejpam-5960 382 3 𝑆(π‘š 𝑆(π‘š SYM ejpam-5960 382 4 ) ) PUNCT ejpam-5960 382 5 βˆ‰ βˆ‰ PROPN ejpam-5960 382 6 πœ‚. πœ‚. NOUN ejpam-5960 382 7 hence hence ADV ejpam-5960 382 8 , , PUNCT ejpam-5960 382 9 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 382 10 ∈ ∈ PROPN ejpam-5960 382 11 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 382 12 lim(𝑆 lim(𝑆 PROPN ejpam-5960 382 13 ) ) PUNCT ejpam-5960 382 14 . . PUNCT ejpam-5960 383 1 so so ADV ejpam-5960 383 2 lim(𝑆 lim(𝑆 NUM ejpam-5960 383 3 ) ) PUNCT ejpam-5960 383 4 ≀ ≀ PUNCT ejpam-5960 383 5 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 383 6 lim(𝑆 lim(𝑆 PROPN ejpam-5960 383 7 ) ) PUNCT ejpam-5960 383 8 . . PUNCT ejpam-5960 384 1 ( ( PUNCT ejpam-5960 384 2 iv iv X ejpam-5960 384 3 ) ) PUNCT ejpam-5960 384 4 let let VERB ejpam-5960 384 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 384 6 ∈ ∈ PROPN ejpam-5960 384 7 π‘Žπ‘‘β„Ž(𝑆 π‘Žπ‘‘β„Ž(𝑆 NUM ejpam-5960 384 8 ) ) PUNCT ejpam-5960 384 9 and and CCONJ ejpam-5960 384 10 let let VERB ejpam-5960 384 11 πœ‚ πœ‚ X ejpam-5960 384 12 ∈ ∈ PROPN ejpam-5960 384 13 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 384 14 . . PUNCT ejpam-5960 385 1 since since SCONJ ejpam-5960 385 2 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 385 3 βŠ† βŠ† NUM ejpam-5960 385 4 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 385 5 , , PUNCT ejpam-5960 385 6 then then ADV ejpam-5960 385 7 πœ‚ πœ‚ PROPN ejpam-5960 385 8 ∈ ∈ PROPN ejpam-5960 385 9 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 385 10 . . PUNCT ejpam-5960 386 1 and and CCONJ ejpam-5960 386 2 since since SCONJ ejpam-5960 386 3 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 386 4 ∈ ∈ PROPN ejpam-5960 386 5 π‘Žπ‘‘β„Ž(𝑆 π‘Žπ‘‘β„Ž(𝑆 PROPN ejpam-5960 386 6 ) ) PUNCT ejpam-5960 386 7 , , PUNCT ejpam-5960 386 8 then then ADV ejpam-5960 386 9 , , PUNCT ejpam-5960 386 10 for for ADP ejpam-5960 386 11 each each DET ejpam-5960 386 12 πœ† πœ† DET ejpam-5960 386 13 ∈ ∈ PROPN ejpam-5960 386 14 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 386 15 and and CCONJ ejpam-5960 386 16 for for ADP ejpam-5960 386 17 each each DET ejpam-5960 386 18 𝑛 𝑛 PRON ejpam-5960 386 19 ∈ ∈ PROPN ejpam-5960 386 20 𝐷 𝐷 PROPN ejpam-5960 386 21 there there PRON ejpam-5960 386 22 exists exist VERB ejpam-5960 386 23 π‘š π‘š PROPN ejpam-5960 386 24 ∈ ∈ PROPN ejpam-5960 386 25 𝐷 𝐷 NOUN ejpam-5960 386 26 such such ADJ ejpam-5960 386 27 that that SCONJ ejpam-5960 386 28 π‘š π‘š PROPN ejpam-5960 386 29 β‰₯ β‰₯ PRON ejpam-5960 386 30 𝑛 𝑛 ADP ejpam-5960 386 31 we we PRON ejpam-5960 386 32 have have VERB ejpam-5960 386 33 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 386 34 ) ) PUNCT ejpam-5960 386 35 βˆ‰ βˆ‰ PROPN ejpam-5960 386 36 πœ†. πœ†. PROPN ejpam-5960 386 37 and and CCONJ ejpam-5960 386 38 so so ADV ejpam-5960 386 39 𝑆(π‘š 𝑆(π‘š SYM ejpam-5960 386 40 ) ) PUNCT ejpam-5960 386 41 βˆ‰ βˆ‰ PROPN ejpam-5960 386 42 πœ‚. πœ‚. NOUN ejpam-5960 386 43 hence hence ADV ejpam-5960 386 44 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 386 45 ∈ ∈ PROPN ejpam-5960 386 46 𝑁𝛼𝐡.(𝑆 𝑁𝛼𝐡.(𝑆 NOUN ejpam-5960 386 47 ) ) PUNCT ejpam-5960 386 48 . . PUNCT ejpam-5960 387 1 so so ADV ejpam-5960 387 2 π‘Žπ‘‘β„Ž(𝑆 π‘Žπ‘‘β„Ž(𝑆 NOUN ejpam-5960 387 3 ) ) PUNCT ejpam-5960 387 4 ≀ ≀ NOUN ejpam-5960 387 5 𝑁𝛼𝐡.(𝑆 𝑁𝛼𝐡.(𝑆 NOUN ejpam-5960 387 6 ) ) PUNCT ejpam-5960 387 7 . . PUNCT ejpam-5960 388 1 ( ( PUNCT ejpam-5960 388 2 v v X ejpam-5960 388 3 ) ) PUNCT ejpam-5960 388 4 let let VERB ejpam-5960 388 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 388 6 ∈ ∈ NOUN ejpam-5960 388 7 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 388 8 ( ( PUNCT ejpam-5960 388 9 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 388 10 lim(𝑆 lim(𝑆 PROPN ejpam-5960 388 11 ) ) PUNCT ejpam-5960 388 12 ) ) PUNCT ejpam-5960 388 13 , , PUNCT ejpam-5960 388 14 then then ADV ejpam-5960 388 15 𝑁𝛼𝐡. 𝑁𝛼𝐡. PROPN ejpam-5960 388 16 lim(𝑆 lim(𝑆 PROPN ejpam-5960 388 17 ) ) PUNCT ejpam-5960 388 18 β‰° β‰° PROPN ejpam-5960 388 19 πœ† πœ† PRON ejpam-5960 388 20 for for ADP ejpam-5960 388 21 each each DET ejpam-5960 388 22 πœ† πœ† PRON ejpam-5960 388 23 ∈ ∈ PROPN ejpam-5960 388 24 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 388 25 and and CCONJ ejpam-5960 388 26 then then ADV ejpam-5960 388 27 there there PRON ejpam-5960 388 28 exists exist VERB ejpam-5960 388 29 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 388 30 ∈ ∈ PROPN ejpam-5960 388 31 𝑀 𝑀 PROPN ejpam-5960 388 32 ( ( PUNCT ejpam-5960 388 33 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 388 34 ) ) PUNCT ejpam-5960 388 35 such such ADJ ejpam-5960 389 1 that that SCONJ ejpam-5960 389 2 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 389 3 ∈ ∈ PROPN ejpam-5960 389 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 389 5 lim(𝑆 lim(𝑆 PROPN ejpam-5960 389 6 ) ) PUNCT ejpam-5960 389 7 and and CCONJ ejpam-5960 389 8 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 389 9 βˆ‰ βˆ‰ PROPN ejpam-5960 389 10 πœ†. πœ†. PROPN ejpam-5960 389 11 then then ADV ejpam-5960 389 12 for for ADP ejpam-5960 389 13 each each DET ejpam-5960 389 14 n. n. NOUN ejpam-5960 389 15 a. a. NOUN ejpam-5960 389 16 alsaedi alsaedi PROPN ejpam-5960 389 17 / / SYM ejpam-5960 389 18 eur eur PROPN ejpam-5960 389 19 . . PUNCT ejpam-5960 390 1 j. j. PROPN ejpam-5960 390 2 pure pure PROPN ejpam-5960 390 3 appl appl PROPN ejpam-5960 390 4 . . PROPN ejpam-5960 390 5 math math PROPN ejpam-5960 390 6 , , PUNCT ejpam-5960 390 7 18 18 NUM ejpam-5960 390 8 ( ( PUNCT ejpam-5960 390 9 4 4 NUM ejpam-5960 390 10 ) ) PUNCT ejpam-5960 390 11 ( ( PUNCT ejpam-5960 390 12 2025 2025 NUM ejpam-5960 390 13 ) ) PUNCT ejpam-5960 390 14 , , PUNCT ejpam-5960 390 15 5960 5960 NUM ejpam-5960 390 16 13 13 NUM ejpam-5960 390 17 of of ADP ejpam-5960 390 18 22 22 NUM ejpam-5960 390 19 πœ‡ πœ‡ ADP ejpam-5960 390 20 ∈ ∈ PROPN ejpam-5960 390 21 𝑁𝛼𝐡𝑅𝑦𝛾 𝑁𝛼𝐡𝑅𝑦𝛾 X ejpam-5960 390 22 , , PUNCT ejpam-5960 390 23 there there PRON ejpam-5960 390 24 is be VERB ejpam-5960 390 25 𝑛 𝑛 DET ejpam-5960 390 26 ∈ ∈ PROPN ejpam-5960 390 27 𝐷 𝐷 NOUN ejpam-5960 390 28 such such ADJ ejpam-5960 390 29 that that PRON ejpam-5960 390 30 for for ADP ejpam-5960 390 31 each each DET ejpam-5960 390 32 π‘š π‘š PROPN ejpam-5960 390 33 ∈ ∈ PROPN ejpam-5960 390 34 𝐷 𝐷 NOUN ejpam-5960 390 35 and and CCONJ ejpam-5960 390 36 π‘š π‘š ADP ejpam-5960 390 37 β‰₯ β‰₯ NOUN ejpam-5960 390 38 𝑛 𝑛 ADP ejpam-5960 390 39 we we PRON ejpam-5960 390 40 have have VERB ejpam-5960 390 41 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 390 42 ) ) PUNCT ejpam-5960 390 43 βˆ‰ βˆ‰ PROPN ejpam-5960 390 44 πœ‡ πœ‡ ADP ejpam-5960 390 45 , , PUNCT ejpam-5960 390 46 and and CCONJ ejpam-5960 390 47 so so ADV ejpam-5960 390 48 𝑆(π‘š 𝑆(π‘š SYM ejpam-5960 390 49 ) ) PUNCT ejpam-5960 390 50 βˆ‰ βˆ‰ PROPN ejpam-5960 390 51 πœ†. πœ†. PROPN ejpam-5960 390 52 hence hence ADV ejpam-5960 390 53 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 390 54 ∈ ∈ PROPN ejpam-5960 390 55 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 390 56 lim(𝑆 lim(𝑆 PROPN ejpam-5960 390 57 ) ) PUNCT ejpam-5960 390 58 . . PUNCT ejpam-5960 391 1 thus thus ADV ejpam-5960 391 2 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 VERB ejpam-5960 391 3 ( ( PUNCT ejpam-5960 391 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 391 5 lim(𝑆 lim(𝑆 PROPN ejpam-5960 391 6 ) ) PUNCT ejpam-5960 391 7 ) ) PUNCT ejpam-5960 391 8 ≀ ≀ PUNCT ejpam-5960 392 1 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 392 2 lim(𝑆 lim(𝑆 PROPN ejpam-5960 392 3 ) ) PUNCT ejpam-5960 392 4 and and CCONJ ejpam-5960 392 5 so so ADV ejpam-5960 392 6 𝑁𝛼𝐡. 𝑁𝛼𝐡. PROPN ejpam-5960 392 7 lim(𝑆 lim(𝑆 PROPN ejpam-5960 392 8 ) ) PUNCT ejpam-5960 392 9 is be AUX ejpam-5960 392 10 a a DET ejpam-5960 392 11 𝑁𝛼𝐡–closed 𝑁𝛼𝐡–closed PROPN ejpam-5960 392 12 set set NOUN ejpam-5960 392 13 . . PUNCT ejpam-5960 393 1 similarly similarly ADV ejpam-5960 393 2 , , PUNCT ejpam-5960 393 3 one one PRON ejpam-5960 393 4 can can AUX ejpam-5960 393 5 easily easily ADV ejpam-5960 393 6 verify verify VERB ejpam-5960 393 7 that that DET ejpam-5960 393 8 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 393 9 ( ( PUNCT ejpam-5960 393 10 𝑁𝛼𝐡.(𝑆 𝑁𝛼𝐡.(𝑆 NOUN ejpam-5960 393 11 ) ) PUNCT ejpam-5960 393 12 ) ) PUNCT ejpam-5960 393 13 ≀ ≀ NOUN ejpam-5960 393 14 𝑁𝛼𝐡.(𝑆 𝑁𝛼𝐡.(𝑆 NOUN ejpam-5960 393 15 ) ) PUNCT ejpam-5960 393 16 . . PUNCT ejpam-5960 394 1 ( ( PUNCT ejpam-5960 394 2 vi vi X ejpam-5960 394 3 ) ) PUNCT ejpam-5960 394 4 let let VERB ejpam-5960 394 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 394 6 ∈ ∈ PROPN ejpam-5960 394 7 𝑀 𝑀 PROPN ejpam-5960 394 8 ( ( PUNCT ejpam-5960 394 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 394 10 ) ) PUNCT ejpam-5960 394 11 such such ADJ ejpam-5960 394 12 that that SCONJ ejpam-5960 394 13 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 394 14 ∈ ∈ PROPN ejpam-5960 394 15 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 394 16 ( ( PUNCT ejpam-5960 394 17 πœ‡ πœ‡ NOUN ejpam-5960 394 18 ) ) PUNCT ejpam-5960 394 19 , , PUNCT ejpam-5960 394 20 then then ADV ejpam-5960 394 21 πœ‡ πœ‡ SCONJ ejpam-5960 394 22 β‰° β‰° PROPN ejpam-5960 394 23 πœ† πœ† PRON ejpam-5960 394 24 for for ADP ejpam-5960 394 25 each each DET ejpam-5960 394 26 πœ† πœ† PRON ejpam-5960 394 27 ∈ ∈ PROPN ejpam-5960 394 28 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 394 29 . . PUNCT ejpam-5960 395 1 since since SCONJ ejpam-5960 395 2 πœ‡ πœ‡ ADP ejpam-5960 395 3 β‰° β‰° PROPN ejpam-5960 395 4 πœ† πœ† PUNCT ejpam-5960 395 5 then then ADV ejpam-5960 395 6 , , PUNCT ejpam-5960 395 7 there there PRON ejpam-5960 395 8 exists exist VERB ejpam-5960 395 9 𝛼(πœ‡ 𝛼(πœ‡ NOUN ejpam-5960 395 10 , , PUNCT ejpam-5960 395 11 πœ† πœ† NOUN ejpam-5960 395 12 ) ) PUNCT ejpam-5960 395 13 ∈ ∈ PROPN ejpam-5960 395 14 𝑀 𝑀 PROPN ejpam-5960 395 15 ( ( PUNCT ejpam-5960 395 16 𝐿 𝐿 PROPN ejpam-5960 395 17 ) ) PUNCT ejpam-5960 395 18 such such ADJ ejpam-5960 395 19 that that DET ejpam-5960 395 20 π‘₯𝛼(πœ‡,πœ† π‘₯𝛼(πœ‡,πœ† NOUN ejpam-5960 395 21 ) ) PUNCT ejpam-5960 395 22 ∈ ∈ NOUN ejpam-5960 395 23 πœ‡ πœ‡ ADP ejpam-5960 395 24 with with ADP ejpam-5960 395 25 π‘₯𝛼(πœ‡,πœ† π‘₯𝛼(πœ‡,πœ† NOUN ejpam-5960 395 26 ) ) PUNCT ejpam-5960 395 27 βˆ‰ βˆ‰ PROPN ejpam-5960 395 28 πœ†. πœ†. PROPN ejpam-5960 395 29 since since SCONJ ejpam-5960 395 30 the the DET ejpam-5960 395 31 pair pair NOUN ejpam-5960 395 32 ( ( PUNCT ejpam-5960 395 33 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 395 34 , , PUNCT ejpam-5960 395 35 β‰₯ β‰₯ NUM ejpam-5960 395 36 ) ) PUNCT ejpam-5960 395 37 is be AUX ejpam-5960 395 38 a a DET ejpam-5960 395 39 directed direct VERB ejpam-5960 395 40 set set NOUN ejpam-5960 395 41 so so SCONJ ejpam-5960 395 42 we we PRON ejpam-5960 395 43 can can AUX ejpam-5960 395 44 define define VERB ejpam-5960 395 45 a a DET ejpam-5960 395 46 molecular molecular ADJ ejpam-5960 395 47 net net ADJ ejpam-5960 395 48 𝑆 𝑆 PROPN ejpam-5960 395 49 : : PUNCT ejpam-5960 395 50 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 395 51 β†’ β†’ SYM ejpam-5960 395 52 𝑀 𝑀 PROPN ejpam-5960 395 53 ( ( PUNCT ejpam-5960 395 54 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 395 55 ) ) PUNCT ejpam-5960 395 56 as as SCONJ ejpam-5960 395 57 follows follow VERB ejpam-5960 395 58 𝑆(πœ† 𝑆(πœ† NOUN ejpam-5960 395 59 ) ) PUNCT ejpam-5960 395 60 = = SYM ejpam-5960 395 61 π‘₯𝛼(πœ‡,πœ† π‘₯𝛼(πœ‡,πœ† NOUN ejpam-5960 395 62 ) ) PUNCT ejpam-5960 395 63 for for ADP ejpam-5960 395 64 each each DET ejpam-5960 395 65 πœ† πœ† PRON ejpam-5960 395 66 ∈ ∈ PROPN ejpam-5960 395 67 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 395 68 . . PUNCT ejpam-5960 396 1 hence hence ADV ejpam-5960 396 2 𝑆 𝑆 PROPN ejpam-5960 396 3 is be AUX ejpam-5960 396 4 a a DET ejpam-5960 396 5 molecular molecular ADJ ejpam-5960 396 6 net net NOUN ejpam-5960 396 7 in in ADP ejpam-5960 396 8 πœ‡. πœ‡. NOUN ejpam-5960 396 9 now now ADV ejpam-5960 396 10 let let VERB ejpam-5960 396 11 πœ‚ πœ‚ X ejpam-5960 396 12 ∈ ∈ NOUN ejpam-5960 396 13 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 396 14 such such ADJ ejpam-5960 396 15 that that SCONJ ejpam-5960 396 16 πœ† πœ† DET ejpam-5960 396 17 ≀ ≀ NUM ejpam-5960 396 18 πœ‚ πœ‚ NOUN ejpam-5960 396 19 , , PUNCT ejpam-5960 396 20 so so SCONJ ejpam-5960 396 21 we we PRON ejpam-5960 396 22 have have AUX ejpam-5960 396 23 there there PRON ejpam-5960 396 24 exists exist VERB ejpam-5960 396 25 𝑆(πœ‚ 𝑆(πœ‚ PRON ejpam-5960 396 26 ) ) PUNCT ejpam-5960 396 27 = = PUNCT ejpam-5960 396 28 π‘₯𝛼(πœ‡,πœ‚ π‘₯𝛼(πœ‡,πœ‚ NOUN ejpam-5960 396 29 ) ) PUNCT ejpam-5960 396 30 βˆ‰ βˆ‰ PROPN ejpam-5960 396 31 πœ‚ πœ‚ PROPN ejpam-5960 396 32 and and CCONJ ejpam-5960 396 33 so so ADV ejpam-5960 396 34 𝑆(πœ‚ 𝑆(πœ‚ PROPN ejpam-5960 396 35 ) ) PUNCT ejpam-5960 396 36 = = PUNCT ejpam-5960 396 37 π‘₯𝛼(πœ‡,πœ‚ π‘₯𝛼(πœ‡,πœ‚ NOUN ejpam-5960 396 38 ) ) PUNCT ejpam-5960 396 39 βˆ‰ βˆ‰ PROPN ejpam-5960 396 40 πœ†. πœ†. PROPN ejpam-5960 396 41 hence hence ADV ejpam-5960 396 42 𝑆 𝑆 PROPN ejpam-5960 396 43 is be AUX ejpam-5960 396 44 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 396 45 to to PART ejpam-5960 396 46 π‘₯𝛼. π‘₯𝛼. VERB ejpam-5960 396 47 conversely conversely ADV ejpam-5960 396 48 , , PUNCT ejpam-5960 396 49 let let VERB ejpam-5960 396 50 𝑆 𝑆 PROPN ejpam-5960 396 51 be be AUX ejpam-5960 396 52 a a DET ejpam-5960 396 53 molecular molecular ADJ ejpam-5960 396 54 net net NOUN ejpam-5960 396 55 in in ADP ejpam-5960 396 56 πœ‡ πœ‡ ADP ejpam-5960 396 57 such such ADJ ejpam-5960 396 58 that that SCONJ ejpam-5960 396 59 𝑆 𝑆 PROPN ejpam-5960 396 60 is be AUX ejpam-5960 396 61 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 396 62 to to ADP ejpam-5960 396 63 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 396 64 , , PUNCT ejpam-5960 396 65 then then ADV ejpam-5960 396 66 for for ADP ejpam-5960 396 67 each each DET ejpam-5960 396 68 πœ† πœ† PRON ejpam-5960 396 69 ∈ ∈ PROPN ejpam-5960 396 70 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 396 71 there there PRON ejpam-5960 396 72 is be VERB ejpam-5960 396 73 𝑛 𝑛 DET ejpam-5960 396 74 ∈ ∈ PROPN ejpam-5960 396 75 𝐷 𝐷 NOUN ejpam-5960 396 76 such such ADJ ejpam-5960 396 77 for for ADP ejpam-5960 396 78 each each DET ejpam-5960 396 79 π‘š π‘š PROPN ejpam-5960 396 80 ∈ ∈ PROPN ejpam-5960 396 81 𝐷 𝐷 NOUN ejpam-5960 396 82 and and CCONJ ejpam-5960 396 83 π‘š π‘š ADP ejpam-5960 396 84 β‰₯ β‰₯ NUM ejpam-5960 396 85 𝑛 𝑛 NOUN ejpam-5960 396 86 , , PUNCT ejpam-5960 396 87 we we PRON ejpam-5960 396 88 have have VERB ejpam-5960 396 89 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 396 90 ) ) PUNCT ejpam-5960 396 91 βˆ‰ βˆ‰ PROPN ejpam-5960 396 92 πœ†. πœ†. PROPN ejpam-5960 396 93 since since SCONJ ejpam-5960 396 94 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 396 95 ) ) PUNCT ejpam-5960 396 96 ∈ ∈ NOUN ejpam-5960 396 97 πœ‡ πœ‡ ADP ejpam-5960 396 98 for for ADP ejpam-5960 396 99 each each DET ejpam-5960 396 100 𝑛 𝑛 PRON ejpam-5960 396 101 ∈ ∈ PROPN ejpam-5960 396 102 𝐷 𝐷 PROPN ejpam-5960 396 103 , , PUNCT ejpam-5960 396 104 π‘š π‘š PROPN ejpam-5960 396 105 ∈ ∈ PROPN ejpam-5960 396 106 𝐷. 𝐷. NOUN ejpam-5960 396 107 so so ADV ejpam-5960 396 108 𝑆(π‘š 𝑆(π‘š SYM ejpam-5960 396 109 ) ) PUNCT ejpam-5960 396 110 ∈ ∈ NOUN ejpam-5960 396 111 πœ‡ πœ‡ ADP ejpam-5960 396 112 and and CCONJ ejpam-5960 396 113 πœ‡ πœ‡ X ejpam-5960 396 114 β‰₯ β‰₯ NOUN ejpam-5960 396 115 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 396 116 ) ) PUNCT ejpam-5960 396 117 > > X ejpam-5960 397 1 πœ† πœ† ADP ejpam-5960 397 2 hence hence ADV ejpam-5960 397 3 πœ‡ πœ‡ SCONJ ejpam-5960 397 4 β‰° β‰° PROPN ejpam-5960 397 5 πœ† πœ† PRON ejpam-5960 397 6 for for ADP ejpam-5960 397 7 each each DET ejpam-5960 397 8 πœ† πœ† PRON ejpam-5960 397 9 ∈ ∈ PROPN ejpam-5960 397 10 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 397 11 . . PUNCT ejpam-5960 398 1 this this PRON ejpam-5960 398 2 means mean VERB ejpam-5960 398 3 that that SCONJ ejpam-5960 398 4 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 398 5 ∈ ∈ PROPN ejpam-5960 398 6 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 398 7 ( ( PUNCT ejpam-5960 398 8 πœ‡ πœ‡ NOUN ejpam-5960 398 9 ) ) PUNCT ejpam-5960 398 10 . . PUNCT ejpam-5960 399 1 definition definition NOUN ejpam-5960 399 2 3.23 3.23 NUM ejpam-5960 399 3 . . PUNCT ejpam-5960 400 1 let let VERB ejpam-5960 400 2 ( ( PUNCT ejpam-5960 400 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 400 4 , , PUNCT ejpam-5960 400 5 𝜏 𝜏 NOUN ejpam-5960 400 6 ) ) PUNCT ejpam-5960 400 7 be be VERB ejpam-5960 400 8 an an DET ejpam-5960 400 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 400 10 and and CCONJ ejpam-5960 400 11 𝐼 𝐼 PROPN ejpam-5960 400 12 be be VERB ejpam-5960 400 13 an an DET ejpam-5960 400 14 ideal ideal NOUN ejpam-5960 400 15 in in ADP ejpam-5960 400 16 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 400 17 then then ADV ejpam-5960 400 18 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 400 19 ∈ ∈ PROPN ejpam-5960 400 20 𝑀 𝑀 PROPN ejpam-5960 400 21 ( ( PUNCT ejpam-5960 400 22 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 400 23 ) ) PUNCT ejpam-5960 400 24 is be AUX ejpam-5960 400 25 called call VERB ejpam-5960 400 26 : : PUNCT ejpam-5960 400 27 ( ( PUNCT ejpam-5960 400 28 i i NOUN ejpam-5960 400 29 ) ) PUNCT ejpam-5960 400 30 limit limit NOUN ejpam-5960 400 31 point point NOUN ejpam-5960 400 32 of of ADP ejpam-5960 400 33 𝐼 𝐼 PROPN ejpam-5960 401 1 [ [ X ejpam-5960 401 2 25 25 NUM ejpam-5960 401 3 ] ] PUNCT ejpam-5960 401 4 , , PUNCT ejpam-5960 401 5 ( ( PUNCT ejpam-5960 401 6 or or CCONJ ejpam-5960 401 7 𝐼 𝐼 ADP ejpam-5960 401 8 converges converge NOUN ejpam-5960 401 9 to to ADP ejpam-5960 401 10 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 401 11 ) ) PUNCT ejpam-5960 401 12 in in ADP ejpam-5960 401 13 symbol symbol NOUN ejpam-5960 401 14 𝐼 𝐼 PROPN ejpam-5960 401 15 β†’ β†’ PUNCT ejpam-5960 401 16 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 401 17 if if SCONJ ejpam-5960 401 18 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 401 19 βŠ† βŠ† NUM ejpam-5960 401 20 𝐼. 𝐼. PROPN ejpam-5960 401 21 the the DET ejpam-5960 401 22 union union NOUN ejpam-5960 401 23 of of ADP ejpam-5960 401 24 all all DET ejpam-5960 401 25 limit limit NOUN ejpam-5960 401 26 points point NOUN ejpam-5960 401 27 of of ADP ejpam-5960 401 28 𝐼 𝐼 PROPN ejpam-5960 401 29 is be AUX ejpam-5960 401 30 denoted denote VERB ejpam-5960 401 31 by by ADP ejpam-5960 401 32 lim(𝐼 lim(𝐼 PROPN ejpam-5960 401 33 ) ) PUNCT ejpam-5960 401 34 . . PUNCT ejpam-5960 402 1 ( ( PUNCT ejpam-5960 402 2 ii ii NOUN ejpam-5960 402 3 ) ) PUNCT ejpam-5960 402 4 𝑁𝛼𝐡–bounded 𝑁𝛼𝐡–bounded PUNCT ejpam-5960 402 5 limit limit NOUN ejpam-5960 402 6 point point NOUN ejpam-5960 402 7 of of ADP ejpam-5960 402 8 𝐼 𝐼 PROPN ejpam-5960 402 9 , , PUNCT ejpam-5960 402 10 ( ( PUNCT ejpam-5960 402 11 or or CCONJ ejpam-5960 402 12 𝐼 𝐼 PROPN ejpam-5960 402 13 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 402 14 to to PART ejpam-5960 402 15 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 402 16 ) ) PUNCT ejpam-5960 402 17 in in ADP ejpam-5960 402 18 symbol symbol NOUN ejpam-5960 402 19 𝐼 𝐼 ADP ejpam-5960 402 20 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 402 21 π‘₯𝛼 π‘₯𝛼 INTJ ejpam-5960 402 22 if if SCONJ ejpam-5960 402 23 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 402 24 βŠ† βŠ† X ejpam-5960 402 25 𝐼. 𝐼. PROPN ejpam-5960 402 26 the the DET ejpam-5960 402 27 union union NOUN ejpam-5960 402 28 of of ADP ejpam-5960 402 29 all all DET ejpam-5960 402 30 𝑁𝛼𝐡–bounded 𝑁𝛼𝐡–bounded ADJ ejpam-5960 402 31 limit limit NOUN ejpam-5960 402 32 points point NOUN ejpam-5960 402 33 of of ADP ejpam-5960 402 34 𝐼 𝐼 PROPN ejpam-5960 402 35 is be AUX ejpam-5960 402 36 denoted denote VERB ejpam-5960 402 37 by by ADP ejpam-5960 402 38 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 402 39 lim(𝐼 lim(𝐼 PROPN ejpam-5960 402 40 ) ) PUNCT ejpam-5960 402 41 . . PUNCT ejpam-5960 403 1 definition definition NOUN ejpam-5960 403 2 3.24 3.24 NUM ejpam-5960 403 3 . . PUNCT ejpam-5960 404 1 let let VERB ejpam-5960 404 2 ( ( PUNCT ejpam-5960 404 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 404 4 , , PUNCT ejpam-5960 404 5 𝜏 𝜏 NOUN ejpam-5960 404 6 ) ) PUNCT ejpam-5960 404 7 be be VERB ejpam-5960 404 8 an an DET ejpam-5960 404 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 404 10 and and CCONJ ejpam-5960 404 11 𝐼 𝐼 PROPN ejpam-5960 404 12 be be VERB ejpam-5960 404 13 an an DET ejpam-5960 404 14 ideal ideal NOUN ejpam-5960 404 15 in in ADP ejpam-5960 404 16 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 404 17 then then ADV ejpam-5960 404 18 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 404 19 ∈ ∈ PROPN ejpam-5960 404 20 𝑀 𝑀 PROPN ejpam-5960 404 21 ( ( PUNCT ejpam-5960 404 22 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 404 23 ) ) PUNCT ejpam-5960 404 24 is be AUX ejpam-5960 404 25 called call VERB ejpam-5960 404 26 : : PUNCT ejpam-5960 404 27 ( ( PUNCT ejpam-5960 404 28 i i NOUN ejpam-5960 404 29 ) ) PUNCT ejpam-5960 404 30 cluster cluster NOUN ejpam-5960 404 31 point point NOUN ejpam-5960 404 32 of of ADP ejpam-5960 404 33 𝐼 𝐼 PROPN ejpam-5960 405 1 [ [ X ejpam-5960 405 2 25 25 NUM ejpam-5960 405 3 ] ] PUNCT ejpam-5960 405 4 , , PUNCT ejpam-5960 405 5 in in ADP ejpam-5960 405 6 symbol symbol NOUN ejpam-5960 405 7 𝐼 𝐼 ADP ejpam-5960 405 8 ∝ ∝ PROPN ejpam-5960 405 9 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 405 10 if if SCONJ ejpam-5960 405 11 for for ADP ejpam-5960 405 12 every every DET ejpam-5960 405 13 πœ‡ πœ‡ PRON ejpam-5960 405 14 ∈ ∈ PROPN ejpam-5960 405 15 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 405 16 and and CCONJ ejpam-5960 405 17 every every DET ejpam-5960 405 18 πœ† πœ† PROPN ejpam-5960 405 19 ∈ ∈ PROPN ejpam-5960 405 20 𝐼 𝐼 PROPN ejpam-5960 405 21 , , PUNCT ejpam-5960 405 22 πœ†βˆ¨πœ‡ πœ†βˆ¨πœ‡ ADP ejpam-5960 405 23 β‰  β‰  PROPN ejpam-5960 405 24 1𝑋. 1𝑋. NUM ejpam-5960 405 25 the the DET ejpam-5960 405 26 union union NOUN ejpam-5960 405 27 of of ADP ejpam-5960 405 28 all all DET ejpam-5960 405 29 cluster cluster NOUN ejpam-5960 405 30 points point NOUN ejpam-5960 405 31 of of ADP ejpam-5960 405 32 𝐼 𝐼 PROPN ejpam-5960 405 33 is be AUX ejpam-5960 405 34 denoted denote VERB ejpam-5960 405 35 by by ADP ejpam-5960 405 36 adh(𝐼 adh(𝐼 PROPN ejpam-5960 405 37 ) ) PUNCT ejpam-5960 405 38 . . PUNCT ejpam-5960 406 1 ( ( PUNCT ejpam-5960 406 2 ii ii NOUN ejpam-5960 406 3 ) ) PUNCT ejpam-5960 406 4 𝑁𝛼𝐡–bounded 𝑁𝛼𝐡–bounded ADJ ejpam-5960 406 5 cluster cluster NOUN ejpam-5960 406 6 point point NOUN ejpam-5960 406 7 of of ADP ejpam-5960 406 8 𝐼 𝐼 PROPN ejpam-5960 406 9 , , PUNCT ejpam-5960 406 10 in in ADP ejpam-5960 406 11 symbol symbol NOUN ejpam-5960 406 12 𝐼 𝐼 SCONJ ejpam-5960 406 13 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ X ejpam-5960 406 14 π‘₯𝛼 π‘₯𝛼 INTJ ejpam-5960 406 15 if if SCONJ ejpam-5960 406 16 for for ADP ejpam-5960 406 17 every every DET ejpam-5960 406 18 πœ‡ πœ‡ ADP ejpam-5960 406 19 ∈ ∈ NOUN ejpam-5960 406 20 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 406 21 and and CCONJ ejpam-5960 406 22 every every DET ejpam-5960 406 23 πœ† πœ† PROPN ejpam-5960 406 24 ∈ ∈ PROPN ejpam-5960 406 25 𝐼 𝐼 PROPN ejpam-5960 406 26 , , PUNCT ejpam-5960 406 27 πœ† πœ† DET ejpam-5960 406 28 ∨ ∨ NOUN ejpam-5960 406 29 πœ‡ πœ‡ ADP ejpam-5960 406 30 β‰  β‰  PROPN ejpam-5960 406 31 1𝑋. 1𝑋. NUM ejpam-5960 406 32 the the DET ejpam-5960 406 33 union union NOUN ejpam-5960 406 34 of of ADP ejpam-5960 406 35 all all DET ejpam-5960 406 36 𝑁𝛼𝐡–bounded 𝑁𝛼𝐡–bounded ADJ ejpam-5960 406 37 cluster cluster NOUN ejpam-5960 406 38 points point NOUN ejpam-5960 406 39 of of ADP ejpam-5960 406 40 𝐼 𝐼 PROPN ejpam-5960 406 41 is be AUX ejpam-5960 406 42 denoted denote VERB ejpam-5960 406 43 by by ADP ejpam-5960 406 44 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 406 45 adh(𝐼 adh(𝐼 PROPN ejpam-5960 406 46 ) ) PUNCT ejpam-5960 406 47 . . PUNCT ejpam-5960 407 1 theorem theorem VERB ejpam-5960 407 2 3.25 3.25 NUM ejpam-5960 407 3 . . PUNCT ejpam-5960 408 1 suppose suppose VERB ejpam-5960 408 2 that that SCONJ ejpam-5960 408 3 𝐼 𝐼 PROPN ejpam-5960 408 4 is be AUX ejpam-5960 408 5 an an DET ejpam-5960 408 6 ideal ideal NOUN ejpam-5960 408 7 in in ADP ejpam-5960 408 8 ( ( PUNCT ejpam-5960 408 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 408 10 , , PUNCT ejpam-5960 408 11 𝜏 𝜏 NOUN ejpam-5960 408 12 ) ) PUNCT ejpam-5960 408 13 , , PUNCT ejpam-5960 408 14 πœ‡ πœ‡ ADP ejpam-5960 408 15 ∈ ∈ PROPN ejpam-5960 408 16 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 408 17 and and CCONJ ejpam-5960 408 18 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 408 19 ∈ ∈ PROPN ejpam-5960 408 20 𝑀 𝑀 PROPN ejpam-5960 408 21 ( ( PUNCT ejpam-5960 408 22 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 408 23 ) ) PUNCT ejpam-5960 408 24 . . PUNCT ejpam-5960 409 1 then then ADV ejpam-5960 409 2 the the DET ejpam-5960 409 3 following following ADJ ejpam-5960 409 4 statements statement NOUN ejpam-5960 409 5 hold hold VERB ejpam-5960 409 6 : : PUNCT ejpam-5960 409 7 ( ( PUNCT ejpam-5960 409 8 i i NOUN ejpam-5960 409 9 ) ) PUNCT ejpam-5960 409 10 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 409 11 ∈ ∈ PROPN ejpam-5960 409 12 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 409 13 lim(𝐼 lim(𝐼 PROPN ejpam-5960 409 14 ) ) PUNCT ejpam-5960 409 15 iff iff PROPN ejpam-5960 409 16 𝐼 𝐼 PROPN ejpam-5960 409 17 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 409 18 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 409 19 ( ( PUNCT ejpam-5960 409 20 ii ii NOUN ejpam-5960 409 21 ) ) PUNCT ejpam-5960 409 22 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 409 23 ∈ ∈ PROPN ejpam-5960 409 24 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 PRON ejpam-5960 409 25 ) ) PUNCT ejpam-5960 410 1 iff iff PROPN ejpam-5960 410 2 𝐼 𝐼 PROPN ejpam-5960 410 3 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 410 4 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 410 5 ( ( PUNCT ejpam-5960 410 6 iii iii NOUN ejpam-5960 410 7 ) ) PUNCT ejpam-5960 410 8 lim(𝐼 lim(𝐼 NUM ejpam-5960 410 9 ) ) PUNCT ejpam-5960 410 10 ≀ ≀ NOUN ejpam-5960 410 11 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 410 12 lim(𝐼 lim(𝐼 NUM ejpam-5960 410 13 ) ) PUNCT ejpam-5960 410 14 ≀ ≀ NUM ejpam-5960 410 15 𝛼𝐡. 𝛼𝐡. NOUN ejpam-5960 410 16 lim(𝐼 lim(𝐼 PROPN ejpam-5960 410 17 ) ) PUNCT ejpam-5960 410 18 . . PUNCT ejpam-5960 411 1 ( ( PUNCT ejpam-5960 411 2 iv iv X ejpam-5960 411 3 ) ) PUNCT ejpam-5960 411 4 adh(𝐼 adh(𝐼 PROPN ejpam-5960 411 5 ) ) PUNCT ejpam-5960 411 6 ≀ ≀ NOUN ejpam-5960 411 7 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 NUM ejpam-5960 411 8 ) ) PUNCT ejpam-5960 411 9 ≀ ≀ NUM ejpam-5960 411 10 𝛼𝐡.adh(𝐼 𝛼𝐡.adh(𝐼 NUM ejpam-5960 411 11 ) ) PUNCT ejpam-5960 411 12 . . PUNCT ejpam-5960 412 1 ( ( PUNCT ejpam-5960 412 2 v v NOUN ejpam-5960 412 3 ) ) PUNCT ejpam-5960 412 4 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 412 5 ∈ ∈ PROPN ejpam-5960 412 6 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 412 7 ( ( PUNCT ejpam-5960 412 8 πœ‡ πœ‡ NOUN ejpam-5960 412 9 ) ) PUNCT ejpam-5960 412 10 iff iff PROPN ejpam-5960 412 11 there there PRON ejpam-5960 412 12 exists exist VERB ejpam-5960 412 13 an an DET ejpam-5960 412 14 ideal ideal NOUN ejpam-5960 412 15 𝐼 𝐼 NOUN ejpam-5960 412 16 in in ADP ejpam-5960 412 17 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 412 18 such such ADJ ejpam-5960 412 19 that that SCONJ ejpam-5960 412 20 𝐼 𝐼 PROPN ejpam-5960 412 21 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 412 22 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 412 23 and and CCONJ ejpam-5960 412 24 πœ‡ πœ‡ ADP ejpam-5960 412 25 βˆ‰ βˆ‰ PROPN ejpam-5960 412 26 𝐼. 𝐼. PROPN ejpam-5960 412 27 ( ( PUNCT ejpam-5960 412 28 vi vi NOUN ejpam-5960 412 29 ) ) PUNCT ejpam-5960 412 30 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 412 31 lim(𝐼 lim(𝐼 PROPN ejpam-5960 412 32 ) ) PUNCT ejpam-5960 412 33 and and CCONJ ejpam-5960 412 34 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 PRON ejpam-5960 412 35 ) ) PUNCT ejpam-5960 412 36 are be AUX ejpam-5960 412 37 𝑁𝛼𝐡–closed 𝑁𝛼𝐡–close VERB ejpam-5960 412 38 set set VERB ejpam-5960 412 39 in in ADP ejpam-5960 412 40 𝐿𝑋. 𝐿𝑋. X ejpam-5960 412 41 n. n. NOUN ejpam-5960 412 42 a. a. NOUN ejpam-5960 412 43 alsaedi alsaedi PROPN ejpam-5960 412 44 / / SYM ejpam-5960 412 45 eur eur PROPN ejpam-5960 412 46 . . PUNCT ejpam-5960 413 1 j. j. PROPN ejpam-5960 413 2 pure pure PROPN ejpam-5960 413 3 appl appl PROPN ejpam-5960 413 4 . . PROPN ejpam-5960 413 5 math math PROPN ejpam-5960 413 6 , , PUNCT ejpam-5960 413 7 18 18 NUM ejpam-5960 413 8 ( ( PUNCT ejpam-5960 413 9 4 4 NUM ejpam-5960 413 10 ) ) PUNCT ejpam-5960 413 11 ( ( PUNCT ejpam-5960 413 12 2025 2025 NUM ejpam-5960 413 13 ) ) PUNCT ejpam-5960 413 14 , , PUNCT ejpam-5960 413 15 5960 5960 NUM ejpam-5960 413 16 14 14 NUM ejpam-5960 413 17 of of ADP ejpam-5960 413 18 22 22 NUM ejpam-5960 413 19 proof proof NOUN ejpam-5960 413 20 . . PUNCT ejpam-5960 414 1 ( ( PUNCT ejpam-5960 414 2 i i NOUN ejpam-5960 414 3 ) ) PUNCT ejpam-5960 414 4 let let VERB ejpam-5960 414 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 414 6 ∈ ∈ VERB ejpam-5960 414 7 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 414 8 lim(𝐼 lim(𝐼 PROPN ejpam-5960 414 9 ) ) PUNCT ejpam-5960 414 10 and and CCONJ ejpam-5960 414 11 let let VERB ejpam-5960 414 12 πœ† πœ† DET ejpam-5960 414 13 ∈ ∈ PROPN ejpam-5960 414 14 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 414 15 . . PUNCT ejpam-5960 415 1 since since SCONJ ejpam-5960 415 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 415 3 βˆ‰ βˆ‰ PROPN ejpam-5960 415 4 πœ† πœ† PROPN ejpam-5960 415 5 and and CCONJ ejpam-5960 415 6 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 415 7 ∈ ∈ NOUN ejpam-5960 415 8 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 415 9 lim(𝐼 lim(𝐼 PROPN ejpam-5960 415 10 ) ) PUNCT ejpam-5960 415 11 , , PUNCT ejpam-5960 415 12 then then ADV ejpam-5960 415 13 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 415 14 lim(𝐼 lim(𝐼 PROPN ejpam-5960 415 15 ) ) PUNCT ejpam-5960 415 16 β‰° β‰° PROPN ejpam-5960 415 17 πœ†. πœ†. VERB ejpam-5960 415 18 therefore therefore ADV ejpam-5960 415 19 there there PRON ejpam-5960 415 20 exists exist VERB ejpam-5960 415 21 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 415 22 ∈ ∈ PROPN ejpam-5960 415 23 𝑀 𝑀 PROPN ejpam-5960 415 24 ( ( PUNCT ejpam-5960 415 25 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 415 26 ) ) PUNCT ejpam-5960 415 27 such such ADJ ejpam-5960 416 1 that that SCONJ ejpam-5960 416 2 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 416 3 ∈ ∈ NOUN ejpam-5960 416 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 416 5 lim(𝐼 lim(𝐼 PROPN ejpam-5960 416 6 ) ) PUNCT ejpam-5960 416 7 and and CCONJ ejpam-5960 416 8 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 416 9 βˆ‰ βˆ‰ PROPN ejpam-5960 416 10 πœ†. πœ†. PROPN ejpam-5960 416 11 then then ADV ejpam-5960 416 12 πœ† πœ† PROPN ejpam-5960 416 13 ∈ ∈ PROPN ejpam-5960 416 14 𝑁𝛼𝐡𝑅𝑦𝛾 𝑁𝛼𝐡𝑅𝑦𝛾 PROPN ejpam-5960 416 15 and and CCONJ ejpam-5960 416 16 so so ADV ejpam-5960 416 17 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 416 18 βŠ† βŠ† NUM ejpam-5960 416 19 𝑁𝛼𝐡𝑅𝑦𝛾 𝑁𝛼𝐡𝑅𝑦𝛾 X ejpam-5960 416 20 βŠ† βŠ† NUM ejpam-5960 416 21 𝐼 𝐼 PROPN ejpam-5960 416 22 hence hence ADV ejpam-5960 416 23 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 416 24 βŠ† βŠ† X ejpam-5960 416 25 𝐼. 𝐼. NOUN ejpam-5960 416 26 thus thus ADV ejpam-5960 416 27 𝐼 𝐼 PROPN ejpam-5960 416 28 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 416 29 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 416 30 conversely conversely ADV ejpam-5960 416 31 , , PUNCT ejpam-5960 416 32 let let VERB ejpam-5960 416 33 𝐼 𝐼 PRON ejpam-5960 416 34 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ VERB ejpam-5960 416 35 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 416 36 , , PUNCT ejpam-5960 416 37 then then ADV ejpam-5960 416 38 by by ADP ejpam-5960 416 39 definition definition NOUN ejpam-5960 416 40 3.23 3.23 NUM ejpam-5960 416 41 ( ( PUNCT ejpam-5960 416 42 ii ii NOUN ejpam-5960 416 43 ) ) PUNCT ejpam-5960 416 44 we we PRON ejpam-5960 416 45 have have VERB ejpam-5960 416 46 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 416 47 ∈ ∈ NOUN ejpam-5960 416 48 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 416 49 lim(𝐼 lim(𝐼 PROPN ejpam-5960 416 50 ) ) PUNCT ejpam-5960 416 51 . . PUNCT ejpam-5960 417 1 ( ( PUNCT ejpam-5960 417 2 ii ii NOUN ejpam-5960 417 3 ) ) PUNCT ejpam-5960 417 4 let let VERB ejpam-5960 417 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 417 6 ∈ ∈ PROPN ejpam-5960 417 7 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 ADV ejpam-5960 417 8 ) ) PUNCT ejpam-5960 417 9 and and CCONJ ejpam-5960 417 10 let let VERB ejpam-5960 417 11 πœ† πœ† DET ejpam-5960 417 12 ∈ ∈ PROPN ejpam-5960 417 13 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 417 14 . . PUNCT ejpam-5960 418 1 since since SCONJ ejpam-5960 418 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 418 3 βˆ‰ βˆ‰ PROPN ejpam-5960 418 4 πœ† πœ† PROPN ejpam-5960 418 5 and and CCONJ ejpam-5960 418 6 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 418 7 ∈ ∈ PROPN ejpam-5960 418 8 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 NOUN ejpam-5960 418 9 ) ) PUNCT ejpam-5960 418 10 , , PUNCT ejpam-5960 418 11 therefore therefore ADV ejpam-5960 418 12 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 ADJ ejpam-5960 418 13 ) ) PUNCT ejpam-5960 418 14 β‰° β‰° PROPN ejpam-5960 418 15 πœ† πœ† PRON ejpam-5960 419 1 and and CCONJ ejpam-5960 419 2 so so ADV ejpam-5960 419 3 there there PRON ejpam-5960 419 4 exists exist VERB ejpam-5960 419 5 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 419 6 ∈ ∈ PROPN ejpam-5960 419 7 𝑀 𝑀 PROPN ejpam-5960 419 8 ( ( PUNCT ejpam-5960 419 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 419 10 ) ) PUNCT ejpam-5960 419 11 such such ADJ ejpam-5960 420 1 that that SCONJ ejpam-5960 420 2 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 420 3 ∈ ∈ NOUN ejpam-5960 420 4 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 NOUN ejpam-5960 420 5 ) ) PUNCT ejpam-5960 420 6 and and CCONJ ejpam-5960 420 7 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 420 8 βˆ‰ βˆ‰ PROPN ejpam-5960 420 9 πœ† πœ† ADP ejpam-5960 420 10 hence hence ADV ejpam-5960 420 11 πœ† πœ† ADP ejpam-5960 420 12 ∈ ∈ ADJ ejpam-5960 420 13 𝑁𝛼𝐡𝑅𝑦𝛾 𝑁𝛼𝐡𝑅𝑦𝛾 PROPN ejpam-5960 420 14 and and CCONJ ejpam-5960 420 15 so so ADV ejpam-5960 420 16 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 420 17 βŠ† βŠ† NUM ejpam-5960 420 18 𝑁𝛼𝐡𝑅𝑦𝛾 𝑁𝛼𝐡𝑅𝑦𝛾 PROPN ejpam-5960 420 19 and and CCONJ ejpam-5960 420 20 πœ‡ πœ‡ ADP ejpam-5960 420 21 ∨ ∨ NUM ejpam-5960 420 22 πœ† πœ† DET ejpam-5960 420 23 β‰  β‰  PROPN ejpam-5960 420 24 1𝑋 1𝑋 NOUN ejpam-5960 420 25 for for ADP ejpam-5960 420 26 each each DET ejpam-5960 420 27 πœ‡ πœ‡ ADP ejpam-5960 420 28 ∈ ∈ NOUN ejpam-5960 420 29 𝐼 𝐼 ADP ejpam-5960 420 30 hence hence ADV ejpam-5960 420 31 𝐼 𝐼 ADP ejpam-5960 420 32 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 420 33 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 420 34 conversely conversely ADV ejpam-5960 420 35 , , PUNCT ejpam-5960 420 36 let let VERB ejpam-5960 420 37 𝐼 𝐼 PROPN ejpam-5960 420 38 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 420 39 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 420 40 , , PUNCT ejpam-5960 420 41 then then ADV ejpam-5960 420 42 by by ADP ejpam-5960 420 43 definition definition NOUN ejpam-5960 420 44 3.24 3.24 NUM ejpam-5960 420 45 ( ( PUNCT ejpam-5960 420 46 ii ii NOUN ejpam-5960 420 47 ) ) PUNCT ejpam-5960 420 48 we we PRON ejpam-5960 420 49 have have VERB ejpam-5960 420 50 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 420 51 ∈ ∈ PROPN ejpam-5960 420 52 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 𝑁𝛼𝐡.π‘Žπ‘‘β„Ž(𝐼 NOUN ejpam-5960 420 53 ) ) PUNCT ejpam-5960 420 54 . . PUNCT ejpam-5960 421 1 ( ( PUNCT ejpam-5960 421 2 iii iii X ejpam-5960 421 3 ) ) PUNCT ejpam-5960 421 4 let let VERB ejpam-5960 421 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 421 6 ∈ ∈ PROPN ejpam-5960 421 7 lim(𝐼 lim(𝐼 PROPN ejpam-5960 421 8 ) ) PUNCT ejpam-5960 421 9 and and CCONJ ejpam-5960 421 10 let let VERB ejpam-5960 421 11 πœ‚ πœ‚ X ejpam-5960 421 12 ∈ ∈ PROPN ejpam-5960 421 13 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 421 14 . . PUNCT ejpam-5960 422 1 since since SCONJ ejpam-5960 422 2 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 422 3 βŠ† βŠ† NUM ejpam-5960 422 4 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 422 5 , , PUNCT ejpam-5960 422 6 then then ADV ejpam-5960 422 7 πœ‚ πœ‚ PROPN ejpam-5960 422 8 ∈ ∈ PROPN ejpam-5960 422 9 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 422 10 . . PUNCT ejpam-5960 423 1 and and CCONJ ejpam-5960 423 2 since since SCONJ ejpam-5960 423 3 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 423 4 ∈ ∈ PROPN ejpam-5960 423 5 lim(𝐼 lim(𝐼 PROPN ejpam-5960 423 6 ) ) PUNCT ejpam-5960 423 7 , , PUNCT ejpam-5960 423 8 then then ADV ejpam-5960 423 9 𝑅π‘₯𝛼 𝑅π‘₯𝛼 VERB ejpam-5960 423 10 βŠ† βŠ† NUM ejpam-5960 423 11 𝐼 𝐼 PROPN ejpam-5960 423 12 so so ADV ejpam-5960 423 13 for for ADP ejpam-5960 423 14 each each DET ejpam-5960 423 15 πœ‚ πœ‚ PROPN ejpam-5960 423 16 ∈ ∈ PROPN ejpam-5960 423 17 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 423 18 , , PUNCT ejpam-5960 423 19 πœ‚ πœ‚ PROPN ejpam-5960 423 20 ∈ ∈ PROPN ejpam-5960 423 21 𝐼 𝐼 PROPN ejpam-5960 423 22 and and CCONJ ejpam-5960 423 23 since since SCONJ ejpam-5960 423 24 πœ‚ πœ‚ PROPN ejpam-5960 423 25 ∈ ∈ PROPN ejpam-5960 423 26 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 423 27 , , PUNCT ejpam-5960 423 28 so so ADV ejpam-5960 423 29 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 423 30 βŠ† βŠ† X ejpam-5960 423 31 𝐼. 𝐼. PROPN ejpam-5960 423 32 hence hence ADV ejpam-5960 423 33 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 423 34 ∈ ∈ PROPN ejpam-5960 423 35 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 423 36 lim(𝐼 lim(𝐼 PROPN ejpam-5960 423 37 ) ) PUNCT ejpam-5960 423 38 . . PUNCT ejpam-5960 424 1 so so ADV ejpam-5960 424 2 lim(𝐼 lim(𝐼 PROPN ejpam-5960 424 3 ) ) PUNCT ejpam-5960 424 4 ≀ ≀ NOUN ejpam-5960 424 5 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 424 6 lim(𝐼 lim(𝐼 PROPN ejpam-5960 424 7 ) ) PUNCT ejpam-5960 424 8 . . PUNCT ejpam-5960 425 1 let let VERB ejpam-5960 425 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 425 3 ∈ ∈ VERB ejpam-5960 425 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 425 5 lim(𝐼 lim(𝐼 PROPN ejpam-5960 425 6 ) ) PUNCT ejpam-5960 425 7 and and CCONJ ejpam-5960 425 8 let let VERB ejpam-5960 425 9 πœ‚ πœ‚ NOUN ejpam-5960 425 10 ∈ ∈ NOUN ejpam-5960 425 11 𝛼𝐡𝑅π‘₯𝛼 𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 425 12 . . PUNCT ejpam-5960 426 1 since since SCONJ ejpam-5960 426 2 𝛼𝐡𝑅π‘₯𝛼 𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 426 3 βŠ† βŠ† NUM ejpam-5960 426 4 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 426 5 , , PUNCT ejpam-5960 426 6 then then ADV ejpam-5960 426 7 πœ‚ πœ‚ PROPN ejpam-5960 426 8 ∈ ∈ PROPN ejpam-5960 426 9 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 426 10 . . PUNCT ejpam-5960 427 1 and and CCONJ ejpam-5960 427 2 since since SCONJ ejpam-5960 427 3 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 427 4 ∈ ∈ PROPN ejpam-5960 427 5 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 427 6 lim(𝐼 lim(𝐼 PROPN ejpam-5960 427 7 ) ) PUNCT ejpam-5960 427 8 , , PUNCT ejpam-5960 427 9 then then ADV ejpam-5960 427 10 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 427 11 βŠ† βŠ† NUM ejpam-5960 427 12 𝐼 𝐼 ADP ejpam-5960 427 13 so so ADV ejpam-5960 427 14 for for ADP ejpam-5960 427 15 each each DET ejpam-5960 427 16 πœ‚ πœ‚ PROPN ejpam-5960 427 17 ∈ ∈ PROPN ejpam-5960 427 18 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 427 19 , , PUNCT ejpam-5960 427 20 πœ‚ πœ‚ PROPN ejpam-5960 427 21 ∈ ∈ PROPN ejpam-5960 427 22 𝐼 𝐼 PROPN ejpam-5960 427 23 and and CCONJ ejpam-5960 427 24 since since SCONJ ejpam-5960 427 25 πœ‚ πœ‚ PROPN ejpam-5960 427 26 ∈ ∈ NOUN ejpam-5960 427 27 𝛼𝐡𝑅π‘₯𝛼 𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 427 28 , , PUNCT ejpam-5960 427 29 πœ‚ πœ‚ NOUN ejpam-5960 427 30 ∈ ∈ PROPN ejpam-5960 427 31 𝐼 𝐼 PROPN ejpam-5960 427 32 and and CCONJ ejpam-5960 427 33 since since SCONJ ejpam-5960 427 34 πœ‚ πœ‚ PROPN ejpam-5960 427 35 ∈ ∈ NOUN ejpam-5960 427 36 𝛼𝐡𝑅π‘₯𝛼 𝛼𝐡𝑅π‘₯𝛼 ADJ ejpam-5960 427 37 , , PUNCT ejpam-5960 427 38 so so CCONJ ejpam-5960 427 39 𝛼𝐡𝑅π‘₯𝛼 𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 427 40 βŠ† βŠ† NUM ejpam-5960 427 41 𝐼. 𝐼. NOUN ejpam-5960 427 42 hence hence ADV ejpam-5960 427 43 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 427 44 ∈ ∈ PROPN ejpam-5960 427 45 𝛼𝐡. 𝛼𝐡. NOUN ejpam-5960 427 46 lim(𝐼 lim(𝐼 PROPN ejpam-5960 427 47 ) ) PUNCT ejpam-5960 427 48 . . PUNCT ejpam-5960 428 1 so so ADV ejpam-5960 428 2 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 428 3 lim(𝐼 lim(𝐼 NUM ejpam-5960 428 4 ) ) PUNCT ejpam-5960 428 5 ≀ ≀ NUM ejpam-5960 428 6 𝛼𝐡. 𝛼𝐡. NOUN ejpam-5960 428 7 lim(𝐼 lim(𝐼 PROPN ejpam-5960 428 8 ) ) PUNCT ejpam-5960 428 9 . . PUNCT ejpam-5960 429 1 ( ( PUNCT ejpam-5960 429 2 iv iv X ejpam-5960 429 3 ) ) PUNCT ejpam-5960 429 4 let let VERB ejpam-5960 429 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 429 6 ∈ ∈ PROPN ejpam-5960 429 7 adh(𝐼 adh(𝐼 PROPN ejpam-5960 429 8 ) ) PUNCT ejpam-5960 429 9 and and CCONJ ejpam-5960 429 10 let let VERB ejpam-5960 429 11 πœ‚ πœ‚ X ejpam-5960 429 12 ∈ ∈ PROPN ejpam-5960 429 13 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 429 14 . . PUNCT ejpam-5960 430 1 since since SCONJ ejpam-5960 430 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 430 3 ∈ ∈ PROPN ejpam-5960 430 4 adh(𝐼 adh(𝐼 PROPN ejpam-5960 430 5 ) ) PUNCT ejpam-5960 430 6 , , PUNCT ejpam-5960 430 7 so so SCONJ ejpam-5960 430 8 for for SCONJ ejpam-5960 430 9 each each DET ejpam-5960 430 10 πœ† πœ† DET ejpam-5960 430 11 ∈ ∈ PROPN ejpam-5960 430 12 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 430 13 , , PUNCT ejpam-5960 430 14 πœ† πœ† DET ejpam-5960 430 15 ∈ ∈ NOUN ejpam-5960 430 16 𝐼 𝐼 PROPN ejpam-5960 430 17 and and CCONJ ejpam-5960 430 18 since since SCONJ ejpam-5960 430 19 πœ‚ πœ‚ PROPN ejpam-5960 430 20 ∈ ∈ PROPN ejpam-5960 430 21 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 430 22 so so SCONJ ejpam-5960 430 23 πœ‚ πœ‚ PROPN ejpam-5960 430 24 ∈ ∈ PROPN ejpam-5960 430 25 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 430 26 . . PUNCT ejpam-5960 431 1 hence hence ADV ejpam-5960 431 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 431 3 ∈ ∈ PROPN ejpam-5960 431 4 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 PRON ejpam-5960 431 5 ) ) PUNCT ejpam-5960 431 6 . . PUNCT ejpam-5960 432 1 so so ADV ejpam-5960 432 2 adh(𝐼 adh(𝐼 PROPN ejpam-5960 432 3 ) ) PUNCT ejpam-5960 432 4 ≀ ≀ NUM ejpam-5960 432 5 𝛼𝐡.adh(𝐼 𝛼𝐡.adh(𝐼 NUM ejpam-5960 432 6 ) ) PUNCT ejpam-5960 432 7 . . PUNCT ejpam-5960 433 1 similarly similarly ADV ejpam-5960 433 2 , , PUNCT ejpam-5960 433 3 one one PRON ejpam-5960 433 4 can can AUX ejpam-5960 433 5 easily easily ADV ejpam-5960 433 6 verify verify VERB ejpam-5960 433 7 that that SCONJ ejpam-5960 433 8 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 NOUN ejpam-5960 433 9 ) ) PUNCT ejpam-5960 433 10 ≀ ≀ NUM ejpam-5960 433 11 𝛼𝐡.adh(𝐼 𝛼𝐡.adh(𝐼 NUM ejpam-5960 433 12 ) ) PUNCT ejpam-5960 433 13 . . PUNCT ejpam-5960 434 1 ( ( PUNCT ejpam-5960 434 2 v v X ejpam-5960 434 3 ) ) PUNCT ejpam-5960 434 4 let let VERB ejpam-5960 434 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 434 6 ∈ ∈ PROPN ejpam-5960 434 7 𝑀 𝑀 PROPN ejpam-5960 434 8 ( ( PUNCT ejpam-5960 434 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 434 10 ) ) PUNCT ejpam-5960 434 11 such such ADJ ejpam-5960 434 12 that that SCONJ ejpam-5960 434 13 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 434 14 ∈ ∈ PROPN ejpam-5960 434 15 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 434 16 ( ( PUNCT ejpam-5960 434 17 πœ‡ πœ‡ NOUN ejpam-5960 434 18 ) ) PUNCT ejpam-5960 434 19 . . PUNCT ejpam-5960 435 1 the the DET ejpam-5960 435 2 family family NOUN ejpam-5960 435 3 𝐼 𝐼 PROPN ejpam-5960 435 4 = = PUNCT ejpam-5960 435 5 { { PUNCT ejpam-5960 435 6 𝜌 𝜌 X ejpam-5960 435 7 ∈ ∈ NOUN ejpam-5960 435 8 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 435 9 : : PUNCT ejpam-5960 435 10 βˆƒπœ† βˆƒπœ† PROPN ejpam-5960 435 11 ∈ ∈ PROPN ejpam-5960 435 12 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 435 13 βˆ‹ βˆ‹ NOUN ejpam-5960 435 14 𝜌 𝜌 ADP ejpam-5960 435 15 ≀ ≀ NUM ejpam-5960 435 16 πœ† πœ† X ejpam-5960 435 17 } } PUNCT ejpam-5960 435 18 is be AUX ejpam-5960 435 19 an an DET ejpam-5960 435 20 ideal ideal NOUN ejpam-5960 435 21 in in ADP ejpam-5960 435 22 𝐿𝑋. 𝐿𝑋. NOUN ejpam-5960 435 23 now now ADV ejpam-5960 435 24 we we PRON ejpam-5960 435 25 show show VERB ejpam-5960 435 26 that that SCONJ ejpam-5960 435 27 πœ‡ πœ‡ ADP ejpam-5960 435 28 βˆ‰ βˆ‰ PROPN ejpam-5960 435 29 𝐼. 𝐼. PROPN ejpam-5960 435 30 since since SCONJ ejpam-5960 435 31 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 435 32 ∈ ∈ PROPN ejpam-5960 435 33 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 435 34 ( ( PUNCT ejpam-5960 435 35 πœ‡ πœ‡ NOUN ejpam-5960 435 36 ) ) PUNCT ejpam-5960 435 37 , , PUNCT ejpam-5960 435 38 then then ADV ejpam-5960 435 39 for for ADP ejpam-5960 435 40 each each DET ejpam-5960 435 41 πœ† πœ† PRON ejpam-5960 435 42 ∈ ∈ PROPN ejpam-5960 435 43 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 435 44 , , PUNCT ejpam-5960 435 45 πœ‡ πœ‡ ADP ejpam-5960 435 46 β‰° β‰° PROPN ejpam-5960 435 47 πœ†. πœ†. VERB ejpam-5960 435 48 so so ADV ejpam-5960 435 49 by by ADP ejpam-5960 435 50 definition definition NOUN ejpam-5960 435 51 of of ADP ejpam-5960 435 52 𝐼 𝐼 PROPN ejpam-5960 435 53 we we PRON ejpam-5960 435 54 have have AUX ejpam-5960 435 55 πœ‡ πœ‡ ADP ejpam-5960 435 56 βˆ‰ βˆ‰ PROPN ejpam-5960 435 57 𝐼. 𝐼. PROPN ejpam-5960 435 58 finally finally ADV ejpam-5960 435 59 , , PUNCT ejpam-5960 435 60 we we PRON ejpam-5960 435 61 show show VERB ejpam-5960 435 62 that that SCONJ ejpam-5960 435 63 𝐼 𝐼 PROPN ejpam-5960 435 64 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 435 65 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 435 66 let let VERB ejpam-5960 435 67 πœ† πœ† DET ejpam-5960 435 68 ∈ ∈ PROPN ejpam-5960 435 69 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 435 70 , , PUNCT ejpam-5960 435 71 since since SCONJ ejpam-5960 435 72 πœ† πœ† DET ejpam-5960 435 73 ≀ ≀ NUM ejpam-5960 435 74 πœ† πœ† ADP ejpam-5960 435 75 , , PUNCT ejpam-5960 435 76 then then ADV ejpam-5960 435 77 πœ† πœ† PROPN ejpam-5960 435 78 ∈ ∈ PROPN ejpam-5960 435 79 𝐼. 𝐼. PROPN ejpam-5960 436 1 so so SCONJ ejpam-5960 436 2 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 436 3 βŠ† βŠ† X ejpam-5960 436 4 𝐼. 𝐼. PROPN ejpam-5960 436 5 thus thus ADV ejpam-5960 436 6 𝐼 𝐼 PROPN ejpam-5960 436 7 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 436 8 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 436 9 conversely conversely ADV ejpam-5960 436 10 , , PUNCT ejpam-5960 436 11 let let VERB ejpam-5960 436 12 𝐼 𝐼 PRON ejpam-5960 436 13 be be AUX ejpam-5960 436 14 an an DET ejpam-5960 436 15 ideal ideal NOUN ejpam-5960 436 16 in in ADP ejpam-5960 436 17 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 436 18 such such ADJ ejpam-5960 436 19 that that SCONJ ejpam-5960 436 20 𝐼 𝐼 PROPN ejpam-5960 436 21 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 436 22 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 436 23 and and CCONJ ejpam-5960 436 24 πœ‡ πœ‡ ADP ejpam-5960 436 25 βˆ‰ βˆ‰ PROPN ejpam-5960 436 26 𝐼. 𝐼. PROPN ejpam-5960 436 27 then then ADV ejpam-5960 436 28 for for ADP ejpam-5960 436 29 each each DET ejpam-5960 436 30 πœ† πœ† PRON ejpam-5960 436 31 ∈ ∈ PROPN ejpam-5960 436 32 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 436 33 , , PUNCT ejpam-5960 436 34 πœ† πœ† PROPN ejpam-5960 436 35 ∈ ∈ PROPN ejpam-5960 436 36 𝐼. 𝐼. PROPN ejpam-5960 436 37 since since SCONJ ejpam-5960 436 38 πœ† πœ† DET ejpam-5960 436 39 ∈ ∈ PROPN ejpam-5960 436 40 𝐼 𝐼 PROPN ejpam-5960 436 41 , , PUNCT ejpam-5960 436 42 πœ‡ πœ‡ ADP ejpam-5960 436 43 βˆ‰ βˆ‰ PROPN ejpam-5960 436 44 𝐼 𝐼 PROPN ejpam-5960 436 45 , , PUNCT ejpam-5960 436 46 πœ‡ πœ‡ ADP ejpam-5960 436 47 β‰° β‰° PROPN ejpam-5960 436 48 πœ† πœ† PRON ejpam-5960 436 49 and and CCONJ ejpam-5960 436 50 so so ADV ejpam-5960 436 51 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 436 52 ∈ ∈ PROPN ejpam-5960 436 53 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 436 54 ( ( PUNCT ejpam-5960 436 55 πœ‡ πœ‡ NOUN ejpam-5960 436 56 ) ) PUNCT ejpam-5960 436 57 . . PUNCT ejpam-5960 437 1 ( ( PUNCT ejpam-5960 437 2 vi vi X ejpam-5960 437 3 ) ) PUNCT ejpam-5960 437 4 let let VERB ejpam-5960 437 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 437 6 ∈ ∈ NOUN ejpam-5960 437 7 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 437 8 ( ( PUNCT ejpam-5960 437 9 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 437 10 lim(𝐼 lim(𝐼 PROPN ejpam-5960 437 11 ) ) PUNCT ejpam-5960 437 12 ) ) PUNCT ejpam-5960 437 13 , , PUNCT ejpam-5960 437 14 then then ADV ejpam-5960 437 15 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 437 16 lim(𝐼 lim(𝐼 PROPN ejpam-5960 437 17 ) ) PUNCT ejpam-5960 437 18 β‰° β‰° PROPN ejpam-5960 437 19 πœ† πœ† PRON ejpam-5960 437 20 for for ADP ejpam-5960 437 21 each each DET ejpam-5960 437 22 πœ† πœ† PRON ejpam-5960 437 23 ∈ ∈ PROPN ejpam-5960 437 24 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 437 25 and and CCONJ ejpam-5960 437 26 then then ADV ejpam-5960 437 27 there there PRON ejpam-5960 437 28 exists exist VERB ejpam-5960 437 29 𝑦𝛾 𝑦𝛾 NOUN ejpam-5960 437 30 ∈ ∈ PROPN ejpam-5960 437 31 𝑀 𝑀 PROPN ejpam-5960 437 32 ( ( PUNCT ejpam-5960 437 33 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 437 34 ) ) PUNCT ejpam-5960 437 35 such such ADJ ejpam-5960 438 1 that that SCONJ ejpam-5960 438 2 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 438 3 ∈ ∈ NOUN ejpam-5960 438 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 438 5 lim(𝐼 lim(𝐼 PROPN ejpam-5960 438 6 ) ) PUNCT ejpam-5960 438 7 and and CCONJ ejpam-5960 438 8 𝑦𝛾 𝑦𝛾 VERB ejpam-5960 438 9 βˆ‰ βˆ‰ PROPN ejpam-5960 438 10 πœ†. πœ†. PROPN ejpam-5960 438 11 since since SCONJ ejpam-5960 438 12 πœ† πœ† DET ejpam-5960 438 13 ∈ ∈ PROPN ejpam-5960 438 14 𝑁𝛼𝐡𝑅𝑦𝛾 𝑁𝛼𝐡𝑅𝑦𝛾 PROPN ejpam-5960 438 15 and and CCONJ ejpam-5960 438 16 𝐼 𝐼 ADP ejpam-5960 438 17 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 438 18 π‘₯𝛼 π‘₯𝛼 INTJ ejpam-5960 438 19 then then ADV ejpam-5960 438 20 πœ‚ πœ‚ PROPN ejpam-5960 438 21 ∈ ∈ PROPN ejpam-5960 438 22 𝐼 𝐼 PROPN ejpam-5960 438 23 for for ADP ejpam-5960 438 24 each each DET ejpam-5960 438 25 πœ‚ πœ‚ PROPN ejpam-5960 438 26 ∈ ∈ PROPN ejpam-5960 438 27 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 438 28 . . PUNCT ejpam-5960 439 1 since since SCONJ ejpam-5960 439 2 𝑦𝛾 𝑦𝛾 PROPN ejpam-5960 439 3 βˆ‰ βˆ‰ PROPN ejpam-5960 439 4 πœ† πœ† ADP ejpam-5960 439 5 then then ADV ejpam-5960 439 6 πœ† πœ† PROPN ejpam-5960 439 7 ∈ ∈ PROPN ejpam-5960 439 8 𝐼. 𝐼. PROPN ejpam-5960 439 9 but but CCONJ ejpam-5960 439 10 πœ† πœ† DET ejpam-5960 439 11 ∈ ∈ PROPN ejpam-5960 439 12 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 439 13 and and CCONJ ejpam-5960 439 14 so so ADV ejpam-5960 439 15 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 439 16 ∈ ∈ PROPN ejpam-5960 439 17 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 439 18 lim(𝐼 lim(𝐼 PROPN ejpam-5960 439 19 ) ) PUNCT ejpam-5960 439 20 . . PUNCT ejpam-5960 440 1 thus thus ADV ejpam-5960 440 2 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 VERB ejpam-5960 440 3 ( ( PUNCT ejpam-5960 440 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 440 5 lim(𝐼 lim(𝐼 PROPN ejpam-5960 440 6 ) ) PUNCT ejpam-5960 440 7 ) ) PUNCT ejpam-5960 440 8 ≀ ≀ PUNCT ejpam-5960 440 9 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 440 10 lim(𝐼 lim(𝐼 PROPN ejpam-5960 440 11 ) ) PUNCT ejpam-5960 440 12 and and CCONJ ejpam-5960 440 13 so so ADV ejpam-5960 440 14 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 440 15 lim(𝐼 lim(𝐼 PROPN ejpam-5960 440 16 ) ) PUNCT ejpam-5960 440 17 is be AUX ejpam-5960 440 18 a a DET ejpam-5960 440 19 𝑁𝛼𝐡–closed 𝑁𝛼𝐡–closed PROPN ejpam-5960 440 20 set set NOUN ejpam-5960 440 21 . . PUNCT ejpam-5960 441 1 similarly similarly ADV ejpam-5960 441 2 , , PUNCT ejpam-5960 441 3 one one PRON ejpam-5960 441 4 can can AUX ejpam-5960 441 5 easily easily ADV ejpam-5960 441 6 verify verify VERB ejpam-5960 441 7 that that DET ejpam-5960 441 8 𝑁𝛼𝐡.𝑐𝑙 𝑁𝛼𝐡.𝑐𝑙 NOUN ejpam-5960 441 9 ( ( PUNCT ejpam-5960 441 10 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 NOUN ejpam-5960 441 11 ) ) PUNCT ejpam-5960 441 12 ) ) PUNCT ejpam-5960 441 13 ≀ ≀ NUM ejpam-5960 441 14 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 NUM ejpam-5960 441 15 ) ) PUNCT ejpam-5960 441 16 . . PUNCT ejpam-5960 442 1 theorem theorem VERB ejpam-5960 442 2 3.26 3.26 NUM ejpam-5960 442 3 . . PUNCT ejpam-5960 443 1 suppose suppose VERB ejpam-5960 443 2 that that SCONJ ejpam-5960 443 3 𝑆 𝑆 PROPN ejpam-5960 443 4 is be AUX ejpam-5960 443 5 a a DET ejpam-5960 443 6 molecular molecular ADJ ejpam-5960 443 7 net net NOUN ejpam-5960 443 8 in in ADP ejpam-5960 443 9 𝐿–ts 𝐿–ts NOUN ejpam-5960 443 10 ( ( PUNCT ejpam-5960 443 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 443 12 , , PUNCT ejpam-5960 443 13 𝜏 𝜏 NOUN ejpam-5960 443 14 ) ) PUNCT ejpam-5960 443 15 , , PUNCT ejpam-5960 443 16 πœ‡ πœ‡ ADP ejpam-5960 443 17 ∈ ∈ X ejpam-5960 443 18 𝐿𝑋. 𝐿𝑋. PUNCT ejpam-5960 443 19 then then ADV ejpam-5960 443 20 : : PUNCT ejpam-5960 443 21 ( ( PUNCT ejpam-5960 443 22 i i NOUN ejpam-5960 443 23 ) ) PUNCT ejpam-5960 443 24 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 443 25 lim(𝑆 lim(𝑆 PROPN ejpam-5960 443 26 ) ) PUNCT ejpam-5960 444 1 = = PUNCT ejpam-5960 444 2 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 444 3 lim(𝐼 lim(𝐼 PROPN ejpam-5960 444 4 ( ( PUNCT ejpam-5960 444 5 𝑆 𝑆 PROPN ejpam-5960 444 6 ) ) PUNCT ejpam-5960 444 7 ) ) PUNCT ejpam-5960 444 8 . . PUNCT ejpam-5960 445 1 ( ( PUNCT ejpam-5960 445 2 ii ii NOUN ejpam-5960 445 3 ) ) PUNCT ejpam-5960 445 4 𝑁𝛼𝐡.adh(𝑆 𝑁𝛼𝐡.adh(𝑆 NOUN ejpam-5960 445 5 ) ) PUNCT ejpam-5960 445 6 ≀ ≀ NOUN ejpam-5960 445 7 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 PRON ejpam-5960 445 8 ( ( PUNCT ejpam-5960 445 9 𝑆 𝑆 PROPN ejpam-5960 445 10 ) ) PUNCT ejpam-5960 445 11 ) ) PUNCT ejpam-5960 445 12 . . PUNCT ejpam-5960 446 1 proof proof NOUN ejpam-5960 446 2 . . PUNCT ejpam-5960 447 1 ( ( PUNCT ejpam-5960 447 2 i i NOUN ejpam-5960 447 3 ) ) PUNCT ejpam-5960 447 4 let let VERB ejpam-5960 447 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 447 6 ∈ ∈ PROPN ejpam-5960 447 7 𝑀 𝑀 PROPN ejpam-5960 447 8 ( ( PUNCT ejpam-5960 447 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 447 10 ) ) PUNCT ejpam-5960 447 11 such such ADJ ejpam-5960 447 12 that that SCONJ ejpam-5960 447 13 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 447 14 ∈ ∈ PROPN ejpam-5960 447 15 𝑁𝛼𝐡. 𝑁𝛼𝐡. PROPN ejpam-5960 447 16 lim(𝑆 lim(𝑆 PROPN ejpam-5960 447 17 ) ) PUNCT ejpam-5960 447 18 , , PUNCT ejpam-5960 447 19 by by ADP ejpam-5960 447 20 theorem theorem NOUN ejpam-5960 447 21 3.25 3.25 NUM ejpam-5960 447 22 ( ( PUNCT ejpam-5960 447 23 i i NOUN ejpam-5960 447 24 ) ) PUNCT ejpam-5960 447 25 , , PUNCT ejpam-5960 447 26 we we PRON ejpam-5960 447 27 have have VERB ejpam-5960 447 28 𝑆 𝑆 PROPN ejpam-5960 447 29 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 447 30 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 447 31 then then ADV ejpam-5960 447 32 for for ADP ejpam-5960 447 33 each each DET ejpam-5960 447 34 πœ† πœ† PRON ejpam-5960 447 35 ∈ ∈ PROPN ejpam-5960 447 36 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 447 37 there there PRON ejpam-5960 447 38 is be VERB ejpam-5960 447 39 𝑛 𝑛 DET ejpam-5960 447 40 ∈ ∈ PROPN ejpam-5960 447 41 𝐷 𝐷 NOUN ejpam-5960 447 42 such such ADJ ejpam-5960 447 43 that that PRON ejpam-5960 447 44 for for ADP ejpam-5960 447 45 each each DET ejpam-5960 447 46 π‘š π‘š PROPN ejpam-5960 447 47 ∈ ∈ PROPN ejpam-5960 447 48 𝐷 𝐷 NOUN ejpam-5960 447 49 and and CCONJ ejpam-5960 447 50 π‘š π‘š ADP ejpam-5960 447 51 β‰₯ β‰₯ NOUN ejpam-5960 447 52 𝑛 𝑛 ADP ejpam-5960 447 53 we we PRON ejpam-5960 447 54 have have VERB ejpam-5960 447 55 𝑆(π‘š 𝑆(π‘š NOUN ejpam-5960 447 56 ) ) PUNCT ejpam-5960 447 57 βˆ‰ βˆ‰ PROPN ejpam-5960 447 58 πœ† πœ† NOUN ejpam-5960 447 59 , , PUNCT ejpam-5960 447 60 and and CCONJ ejpam-5960 447 61 hence hence ADV ejpam-5960 447 62 by by ADP ejpam-5960 447 63 the the DET ejpam-5960 447 64 definition definition NOUN ejpam-5960 447 65 of of ADP ejpam-5960 447 66 𝐼 𝐼 PROPN ejpam-5960 447 67 ( ( PUNCT ejpam-5960 447 68 𝑆 𝑆 PROPN ejpam-5960 447 69 ) ) PUNCT ejpam-5960 447 70 we we PRON ejpam-5960 447 71 have have VERB ejpam-5960 447 72 n. n. PROPN ejpam-5960 447 73 a. a. NOUN ejpam-5960 447 74 alsaedi alsaedi PROPN ejpam-5960 447 75 / / SYM ejpam-5960 447 76 eur eur PROPN ejpam-5960 447 77 . . PUNCT ejpam-5960 448 1 j. j. PROPN ejpam-5960 448 2 pure pure PROPN ejpam-5960 448 3 appl appl PROPN ejpam-5960 448 4 . . PROPN ejpam-5960 448 5 math math PROPN ejpam-5960 448 6 , , PUNCT ejpam-5960 448 7 18 18 NUM ejpam-5960 448 8 ( ( PUNCT ejpam-5960 448 9 4 4 NUM ejpam-5960 448 10 ) ) PUNCT ejpam-5960 448 11 ( ( PUNCT ejpam-5960 448 12 2025 2025 NUM ejpam-5960 448 13 ) ) PUNCT ejpam-5960 448 14 , , PUNCT ejpam-5960 448 15 5960 5960 NUM ejpam-5960 448 16 15 15 NUM ejpam-5960 448 17 of of ADP ejpam-5960 448 18 22 22 NUM ejpam-5960 448 19 πœ† πœ† PRON ejpam-5960 448 20 ∈ ∈ PROPN ejpam-5960 448 21 𝐼 𝐼 PROPN ejpam-5960 448 22 ( ( PUNCT ejpam-5960 448 23 𝑆 𝑆 PROPN ejpam-5960 448 24 ) ) PUNCT ejpam-5960 448 25 for for ADP ejpam-5960 448 26 each each DET ejpam-5960 448 27 πœ† πœ† PRON ejpam-5960 448 28 ∈ ∈ PROPN ejpam-5960 448 29 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 448 30 and and CCONJ ejpam-5960 448 31 so so ADV ejpam-5960 448 32 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 448 33 βŠ† βŠ† NUM ejpam-5960 448 34 𝐼 𝐼 PROPN ejpam-5960 448 35 ( ( PUNCT ejpam-5960 448 36 𝑆 𝑆 PROPN ejpam-5960 448 37 ) ) PUNCT ejpam-5960 448 38 . . PUNCT ejpam-5960 449 1 thus thus ADV ejpam-5960 449 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 449 3 ∈ ∈ NOUN ejpam-5960 449 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 449 5 lim(𝐼 lim(𝐼 PROPN ejpam-5960 449 6 ( ( PUNCT ejpam-5960 449 7 𝑆 𝑆 PROPN ejpam-5960 449 8 ) ) PUNCT ejpam-5960 449 9 ) ) PUNCT ejpam-5960 449 10 , , PUNCT ejpam-5960 449 11 i.e. i.e. X ejpam-5960 449 12 , , PUNCT ejpam-5960 449 13 𝑁𝛼𝐡. 𝑁𝛼𝐡. PROPN ejpam-5960 449 14 lim(𝑆 lim(𝑆 PROPN ejpam-5960 449 15 ) ) PUNCT ejpam-5960 449 16 ≀ ≀ PUNCT ejpam-5960 450 1 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 450 2 lim(𝐼 lim(𝐼 PROPN ejpam-5960 450 3 ( ( PUNCT ejpam-5960 450 4 𝑆 𝑆 PROPN ejpam-5960 450 5 ) ) PUNCT ejpam-5960 450 6 ) ) PUNCT ejpam-5960 450 7 . . PUNCT ejpam-5960 451 1 conversely conversely ADV ejpam-5960 451 2 , , PUNCT ejpam-5960 451 3 let let VERB ejpam-5960 451 4 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 451 5 ∈ ∈ NOUN ejpam-5960 451 6 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 451 7 lim(𝐼 lim(𝐼 PROPN ejpam-5960 451 8 ( ( PUNCT ejpam-5960 451 9 𝑆 𝑆 PROPN ejpam-5960 451 10 ) ) PUNCT ejpam-5960 451 11 ) ) PUNCT ejpam-5960 451 12 , , PUNCT ejpam-5960 451 13 then then ADV ejpam-5960 451 14 πœ† πœ† PROPN ejpam-5960 451 15 ∈ ∈ PROPN ejpam-5960 451 16 𝐼 𝐼 PROPN ejpam-5960 451 17 ( ( PUNCT ejpam-5960 451 18 𝑆 𝑆 PROPN ejpam-5960 451 19 ) ) PUNCT ejpam-5960 451 20 for for ADP ejpam-5960 451 21 each each DET ejpam-5960 451 22 πœ† πœ† PRON ejpam-5960 451 23 ∈ ∈ PROPN ejpam-5960 451 24 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 451 25 , , PUNCT ejpam-5960 451 26 and and CCONJ ejpam-5960 451 27 then then ADV ejpam-5960 451 28 there there PRON ejpam-5960 451 29 is be VERB ejpam-5960 451 30 𝑛 𝑛 DET ejpam-5960 451 31 ∈ ∈ PROPN ejpam-5960 451 32 𝐷 𝐷 NOUN ejpam-5960 451 33 such such ADJ ejpam-5960 451 34 that that PRON ejpam-5960 451 35 for for ADP ejpam-5960 451 36 each each DET ejpam-5960 451 37 π‘š π‘š PROPN ejpam-5960 451 38 ∈ ∈ PROPN ejpam-5960 451 39 𝐷 𝐷 NOUN ejpam-5960 451 40 and and CCONJ ejpam-5960 451 41 π‘š π‘š ADP ejpam-5960 451 42 β‰₯ β‰₯ NOUN ejpam-5960 451 43 𝑛 𝑛 ADP ejpam-5960 451 44 we we PRON ejpam-5960 451 45 have have VERB ejpam-5960 451 46 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 451 47 ) ) PUNCT ejpam-5960 451 48 βˆ‰ βˆ‰ PROPN ejpam-5960 451 49 πœ†. πœ†. PROPN ejpam-5960 452 1 therefore therefore ADV ejpam-5960 452 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 452 3 ∈ ∈ PROPN ejpam-5960 452 4 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 452 5 lim(𝑆 lim(𝑆 PROPN ejpam-5960 452 6 ) ) PUNCT ejpam-5960 452 7 . . PUNCT ejpam-5960 453 1 thus thus ADV ejpam-5960 453 2 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 453 3 lim(𝐼 lim(𝐼 PROPN ejpam-5960 453 4 ( ( PUNCT ejpam-5960 453 5 𝑆 𝑆 PROPN ejpam-5960 453 6 ) ) PUNCT ejpam-5960 453 7 ) ) PUNCT ejpam-5960 453 8 ≀ ≀ PUNCT ejpam-5960 453 9 𝑁𝛼𝐡. 𝑁𝛼𝐡. ADP ejpam-5960 453 10 lim(𝑆 lim(𝑆 PROPN ejpam-5960 453 11 ) ) PUNCT ejpam-5960 453 12 . . PUNCT ejpam-5960 454 1 hence hence ADV ejpam-5960 454 2 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 454 3 lim(𝑆 lim(𝑆 PROPN ejpam-5960 454 4 ) ) PUNCT ejpam-5960 455 1 = = PUNCT ejpam-5960 455 2 𝑁𝛼𝐡. 𝑁𝛼𝐡. NOUN ejpam-5960 455 3 lim(𝐼 lim(𝐼 PROPN ejpam-5960 455 4 ( ( PUNCT ejpam-5960 455 5 𝑆 𝑆 PROPN ejpam-5960 455 6 ) ) PUNCT ejpam-5960 455 7 ) ) PUNCT ejpam-5960 455 8 . . PUNCT ejpam-5960 456 1 ( ( PUNCT ejpam-5960 456 2 ii ii X ejpam-5960 456 3 ) ) PUNCT ejpam-5960 456 4 let let VERB ejpam-5960 456 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 456 6 ∈ ∈ PROPN ejpam-5960 456 7 𝑁𝛼𝐡.adh(𝑆 𝑁𝛼𝐡.adh(𝑆 PROPN ejpam-5960 456 8 ) ) PUNCT ejpam-5960 456 9 , , PUNCT ejpam-5960 456 10 then then ADV ejpam-5960 456 11 for for ADP ejpam-5960 456 12 each each DET ejpam-5960 456 13 πœ† πœ† PRON ejpam-5960 456 14 ∈ ∈ PROPN ejpam-5960 456 15 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 456 16 and and CCONJ ejpam-5960 456 17 for for ADP ejpam-5960 456 18 each each DET ejpam-5960 456 19 𝑛 𝑛 PRON ejpam-5960 456 20 ∈ ∈ PROPN ejpam-5960 456 21 𝐷 𝐷 NOUN ejpam-5960 456 22 such such ADJ ejpam-5960 456 23 there there PRON ejpam-5960 456 24 is be VERB ejpam-5960 456 25 π‘š π‘š DET ejpam-5960 456 26 ∈ ∈ PROPN ejpam-5960 456 27 𝐷 𝐷 NOUN ejpam-5960 456 28 and and CCONJ ejpam-5960 456 29 π‘š π‘š ADP ejpam-5960 456 30 β‰₯ β‰₯ NOUN ejpam-5960 456 31 𝑛 𝑛 ADP ejpam-5960 456 32 we we PRON ejpam-5960 456 33 have have VERB ejpam-5960 456 34 𝑆(π‘š 𝑆(π‘š NUM ejpam-5960 456 35 ) ) PUNCT ejpam-5960 456 36 βˆ‰ βˆ‰ PROPN ejpam-5960 456 37 πœ†. πœ†. VERB ejpam-5960 457 1 if if SCONJ ejpam-5960 457 2 πœ‡ πœ‡ ADP ejpam-5960 457 3 ∈ ∈ PROPN ejpam-5960 457 4 𝐼 𝐼 PROPN ejpam-5960 457 5 ( ( PUNCT ejpam-5960 457 6 𝑆 𝑆 PROPN ejpam-5960 457 7 ) ) PUNCT ejpam-5960 457 8 , , PUNCT ejpam-5960 457 9 then then ADV ejpam-5960 457 10 there there PRON ejpam-5960 457 11 is be VERB ejpam-5960 457 12 𝑛 𝑛 DET ejpam-5960 457 13 ∈ ∈ PROPN ejpam-5960 457 14 𝐷 𝐷 NOUN ejpam-5960 457 15 such such ADJ ejpam-5960 457 16 that that PRON ejpam-5960 457 17 for for ADP ejpam-5960 457 18 each each DET ejpam-5960 457 19 π‘š π‘š PROPN ejpam-5960 457 20 ∈ ∈ PROPN ejpam-5960 457 21 𝐷 𝐷 NOUN ejpam-5960 457 22 and and CCONJ ejpam-5960 457 23 π‘š π‘š ADP ejpam-5960 457 24 β‰₯ β‰₯ NOUN ejpam-5960 457 25 𝑛 𝑛 PRON ejpam-5960 457 26 then then ADV ejpam-5960 457 27 𝑆(π‘š 𝑆(π‘š X ejpam-5960 457 28 ) ) PUNCT ejpam-5960 457 29 βˆ‰ βˆ‰ PROPN ejpam-5960 457 30 πœ‡. πœ‡. NOUN ejpam-5960 457 31 thus thus ADV ejpam-5960 457 32 πœ‡ πœ‡ ADP ejpam-5960 457 33 ∨ ∨ NOUN ejpam-5960 457 34 πœ† πœ† DET ejpam-5960 457 35 β‰  β‰  PROPN ejpam-5960 457 36 1𝑋 1𝑋 NOUN ejpam-5960 457 37 for for ADP ejpam-5960 457 38 each each DET ejpam-5960 457 39 πœ‡ πœ‡ ADP ejpam-5960 457 40 ∈ ∈ NOUN ejpam-5960 457 41 𝐼 𝐼 PROPN ejpam-5960 457 42 ( ( PUNCT ejpam-5960 457 43 𝑆 𝑆 PROPN ejpam-5960 457 44 ) ) PUNCT ejpam-5960 457 45 and and CCONJ ejpam-5960 457 46 for for ADP ejpam-5960 457 47 each each DET ejpam-5960 457 48 πœ† πœ† DET ejpam-5960 457 49 ∈ ∈ PROPN ejpam-5960 457 50 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 457 51 , , PUNCT ejpam-5960 457 52 that that ADV ejpam-5960 457 53 is is ADV ejpam-5960 457 54 , , PUNCT ejpam-5960 457 55 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 457 56 ∈ ∈ PROPN ejpam-5960 457 57 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 PRON ejpam-5960 457 58 ( ( PUNCT ejpam-5960 457 59 𝑆 𝑆 PROPN ejpam-5960 457 60 ) ) PUNCT ejpam-5960 457 61 ) ) PUNCT ejpam-5960 457 62 . . PUNCT ejpam-5960 458 1 this this PRON ejpam-5960 458 2 implies imply VERB ejpam-5960 458 3 that that SCONJ ejpam-5960 458 4 𝑁𝛼𝐡.adh(𝑆 𝑁𝛼𝐡.adh(𝑆 NOUN ejpam-5960 458 5 ) ) PUNCT ejpam-5960 458 6 ≀ ≀ NOUN ejpam-5960 458 7 𝑁𝛼𝐡.adh(𝐼 𝑁𝛼𝐡.adh(𝐼 PRON ejpam-5960 458 8 ( ( PUNCT ejpam-5960 458 9 𝑆 𝑆 PROPN ejpam-5960 458 10 ) ) PUNCT ejpam-5960 458 11 ) ) PUNCT ejpam-5960 458 12 . . PUNCT ejpam-5960 459 1 theorem theorem VERB ejpam-5960 459 2 3.27 3.27 NUM ejpam-5960 459 3 . . PUNCT ejpam-5960 460 1 let let VERB ejpam-5960 460 2 ( ( PUNCT ejpam-5960 460 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 460 4 , , PUNCT ejpam-5960 460 5 𝜏 𝜏 NOUN ejpam-5960 460 6 ) ) PUNCT ejpam-5960 460 7 be be AUX ejpam-5960 460 8 a a DET ejpam-5960 460 9 πΏβˆ’ts πΏβˆ’t NOUN ejpam-5960 460 10 . . PUNCT ejpam-5960 461 1 and and CCONJ ejpam-5960 461 2 let let VERB ejpam-5960 461 3 πœ‡ πœ‡ ADP ejpam-5960 461 4 ∈ ∈ X ejpam-5960 461 5 𝐿𝑋. 𝐿𝑋. PUNCT ejpam-5960 461 6 then then ADV ejpam-5960 461 7 : : PUNCT ejpam-5960 461 8 ( ( PUNCT ejpam-5960 461 9 i i NOUN ejpam-5960 461 10 ) ) PUNCT ejpam-5960 461 11 if if SCONJ ejpam-5960 461 12 1𝑋 1𝑋 PROPN ejpam-5960 461 13 is be AUX ejpam-5960 461 14 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 461 15 iff iff PROPN ejpam-5960 461 16 1𝑋 1𝑋 PROPN ejpam-5960 461 17 is be AUX ejpam-5960 461 18 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 461 19 . . PUNCT ejpam-5960 462 1 ( ( PUNCT ejpam-5960 462 2 ii ii NOUN ejpam-5960 462 3 ) ) PUNCT ejpam-5960 462 4 if if SCONJ ejpam-5960 462 5 πœ‡ πœ‡ PRON ejpam-5960 462 6 is be AUX ejpam-5960 462 7 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 462 8 , , PUNCT ejpam-5960 462 9 then then ADV ejpam-5960 462 10 πœ‡ πœ‡ ADP ejpam-5960 462 11 is be AUX ejpam-5960 462 12 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 462 13 . . PUNCT ejpam-5960 463 1 ( ( PUNCT ejpam-5960 463 2 iii iii X ejpam-5960 463 3 ) ) PUNCT ejpam-5960 463 4 if if SCONJ ejpam-5960 463 5 πœ‚ πœ‚ NOUN ejpam-5960 463 6 is be AUX ejpam-5960 463 7 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 463 8 and and CCONJ ejpam-5960 463 9 πœ‡ πœ‡ ADP ejpam-5960 463 10 ≀ ≀ NOUN ejpam-5960 463 11 πœ‚ πœ‚ NOUN ejpam-5960 463 12 , , PUNCT ejpam-5960 463 13 then then ADV ejpam-5960 463 14 πœ‡ πœ‡ ADP ejpam-5960 463 15 is be AUX ejpam-5960 463 16 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 463 17 . . PUNCT ejpam-5960 464 1 ( ( PUNCT ejpam-5960 464 2 iv iv X ejpam-5960 464 3 ) ) PUNCT ejpam-5960 464 4 if if SCONJ ejpam-5960 464 5 πœ‡1 πœ‡1 ADJ ejpam-5960 464 6 , , PUNCT ejpam-5960 464 7 πœ‡2 πœ‡2 PROPN ejpam-5960 464 8 , , PUNCT ejpam-5960 464 9 ... ... PUNCT ejpam-5960 464 10 , , PUNCT ejpam-5960 464 11 πœ‡π‘š πœ‡π‘š NOUN ejpam-5960 464 12 are be AUX ejpam-5960 464 13 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 464 14 sets set NOUN ejpam-5960 464 15 , , PUNCT ejpam-5960 464 16 then then ADV ejpam-5960 464 17 βˆ¨π‘š βˆ¨π‘š NOUN ejpam-5960 464 18 𝑖=1 𝑖=1 PROPN ejpam-5960 464 19 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 464 20 is be AUX ejpam-5960 464 21 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 464 22 . . PUNCT ejpam-5960 465 1 proof proof NOUN ejpam-5960 465 2 ( ( PUNCT ejpam-5960 465 3 i i NOUN ejpam-5960 465 4 ) ) PUNCT ejpam-5960 465 5 let let VERB ejpam-5960 465 6 ( ( PUNCT ejpam-5960 465 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 465 8 , , PUNCT ejpam-5960 465 9 𝜏 𝜏 NOUN ejpam-5960 465 10 ) ) PUNCT ejpam-5960 465 11 be be VERB ejpam-5960 465 12 a a DET ejpam-5960 465 13 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 465 14 space space NOUN ejpam-5960 465 15 and and CCONJ ejpam-5960 465 16 let let VERB ejpam-5960 465 17 ψ ψ X ejpam-5960 465 18 = = X ejpam-5960 465 19 { { PUNCT ejpam-5960 465 20 πœ†π‘– πœ†π‘– NOUN ejpam-5960 465 21 : : PUNCT ejpam-5960 465 22 𝑖 𝑖 SYM ejpam-5960 465 23 ∈ ∈ PROPN ejpam-5960 465 24 𝐼 𝐼 PROPN ejpam-5960 465 25 } } PUNCT ejpam-5960 465 26 βŠ† βŠ† NUM ejpam-5960 465 27 πœβ€² πœβ€² NOUN ejpam-5960 465 28 be be AUX ejpam-5960 465 29 an an DET ejpam-5960 465 30 π›Όβˆ’rf π›Όβˆ’rf NOUN ejpam-5960 465 31 of of ADP ejpam-5960 465 32 1𝑋. 1𝑋. NUM ejpam-5960 465 33 since since SCONJ ejpam-5960 465 34 ( ( PUNCT ejpam-5960 465 35 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 465 36 , , PUNCT ejpam-5960 465 37 𝜏 𝜏 NOUN ejpam-5960 465 38 ) ) PUNCT ejpam-5960 465 39 is be AUX ejpam-5960 465 40 a a DET ejpam-5960 465 41 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 465 42 space space NOUN ejpam-5960 465 43 , , PUNCT ejpam-5960 465 44 there there PRON ejpam-5960 465 45 exists exist VERB ejpam-5960 465 46 a a DET ejpam-5960 465 47 finite finite NOUN ejpam-5960 465 48 subfamily subfamily ADV ejpam-5960 465 49 ψ ψ X ejpam-5960 465 50 β—¦ β—¦ NOUN ejpam-5960 465 51 = = SYM ejpam-5960 465 52 { { PUNCT ejpam-5960 465 53 πœ†π‘– πœ†π‘– NOUN ejpam-5960 465 54 : : PUNCT ejpam-5960 465 55 𝑖 𝑖 SYM ejpam-5960 465 56 = = SYM ejpam-5960 465 57 1 1 NUM ejpam-5960 465 58 , , PUNCT ejpam-5960 465 59 2 2 NUM ejpam-5960 465 60 , , PUNCT ejpam-5960 465 61 ... ... PUNCT ejpam-5960 465 62 , , PUNCT ejpam-5960 465 63 π‘š π‘š X ejpam-5960 465 64 } } PUNCT ejpam-5960 465 65 ∈ ∈ NOUN ejpam-5960 465 66 2(ψ 2(ψ NUM ejpam-5960 465 67 ) ) PUNCT ejpam-5960 465 68 such such ADJ ejpam-5960 465 69 that that SCONJ ejpam-5960 465 70 ψ ψ X ejpam-5960 465 71 β—¦ β—¦ NOUN ejpam-5960 465 72 is be AUX ejpam-5960 465 73 an an DET ejpam-5960 465 74 π›Όβˆ’rcrf π›Όβˆ’rcrf NOUN ejpam-5960 465 75 of of ADP ejpam-5960 465 76 1𝑋 1𝑋 NOUN ejpam-5960 465 77 and and CCONJ ejpam-5960 465 78 so so ADV ejpam-5960 465 79 1𝑋 1𝑋 PROPN ejpam-5960 465 80 is be AUX ejpam-5960 465 81 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 465 82 set set VERB ejpam-5960 465 83 . . PUNCT ejpam-5960 466 1 conversely conversely ADV ejpam-5960 466 2 , , PUNCT ejpam-5960 466 3 let let VERB ejpam-5960 466 4 1𝑋 1𝑋 PROPN ejpam-5960 466 5 be be AUX ejpam-5960 466 6 a a DET ejpam-5960 466 7 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 466 8 set set NOUN ejpam-5960 466 9 and and CCONJ ejpam-5960 466 10 let let VERB ejpam-5960 466 11 ψ ψ X ejpam-5960 466 12 = = X ejpam-5960 466 13 { { PUNCT ejpam-5960 466 14 πœ†π‘– πœ†π‘– NOUN ejpam-5960 466 15 : : PUNCT ejpam-5960 466 16 𝑖 𝑖 SYM ejpam-5960 466 17 ∈ ∈ PROPN ejpam-5960 466 18 𝐼 𝐼 PROPN ejpam-5960 466 19 } } PUNCT ejpam-5960 466 20 βŠ† βŠ† NUM ejpam-5960 466 21 πœβ€² πœβ€² NOUN ejpam-5960 466 22 be be AUX ejpam-5960 466 23 an an DET ejpam-5960 466 24 π›Όβˆ’rf π›Όβˆ’rf NOUN ejpam-5960 466 25 of of ADP ejpam-5960 466 26 1𝑋. 1𝑋. NUM ejpam-5960 466 27 since since SCONJ ejpam-5960 466 28 1𝑋 1𝑋 PROPN ejpam-5960 466 29 is be AUX ejpam-5960 466 30 a a DET ejpam-5960 466 31 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 466 32 set set NOUN ejpam-5960 466 33 , , PUNCT ejpam-5960 466 34 then then ADV ejpam-5960 466 35 there there PRON ejpam-5960 466 36 exists exist VERB ejpam-5960 466 37 a a DET ejpam-5960 466 38 finite finite NOUN ejpam-5960 466 39 subfamily subfamily ADV ejpam-5960 466 40 ψ ψ X ejpam-5960 466 41 β—¦ β—¦ NOUN ejpam-5960 466 42 = = SYM ejpam-5960 466 43 { { PUNCT ejpam-5960 466 44 πœ†π‘– πœ†π‘– NOUN ejpam-5960 466 45 : : PUNCT ejpam-5960 466 46 𝑖 𝑖 SYM ejpam-5960 466 47 = = SYM ejpam-5960 466 48 1 1 NUM ejpam-5960 466 49 , , PUNCT ejpam-5960 466 50 2 2 NUM ejpam-5960 466 51 , , PUNCT ejpam-5960 466 52 ... ... PUNCT ejpam-5960 466 53 , , PUNCT ejpam-5960 466 54 π‘š π‘š X ejpam-5960 466 55 } } PUNCT ejpam-5960 466 56 ∈ ∈ NOUN ejpam-5960 466 57 2(ψ 2(ψ NUM ejpam-5960 466 58 ) ) PUNCT ejpam-5960 466 59 such such ADJ ejpam-5960 466 60 that that SCONJ ejpam-5960 466 61 ψ ψ X ejpam-5960 466 62 β—¦ β—¦ NOUN ejpam-5960 466 63 is be AUX ejpam-5960 466 64 an an DET ejpam-5960 466 65 π›Όβˆ’rcrf π›Όβˆ’rcrf NOUN ejpam-5960 466 66 of of ADP ejpam-5960 466 67 1𝑋 1𝑋 NOUN ejpam-5960 466 68 and and CCONJ ejpam-5960 466 69 so so ADV ejpam-5960 466 70 1𝑋 1𝑋 PROPN ejpam-5960 466 71 is be AUX ejpam-5960 466 72 a a DET ejpam-5960 466 73 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 466 74 set set NOUN ejpam-5960 466 75 . . PUNCT ejpam-5960 467 1 ( ( PUNCT ejpam-5960 467 2 ii ii NOUN ejpam-5960 467 3 ) ) PUNCT ejpam-5960 467 4 let let AUX ejpam-5960 467 5 πœ‡ πœ‡ PART ejpam-5960 467 6 be be AUX ejpam-5960 467 7 a a DET ejpam-5960 467 8 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 467 9 and and CCONJ ejpam-5960 467 10 let let VERB ejpam-5960 467 11 ψ ψ X ejpam-5960 467 12 = = X ejpam-5960 467 13 { { PUNCT ejpam-5960 467 14 πœ†π‘– πœ†π‘– NOUN ejpam-5960 467 15 : : PUNCT ejpam-5960 467 16 𝑖 𝑖 SYM ejpam-5960 467 17 ∈ ∈ PROPN ejpam-5960 467 18 𝐼 𝐼 PROPN ejpam-5960 467 19 } } PUNCT ejpam-5960 467 20 βŠ† βŠ† NUM ejpam-5960 467 21 πœβ€² πœβ€² NOUN ejpam-5960 467 22 be be AUX ejpam-5960 467 23 an an DET ejpam-5960 467 24 π›Όβˆ’rf π›Όβˆ’rf NOUN ejpam-5960 467 25 of of ADP ejpam-5960 467 26 1𝑋 1𝑋 PROPN ejpam-5960 467 27 and and CCONJ ejpam-5960 467 28 so so ADV ejpam-5960 467 29 ψ ψ NOUN ejpam-5960 467 30 is be AUX ejpam-5960 467 31 an an DET ejpam-5960 467 32 π›Όβˆ’rf π›Όβˆ’rf NOUN ejpam-5960 467 33 of of ADP ejpam-5960 467 34 πœ‡. πœ‡. NOUN ejpam-5960 467 35 since since SCONJ ejpam-5960 467 36 πœ‡ πœ‡ ADV ejpam-5960 467 37 is be AUX ejpam-5960 467 38 a a DET ejpam-5960 467 39 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 467 40 set set NOUN ejpam-5960 467 41 , , PUNCT ejpam-5960 467 42 then then ADV ejpam-5960 467 43 there there PRON ejpam-5960 467 44 exists exist VERB ejpam-5960 467 45 a a DET ejpam-5960 467 46 finite finite NOUN ejpam-5960 467 47 subfamily subfamily ADV ejpam-5960 467 48 ψ ψ X ejpam-5960 467 49 β—¦ β—¦ NOUN ejpam-5960 467 50 = = SYM ejpam-5960 467 51 { { PUNCT ejpam-5960 467 52 πœ†π‘– πœ†π‘– NOUN ejpam-5960 467 53 : : PUNCT ejpam-5960 467 54 𝑖 𝑖 SYM ejpam-5960 467 55 = = SYM ejpam-5960 467 56 1 1 NUM ejpam-5960 467 57 , , PUNCT ejpam-5960 467 58 2 2 NUM ejpam-5960 467 59 , , PUNCT ejpam-5960 467 60 ... ... PUNCT ejpam-5960 467 61 , , PUNCT ejpam-5960 467 62 π‘š π‘š X ejpam-5960 467 63 } } PUNCT ejpam-5960 467 64 ∈ ∈ NOUN ejpam-5960 467 65 2(ψ 2(ψ NUM ejpam-5960 467 66 ) ) PUNCT ejpam-5960 467 67 such such ADJ ejpam-5960 467 68 that that SCONJ ejpam-5960 467 69 ψ ψ X ejpam-5960 467 70 β—¦ β—¦ NOUN ejpam-5960 467 71 is be AUX ejpam-5960 467 72 an an DET ejpam-5960 467 73 π›Όβˆ’rcrf π›Όβˆ’rcrf NOUN ejpam-5960 467 74 of of ADP ejpam-5960 467 75 πœ‡ πœ‡ ADV ejpam-5960 467 76 and and CCONJ ejpam-5960 467 77 so so ADV ejpam-5960 467 78 πœ‡ πœ‡ PRON ejpam-5960 467 79 is be AUX ejpam-5960 467 80 a a DET ejpam-5960 467 81 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 467 82 set set NOUN ejpam-5960 467 83 . . PUNCT ejpam-5960 468 1 ( ( PUNCT ejpam-5960 468 2 iii iii X ejpam-5960 468 3 ) ) PUNCT ejpam-5960 468 4 let let VERB ejpam-5960 468 5 πœ‚ πœ‚ NOUN ejpam-5960 468 6 be be AUX ejpam-5960 468 7 a a DET ejpam-5960 468 8 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 468 9 set set NOUN ejpam-5960 468 10 and and CCONJ ejpam-5960 468 11 πœ‡ πœ‡ ADP ejpam-5960 468 12 ≀ ≀ NUM ejpam-5960 468 13 πœ‚. πœ‚. NOUN ejpam-5960 468 14 let let VERB ejpam-5960 468 15 ψ ψ X ejpam-5960 468 16 = = X ejpam-5960 468 17 { { PUNCT ejpam-5960 468 18 πœ†π‘– πœ†π‘– NOUN ejpam-5960 468 19 : : PUNCT ejpam-5960 468 20 𝑖 𝑖 SYM ejpam-5960 468 21 ∈ ∈ PROPN ejpam-5960 468 22 𝐼 𝐼 PROPN ejpam-5960 468 23 } } PUNCT ejpam-5960 468 24 βŠ† βŠ† NUM ejpam-5960 468 25 πœβ€² πœβ€² NOUN ejpam-5960 468 26 be be AUX ejpam-5960 468 27 an an DET ejpam-5960 468 28 π›Όβˆ’rf π›Όβˆ’rf NOUN ejpam-5960 468 29 of of ADP ejpam-5960 468 30 1𝑋 1𝑋 PROPN ejpam-5960 468 31 and and CCONJ ejpam-5960 468 32 so so ADV ejpam-5960 468 33 ψ ψ NOUN ejpam-5960 468 34 is be AUX ejpam-5960 468 35 π›Όβˆ’rf π›Όβˆ’rf NOUN ejpam-5960 468 36 of of ADP ejpam-5960 468 37 πœ‚. πœ‚. NOUN ejpam-5960 468 38 since since SCONJ ejpam-5960 468 39 πœ‚ πœ‚ NOUN ejpam-5960 468 40 is be AUX ejpam-5960 468 41 a a DET ejpam-5960 468 42 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 468 43 set set NOUN ejpam-5960 468 44 , , PUNCT ejpam-5960 468 45 then then ADV ejpam-5960 468 46 there there PRON ejpam-5960 468 47 exists exist VERB ejpam-5960 468 48 a a DET ejpam-5960 468 49 finite finite NOUN ejpam-5960 468 50 subfamily subfamily ADV ejpam-5960 468 51 ψ ψ X ejpam-5960 468 52 β—¦ β—¦ NOUN ejpam-5960 468 53 = = SYM ejpam-5960 468 54 { { PUNCT ejpam-5960 468 55 πœ†π‘– πœ†π‘– NOUN ejpam-5960 468 56 : : PUNCT ejpam-5960 468 57 𝑖 𝑖 SYM ejpam-5960 468 58 = = SYM ejpam-5960 468 59 1 1 NUM ejpam-5960 468 60 , , PUNCT ejpam-5960 468 61 2 2 NUM ejpam-5960 468 62 , , PUNCT ejpam-5960 468 63 ... ... PUNCT ejpam-5960 468 64 , , PUNCT ejpam-5960 468 65 π‘š π‘š X ejpam-5960 468 66 } } PUNCT ejpam-5960 468 67 ∈ ∈ NOUN ejpam-5960 468 68 2(ψ 2(ψ NUM ejpam-5960 468 69 ) ) PUNCT ejpam-5960 468 70 such such ADJ ejpam-5960 468 71 that that SCONJ ejpam-5960 468 72 ψ ψ X ejpam-5960 468 73 β—¦ β—¦ NOUN ejpam-5960 468 74 is be AUX ejpam-5960 468 75 an an DET ejpam-5960 468 76 π›Όβˆ’rcrf π›Όβˆ’rcrf NOUN ejpam-5960 468 77 of of ADP ejpam-5960 468 78 πœ‚ πœ‚ NOUN ejpam-5960 468 79 , , PUNCT ejpam-5960 468 80 since since SCONJ ejpam-5960 468 81 πœ‡ πœ‡ ADP ejpam-5960 468 82 ≀ ≀ NUM ejpam-5960 468 83 πœ‚ πœ‚ NOUN ejpam-5960 468 84 , , PUNCT ejpam-5960 468 85 then then ADV ejpam-5960 468 86 ψ ψ X ejpam-5960 468 87 β—¦ β—¦ NOUN ejpam-5960 468 88 is be AUX ejpam-5960 468 89 an an DET ejpam-5960 468 90 π›Όβˆ’rcrf π›Όβˆ’rcrf NOUN ejpam-5960 468 91 of of ADP ejpam-5960 468 92 πœ‡ πœ‡ ADV ejpam-5960 468 93 and and CCONJ ejpam-5960 468 94 so so ADV ejpam-5960 468 95 πœ‡ πœ‡ PRON ejpam-5960 468 96 is be AUX ejpam-5960 468 97 a a DET ejpam-5960 468 98 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 468 99 set set NOUN ejpam-5960 468 100 . . PUNCT ejpam-5960 469 1 ( ( PUNCT ejpam-5960 469 2 iv iv X ejpam-5960 469 3 ) ) PUNCT ejpam-5960 469 4 let let VERB ejpam-5960 469 5 πœ‡1 πœ‡1 ADJ ejpam-5960 469 6 , , PUNCT ejpam-5960 469 7 πœ‡2 πœ‡2 PROPN ejpam-5960 469 8 , , PUNCT ejpam-5960 469 9 ... ... PUNCT ejpam-5960 469 10 , , PUNCT ejpam-5960 469 11 πœ‡π‘› πœ‡π‘› INTJ ejpam-5960 469 12 be be AUX ejpam-5960 469 13 a a DET ejpam-5960 469 14 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 469 15 set set NOUN ejpam-5960 469 16 and and CCONJ ejpam-5960 469 17 let let VERB ejpam-5960 469 18 ψ ψ PRON ejpam-5960 469 19 βŠ† βŠ† NUM ejpam-5960 469 20 πœβ€² πœβ€² NOUN ejpam-5960 469 21 be be AUX ejpam-5960 469 22 an an DET ejpam-5960 469 23 π›Όβˆ’rf π›Όβˆ’rf NOUN ejpam-5960 469 24 of of ADP ejpam-5960 469 25 1𝑋. 1𝑋. NUM ejpam-5960 469 26 since since SCONJ ejpam-5960 469 27 πœ‡1 πœ‡1 NOUN ejpam-5960 469 28 , , PUNCT ejpam-5960 469 29 πœ‡2 πœ‡2 PROPN ejpam-5960 469 30 , , PUNCT ejpam-5960 469 31 ... ... PUNCT ejpam-5960 469 32 , , PUNCT ejpam-5960 469 33 πœ‡π‘š πœ‡π‘š NOUN ejpam-5960 469 34 are be AUX ejpam-5960 469 35 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 469 36 sets set NOUN ejpam-5960 469 37 , , PUNCT ejpam-5960 469 38 then then ADV ejpam-5960 469 39 there there PRON ejpam-5960 469 40 exist exist VERB ejpam-5960 469 41 ψ1 ψ1 ADJ ejpam-5960 469 42 β—¦ β—¦ NOUN ejpam-5960 469 43 , , PUNCT ejpam-5960 469 44 ψ ψ NOUN ejpam-5960 469 45 2 2 NUM ejpam-5960 469 46 β—¦ β—¦ NOUN ejpam-5960 469 47 , , PUNCT ejpam-5960 469 48 ... ... PUNCT ejpam-5960 469 49 , , PUNCT ejpam-5960 469 50 ψ ψ X ejpam-5960 469 51 𝑛 𝑛 ADP ejpam-5960 469 52 β—¦ β—¦ NOUN ejpam-5960 469 53 ∈ ∈ NOUN ejpam-5960 469 54 2(ψ 2(ψ NUM ejpam-5960 469 55 ) ) PUNCT ejpam-5960 469 56 such such ADJ ejpam-5960 469 57 that that DET ejpam-5960 469 58 ψ1 ψ1 ADJ ejpam-5960 469 59 β—¦ β—¦ NOUN ejpam-5960 469 60 , , PUNCT ejpam-5960 469 61 ψ ψ NOUN ejpam-5960 469 62 2 2 NUM ejpam-5960 469 63 β—¦ β—¦ NOUN ejpam-5960 469 64 , , PUNCT ejpam-5960 469 65 ... ... PUNCT ejpam-5960 469 66 , , PUNCT ejpam-5960 469 67 ψ ψ X ejpam-5960 469 68 𝑛 𝑛 PRON ejpam-5960 469 69 β—¦ β—¦ NOUN ejpam-5960 469 70 are be AUX ejpam-5960 469 71 π›Όβˆ’rcrf π›Όβˆ’rcrf NUM ejpam-5960 469 72 of of ADP ejpam-5960 469 73 πœ‡1 πœ‡1 NOUN ejpam-5960 469 74 , , PUNCT ejpam-5960 469 75 πœ‡2 πœ‡2 PROPN ejpam-5960 469 76 , , PUNCT ejpam-5960 469 77 ... ... PUNCT ejpam-5960 469 78 , , PUNCT ejpam-5960 469 79 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 469 80 , , PUNCT ejpam-5960 469 81 respectively respectively ADV ejpam-5960 469 82 , , PUNCT ejpam-5960 469 83 and and CCONJ ejpam-5960 469 84 so so ADV ejpam-5960 469 85 for for ADP ejpam-5960 469 86 each each DET ejpam-5960 469 87 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 469 88 ∈ ∈ PROPN ejpam-5960 469 89 πœ‡1 πœ‡1 PROPN ejpam-5960 469 90 there there PRON ejpam-5960 469 91 is be VERB ejpam-5960 469 92 πœ†1 πœ†1 NOUN ejpam-5960 469 93 ∈ ∈ PROPN ejpam-5960 469 94 ψ1 ψ1 NOUN ejpam-5960 469 95 β—¦ β—¦ NOUN ejpam-5960 469 96 such such ADJ ejpam-5960 469 97 that that PRON ejpam-5960 469 98 πœ†1 πœ†1 NOUN ejpam-5960 469 99 ∈ ∈ PROPN ejpam-5960 469 100 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 469 101 , , PUNCT ejpam-5960 469 102 for for ADP ejpam-5960 469 103 each each DET ejpam-5960 469 104 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 469 105 ∈ ∈ PROPN ejpam-5960 469 106 πœ‡2 πœ‡2 NOUN ejpam-5960 469 107 , , PUNCT ejpam-5960 469 108 there there PRON ejpam-5960 469 109 is be VERB ejpam-5960 469 110 πœ†2 πœ†2 NOUN ejpam-5960 469 111 ∈ ∈ NOUN ejpam-5960 469 112 ψ2 ψ2 NOUN ejpam-5960 469 113 β—¦ β—¦ VERB ejpam-5960 469 114 such such ADJ ejpam-5960 469 115 that that DET ejpam-5960 469 116 πœ†2 πœ†2 PROPN ejpam-5960 469 117 ∈ ∈ PROPN ejpam-5960 469 118 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 469 119 , , PUNCT ejpam-5960 469 120 ... ... PUNCT ejpam-5960 469 121 , , PUNCT ejpam-5960 469 122 for for ADP ejpam-5960 469 123 each each DET ejpam-5960 469 124 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 469 125 ∈ ∈ NOUN ejpam-5960 469 126 πœ‡π‘› πœ‡π‘› INTJ ejpam-5960 469 127 there there PRON ejpam-5960 469 128 is be VERB ejpam-5960 469 129 πœ†π‘› πœ†π‘› ADP ejpam-5960 469 130 ∈ ∈ NOUN ejpam-5960 469 131 Οˆπ‘› Οˆπ‘› ADP ejpam-5960 469 132 β—¦ β—¦ VERB ejpam-5960 469 133 such such ADJ ejpam-5960 469 134 that that DET ejpam-5960 469 135 πœ†π‘› πœ†π‘› PROPN ejpam-5960 469 136 ∈ ∈ PROPN ejpam-5960 469 137 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 469 138 . . PUNCT ejpam-5960 470 1 hence hence ADV ejpam-5960 470 2 , , PUNCT ejpam-5960 470 3 for for ADP ejpam-5960 470 4 each each DET ejpam-5960 470 5 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 470 6 ∈ ∈ PROPN ejpam-5960 470 7 πœ‡1 πœ‡1 PROPN ejpam-5960 470 8 ∨ ∨ PROPN ejpam-5960 470 9 πœ‡2 πœ‡2 PROPN ejpam-5960 470 10 ∨ ∨ NUM ejpam-5960 470 11 ... ... PUNCT ejpam-5960 470 12 ∨ ∨ NUM ejpam-5960 470 13 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 470 14 , , PUNCT ejpam-5960 470 15 we we PRON ejpam-5960 470 16 have have VERB ejpam-5960 470 17 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 470 18 ∈ ∈ NOUN ejpam-5960 470 19 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 470 20 for for ADP ejpam-5960 470 21 some some DET ejpam-5960 470 22 𝑖 𝑖 SYM ejpam-5960 470 23 ∈ ∈ PROPN ejpam-5960 470 24 { { PUNCT ejpam-5960 470 25 1 1 NUM ejpam-5960 470 26 , , PUNCT ejpam-5960 470 27 2 2 NUM ejpam-5960 470 28 , , PUNCT ejpam-5960 470 29 ... ... PUNCT ejpam-5960 470 30 , , PUNCT ejpam-5960 470 31 𝑛 𝑛 ADJ ejpam-5960 470 32 } } PUNCT ejpam-5960 470 33 and and CCONJ ejpam-5960 470 34 so so ADV ejpam-5960 470 35 there there PRON ejpam-5960 470 36 is be VERB ejpam-5960 470 37 πœ†π‘– πœ†π‘– PROPN ejpam-5960 470 38 ∈ ∈ PROPN ejpam-5960 470 39 βˆ¨π‘› βˆ¨π‘› VERB ejpam-5960 470 40 𝑖=1 𝑖=1 PUNCT ejpam-5960 470 41 ψ ψ AUX ejpam-5960 470 42 𝑖 𝑖 PUNCT ejpam-5960 470 43 β—¦ β—¦ NOUN ejpam-5960 471 1 such such ADJ ejpam-5960 471 2 that that SCONJ ejpam-5960 471 3 πœ†π‘– πœ†π‘– PROPN ejpam-5960 471 4 ∈ ∈ PROPN ejpam-5960 471 5 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 471 6 and and CCONJ ejpam-5960 471 7 hence hence ADV ejpam-5960 471 8 βˆ¨π‘› βˆ¨π‘› VERB ejpam-5960 471 9 𝑖=1 𝑖=1 PUNCT ejpam-5960 471 10 ψ ψ SYM ejpam-5960 471 11 𝑖 𝑖 X ejpam-5960 471 12 β—¦ β—¦ NOUN ejpam-5960 471 13 is be AUX ejpam-5960 471 14 an an DET ejpam-5960 471 15 π›Όβˆ’rcrf π›Όβˆ’rcrf NOUN ejpam-5960 471 16 of of ADP ejpam-5960 471 17 πœ‡1 πœ‡1 PROPN ejpam-5960 471 18 ∨ ∨ PROPN ejpam-5960 471 19 πœ‡2 πœ‡2 PROPN ejpam-5960 471 20 ∨ ∨ NUM ejpam-5960 471 21 ... ... PUNCT ejpam-5960 471 22 ∨ ∨ NUM ejpam-5960 471 23 πœ‡π‘›. πœ‡π‘›. PROPN ejpam-5960 471 24 thus thus ADV ejpam-5960 471 25 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 471 26 = = PUNCT ejpam-5960 471 27 βˆ¨π‘› βˆ¨π‘› PROPN ejpam-5960 471 28 𝑖=1ψ 𝑖=1ψ VERB ejpam-5960 471 29 π‘œ π‘œ NOUN ejpam-5960 471 30 𝑖 𝑖 VERB ejpam-5960 471 31 is be AUX ejpam-5960 471 32 π›Όβˆ’rcrf π›Όβˆ’rcrf NUM ejpam-5960 471 33 of of ADP ejpam-5960 471 34 πœ‡1 πœ‡1 PROPN ejpam-5960 471 35 ∨ ∨ PROPN ejpam-5960 471 36 πœ‡2 πœ‡2 PROPN ejpam-5960 471 37 ∨ ∨ NOUN ejpam-5960 471 38 . . PUNCT ejpam-5960 471 39 . . PUNCT ejpam-5960 472 1 . . PUNCT ejpam-5960 473 1 ∨ ∨ NUM ejpam-5960 473 2 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 473 3 and and CCONJ ejpam-5960 473 4 so so ADV ejpam-5960 473 5 ∨{πœ‡π‘– ∨{πœ‡π‘– PROPN ejpam-5960 473 6 : : PUNCT ejpam-5960 473 7 𝑖 𝑖 SYM ejpam-5960 473 8 = = SYM ejpam-5960 473 9 1 1 NUM ejpam-5960 473 10 , , PUNCT ejpam-5960 473 11 2 2 NUM ejpam-5960 473 12 , , PUNCT ejpam-5960 473 13 . . PUNCT ejpam-5960 473 14 . . PUNCT ejpam-5960 473 15 . . PUNCT ejpam-5960 474 1 , , PUNCT ejpam-5960 474 2 𝑛 𝑛 X ejpam-5960 474 3 } } PUNCT ejpam-5960 474 4 is be AUX ejpam-5960 474 5 a a DET ejpam-5960 474 6 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 474 7 set set NOUN ejpam-5960 474 8 . . PUNCT ejpam-5960 475 1 the the DET ejpam-5960 475 2 following follow VERB ejpam-5960 475 3 example example NOUN ejpam-5960 475 4 shows show VERB ejpam-5960 475 5 that that SCONJ ejpam-5960 475 6 the the DET ejpam-5960 475 7 converse converse NOUN ejpam-5960 475 8 of of ADP ejpam-5960 475 9 theorem theorem ADJ ejpam-5960 475 10 3.27 3.27 NUM ejpam-5960 475 11 ( ( PUNCT ejpam-5960 475 12 ii ii NOUN ejpam-5960 475 13 ) ) PUNCT ejpam-5960 475 14 is be AUX ejpam-5960 475 15 not not PART ejpam-5960 475 16 true true ADJ ejpam-5960 475 17 in in ADP ejpam-5960 475 18 general general ADJ ejpam-5960 475 19 . . PUNCT ejpam-5960 476 1 n. n. PROPN ejpam-5960 476 2 a. a. PROPN ejpam-5960 476 3 alsaedi alsaedi PROPN ejpam-5960 476 4 / / SYM ejpam-5960 476 5 eur eur PROPN ejpam-5960 476 6 . . PUNCT ejpam-5960 477 1 j. j. PROPN ejpam-5960 477 2 pure pure PROPN ejpam-5960 477 3 appl appl PROPN ejpam-5960 477 4 . . PROPN ejpam-5960 477 5 math math PROPN ejpam-5960 477 6 , , PUNCT ejpam-5960 477 7 18 18 NUM ejpam-5960 477 8 ( ( PUNCT ejpam-5960 477 9 4 4 NUM ejpam-5960 477 10 ) ) PUNCT ejpam-5960 477 11 ( ( PUNCT ejpam-5960 477 12 2025 2025 NUM ejpam-5960 477 13 ) ) PUNCT ejpam-5960 477 14 , , PUNCT ejpam-5960 477 15 5960 5960 NUM ejpam-5960 477 16 16 16 NUM ejpam-5960 477 17 of of ADP ejpam-5960 477 18 22 22 NUM ejpam-5960 477 19 example example NOUN ejpam-5960 477 20 3.28 3.28 NUM ejpam-5960 477 21 . . PUNCT ejpam-5960 478 1 let let VERB ejpam-5960 478 2 𝑋 𝑋 PROPN ejpam-5960 478 3 = = SYM ejpam-5960 478 4 { { PUNCT ejpam-5960 478 5 π‘₯ π‘₯ NOUN ejpam-5960 478 6 } } PUNCT ejpam-5960 478 7 , , PUNCT ejpam-5960 478 8 𝐿 𝐿 PROPN ejpam-5960 478 9 = = SYM ejpam-5960 479 1 [ [ X ejpam-5960 479 2 0 0 NUM ejpam-5960 479 3 , , PUNCT ejpam-5960 479 4 1 1 NUM ejpam-5960 479 5 ] ] PUNCT ejpam-5960 479 6 , , PUNCT ejpam-5960 479 7 and and CCONJ ejpam-5960 479 8 let let VERB ejpam-5960 479 9 𝜏 𝜏 NOUN ejpam-5960 479 10 = = SYM ejpam-5960 479 11 { { PUNCT ejpam-5960 479 12 0𝑋 0𝑋 PROPN ejpam-5960 479 13 , , PUNCT ejpam-5960 479 14 π‘₯ π‘₯ PROPN ejpam-5960 479 15 1 1 NUM ejpam-5960 479 16 4 4 NUM ejpam-5960 479 17 , , PUNCT ejpam-5960 479 18 π‘₯ π‘₯ PROPN ejpam-5960 479 19 8 8 NUM ejpam-5960 479 20 9 9 NUM ejpam-5960 479 21 , , PUNCT ejpam-5960 479 22 1𝑋 1𝑋 PROPN ejpam-5960 479 23 } } PUNCT ejpam-5960 479 24 . . PUNCT ejpam-5960 480 1 then then ADV ejpam-5960 480 2 ( ( PUNCT ejpam-5960 480 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 480 4 , , PUNCT ejpam-5960 480 5 𝜏 𝜏 NOUN ejpam-5960 480 6 ) ) PUNCT ejpam-5960 480 7 is be AUX ejpam-5960 480 8 πΏβˆ’ts πΏβˆ’ts PROPN ejpam-5960 480 9 and and CCONJ ejpam-5960 480 10 πœβ€² πœβ€² NOUN ejpam-5960 480 11 = = PUNCT ejpam-5960 480 12 { { PUNCT ejpam-5960 480 13 0𝑋 0𝑋 PROPN ejpam-5960 480 14 , , PUNCT ejpam-5960 480 15 π‘₯ π‘₯ PROPN ejpam-5960 480 16 3 3 NUM ejpam-5960 480 17 4 4 NUM ejpam-5960 480 18 , , PUNCT ejpam-5960 480 19 π‘₯ π‘₯ PROPN ejpam-5960 480 20 1 1 NUM ejpam-5960 480 21 9 9 NUM ejpam-5960 480 22 , , PUNCT ejpam-5960 480 23 1𝑋 1𝑋 PROPN ejpam-5960 480 24 } } PUNCT ejpam-5960 480 25 . . PUNCT ejpam-5960 481 1 firstly firstly ADV ejpam-5960 481 2 , , PUNCT ejpam-5960 481 3 we we PRON ejpam-5960 481 4 show show VERB ejpam-5960 481 5 that that SCONJ ejpam-5960 481 6 πœ‡ πœ‡ ADP ejpam-5960 481 7 = = X ejpam-5960 481 8 π‘₯ π‘₯ PROPN ejpam-5960 481 9 1 1 NUM ejpam-5960 481 10 2 2 NUM ejpam-5960 481 11 ∈ ∈ PROPN ejpam-5960 481 12 𝑀 𝑀 PROPN ejpam-5960 481 13 ( ( PUNCT ejpam-5960 481 14 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 481 15 ) ) PUNCT ejpam-5960 481 16 is be AUX ejpam-5960 481 17 𝑁𝛼–bounded 𝑁𝛼–bounde VERB ejpam-5960 481 18 set set NOUN ejpam-5960 481 19 . . PUNCT ejpam-5960 482 1 we we PRON ejpam-5960 482 2 suppose suppose VERB ejpam-5960 482 3 that that SCONJ ejpam-5960 482 4 𝑆 𝑆 PROPN ejpam-5960 482 5 = = PRON ejpam-5960 482 6 { { PUNCT ejpam-5960 482 7 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 482 8 : : PUNCT ejpam-5960 482 9 π‘₯ π‘₯ PROPN ejpam-5960 482 10 ∈ ∈ PROPN ejpam-5960 482 11 𝑋 𝑋 PROPN ejpam-5960 482 12 , , PUNCT ejpam-5960 482 13 𝛼 𝛼 PROPN ejpam-5960 482 14 ≀ ≀ NUM ejpam-5960 482 15 1 1 NUM ejpam-5960 482 16 2 2 NUM ejpam-5960 482 17 } } PUNCT ejpam-5960 482 18 is be AUX ejpam-5960 482 19 any any DET ejpam-5960 482 20 constant constant ADJ ejpam-5960 482 21 𝛼–molecular 𝛼–molecular ADJ ejpam-5960 482 22 net net NOUN ejpam-5960 482 23 in in ADP ejpam-5960 482 24 πœ‡. πœ‡. NOUN ejpam-5960 482 25 if if SCONJ ejpam-5960 482 26 𝛼 𝛼 NOUN ejpam-5960 482 27 ≀ ≀ NUM ejpam-5960 482 28 1 1 NUM ejpam-5960 482 29 2 2 NUM ejpam-5960 482 30 , , PUNCT ejpam-5960 482 31 we we PRON ejpam-5960 482 32 take take VERB ejpam-5960 482 33 π‘₯ π‘₯ DET ejpam-5960 482 34 ∈ ∈ NOUN ejpam-5960 482 35 𝑋 𝑋 NOUN ejpam-5960 482 36 so so SCONJ ejpam-5960 482 37 that that PRON ejpam-5960 482 38 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 482 39 ) ) PUNCT ejpam-5960 482 40 = = PUNCT ejpam-5960 482 41 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 482 42 βˆ‰ βˆ‰ PROPN ejpam-5960 482 43 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 482 44 ( ( PUNCT ejpam-5960 482 45 int(πœ† int(πœ† PROPN ejpam-5960 482 46 ) ) PUNCT ejpam-5960 482 47 ) ) PUNCT ejpam-5960 482 48 for for ADP ejpam-5960 482 49 each each DET ejpam-5960 482 50 πœ† πœ† DET ejpam-5960 482 51 ∈ ∈ PROPN ejpam-5960 482 52 𝑅π‘₯0.5 𝑅π‘₯0.5 NOUN ejpam-5960 482 53 = = SYM ejpam-5960 482 54 { { PUNCT ejpam-5960 482 55 0𝑋 0𝑋 PROPN ejpam-5960 482 56 , , PUNCT ejpam-5960 482 57 π‘₯ π‘₯ PROPN ejpam-5960 482 58 1 1 NUM ejpam-5960 482 59 9 9 NUM ejpam-5960 482 60 } } PUNCT ejpam-5960 482 61 , , PUNCT ejpam-5960 482 62 where where SCONJ ejpam-5960 482 63 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 482 64 ( ( PUNCT ejpam-5960 482 65 int(0𝑋 int(0𝑋 PROPN ejpam-5960 482 66 ) ) PUNCT ejpam-5960 482 67 ) ) PUNCT ejpam-5960 483 1 = = SYM ejpam-5960 483 2 0𝑋 0𝑋 NOUN ejpam-5960 483 3 and and CCONJ ejpam-5960 483 4 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 483 5 ( ( PUNCT ejpam-5960 483 6 int(π‘₯ int(π‘₯ PROPN ejpam-5960 483 7 1 1 NUM ejpam-5960 483 8 9 9 NUM ejpam-5960 483 9 ) ) PUNCT ejpam-5960 483 10 ) ) PUNCT ejpam-5960 484 1 = = SYM ejpam-5960 485 1 0𝑋. 0𝑋. NOUN ejpam-5960 485 2 then then ADV ejpam-5960 485 3 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 485 4 is be AUX ejpam-5960 485 5 a a DET ejpam-5960 485 6 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 485 7 point point NOUN ejpam-5960 485 8 of of ADP ejpam-5960 485 9 𝑆 𝑆 PROPN ejpam-5960 485 10 in in ADP ejpam-5960 485 11 1𝑋. 1𝑋. NUM ejpam-5960 485 12 thus thus ADV ejpam-5960 485 13 πœ‡ πœ‡ SCONJ ejpam-5960 485 14 = = X ejpam-5960 485 15 π‘₯0.5 π‘₯0.5 NOUN ejpam-5960 485 16 is be AUX ejpam-5960 485 17 a a DET ejpam-5960 485 18 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 485 19 set set NOUN ejpam-5960 485 20 . . PUNCT ejpam-5960 486 1 now now ADV ejpam-5960 486 2 , , PUNCT ejpam-5960 486 3 we we PRON ejpam-5960 486 4 show show VERB ejpam-5960 486 5 that that SCONJ ejpam-5960 486 6 πœ‡ πœ‡ SCONJ ejpam-5960 486 7 = = X ejpam-5960 486 8 π‘₯0.5 π‘₯0.5 NUM ejpam-5960 486 9 ∈ ∈ PROPN ejpam-5960 486 10 𝑀 𝑀 PROPN ejpam-5960 486 11 ( ( PUNCT ejpam-5960 486 12 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 486 13 ) ) PUNCT ejpam-5960 486 14 is be AUX ejpam-5960 486 15 not not PART ejpam-5960 486 16 a a DET ejpam-5960 486 17 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 486 18 set set VERB ejpam-5960 486 19 . . PUNCT ejpam-5960 487 1 indeed indeed ADV ejpam-5960 487 2 , , PUNCT ejpam-5960 487 3 ( ( PUNCT ejpam-5960 487 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 487 5 , , PUNCT ejpam-5960 487 6 𝜏 𝜏 NOUN ejpam-5960 487 7 ) ) PUNCT ejpam-5960 487 8 is be AUX ejpam-5960 487 9 not not PART ejpam-5960 487 10 𝐿𝑅2 𝐿𝑅2 PROPN ejpam-5960 487 11 – – PUNCT ejpam-5960 487 12 space space NOUN ejpam-5960 487 13 . . PUNCT ejpam-5960 488 1 since since SCONJ ejpam-5960 488 2 there there PRON ejpam-5960 488 3 is be VERB ejpam-5960 488 4 π‘₯0.8 π‘₯0.8 PROPN ejpam-5960 488 5 ∈ ∈ PROPN ejpam-5960 488 6 𝑀 𝑀 PROPN ejpam-5960 488 7 ( ( PUNCT ejpam-5960 488 8 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 488 9 ) ) PUNCT ejpam-5960 488 10 and and CCONJ ejpam-5960 488 11 there there PRON ejpam-5960 488 12 is be VERB ejpam-5960 488 13 πœ† πœ† PRON ejpam-5960 488 14 = = SYM ejpam-5960 488 15 π‘₯ π‘₯ PROPN ejpam-5960 488 16 3 3 NUM ejpam-5960 488 17 4 4 NUM ejpam-5960 488 18 ∈ ∈ NOUN ejpam-5960 488 19 𝑅π‘₯0.8 𝑅π‘₯0.8 NOUN ejpam-5960 488 20 such such ADJ ejpam-5960 488 21 that that PRON ejpam-5960 488 22 for for ADP ejpam-5960 488 23 each each DET ejpam-5960 488 24 πœ‚ πœ‚ PROPN ejpam-5960 488 25 ∈ ∈ PROPN ejpam-5960 488 26 𝑅π‘₯0.8 𝑅π‘₯0.8 NOUN ejpam-5960 488 27 = = SYM ejpam-5960 488 28 { { PUNCT ejpam-5960 488 29 0𝑋 0𝑋 PROPN ejpam-5960 488 30 , , PUNCT ejpam-5960 488 31 π‘₯ π‘₯ PROPN ejpam-5960 488 32 1 1 NUM ejpam-5960 488 33 9 9 NUM ejpam-5960 488 34 , , PUNCT ejpam-5960 488 35 π‘₯ π‘₯ PROPN ejpam-5960 488 36 3 3 NUM ejpam-5960 488 37 4 4 NUM ejpam-5960 488 38 } } PUNCT ejpam-5960 488 39 we we PRON ejpam-5960 488 40 have have VERB ejpam-5960 488 41 πœ† πœ† DET ejpam-5960 488 42 β‰° β‰° PROPN ejpam-5960 488 43 int(πœ‚ int(πœ‚ NOUN ejpam-5960 488 44 ) ) PUNCT ejpam-5960 488 45 , , PUNCT ejpam-5960 488 46 where where SCONJ ejpam-5960 488 47 int(0𝑋 int(0𝑋 PROPN ejpam-5960 488 48 ) ) PUNCT ejpam-5960 489 1 = = SYM ejpam-5960 489 2 0𝑋 0𝑋 PROPN ejpam-5960 489 3 , , PUNCT ejpam-5960 489 4 int(π‘₯ int(π‘₯ PROPN ejpam-5960 489 5 1 1 NUM ejpam-5960 489 6 9 9 NUM ejpam-5960 489 7 ) ) PUNCT ejpam-5960 489 8 = = SYM ejpam-5960 490 1 0𝑋 0𝑋 NOUN ejpam-5960 490 2 and and CCONJ ejpam-5960 490 3 int(π‘₯ int(π‘₯ PROPN ejpam-5960 490 4 3 3 NUM ejpam-5960 490 5 4 4 NUM ejpam-5960 490 6 ) ) PUNCT ejpam-5960 490 7 = = SYM ejpam-5960 491 1 π‘₯ π‘₯ DET ejpam-5960 491 2 1 1 NUM ejpam-5960 491 3 4 4 NUM ejpam-5960 491 4 . . PUNCT ejpam-5960 492 1 hence hence ADV ejpam-5960 492 2 ( ( PUNCT ejpam-5960 492 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 492 4 , , PUNCT ejpam-5960 492 5 𝜏 𝜏 NOUN ejpam-5960 492 6 ) ) PUNCT ejpam-5960 492 7 is be AUX ejpam-5960 492 8 not not PART ejpam-5960 492 9 a a DET ejpam-5960 492 10 𝐿𝑅2 𝐿𝑅2 PROPN ejpam-5960 492 11 – – PUNCT ejpam-5960 492 12 space space NOUN ejpam-5960 492 13 , , PUNCT ejpam-5960 492 14 and and CCONJ ejpam-5960 492 15 so so ADV ejpam-5960 492 16 πœ‡ πœ‡ SCONJ ejpam-5960 492 17 = = X ejpam-5960 492 18 π‘₯0.5 π‘₯0.5 NOUN ejpam-5960 492 19 is be AUX ejpam-5960 492 20 not not PART ejpam-5960 492 21 a a DET ejpam-5960 492 22 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 492 23 set set VERB ejpam-5960 492 24 . . PUNCT ejpam-5960 493 1 theorem theorem VERB ejpam-5960 493 2 3.29 3.29 NUM ejpam-5960 493 3 : : PUNCT ejpam-5960 493 4 every every DET ejpam-5960 493 5 πΏβˆ’subset πΏβˆ’subset ADJ ejpam-5960 493 6 with with ADP ejpam-5960 493 7 finite finite ADJ ejpam-5960 493 8 support support NOUN ejpam-5960 493 9 is be AUX ejpam-5960 493 10 a a DET ejpam-5960 493 11 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 493 12 set set NOUN ejpam-5960 493 13 . . PUNCT ejpam-5960 494 1 proof proof NOUN ejpam-5960 494 2 . . PUNCT ejpam-5960 495 1 let let VERB ejpam-5960 495 2 ( ( PUNCT ejpam-5960 495 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 495 4 , , PUNCT ejpam-5960 495 5 𝜏 𝜏 NOUN ejpam-5960 495 6 ) ) PUNCT ejpam-5960 495 7 be be AUX ejpam-5960 495 8 a a DET ejpam-5960 495 9 πΏβˆ’ts πΏβˆ’t NOUN ejpam-5960 495 10 . . PUNCT ejpam-5960 496 1 and and CCONJ ejpam-5960 496 2 πœ‡ πœ‡ ADP ejpam-5960 496 3 ∈ ∈ PROPN ejpam-5960 496 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 496 5 with with ADP ejpam-5960 496 6 finite finite ADJ ejpam-5960 496 7 support support NOUN ejpam-5960 496 8 . . PUNCT ejpam-5960 497 1 then then ADV ejpam-5960 497 2 by by ADP ejpam-5960 497 3 theorem theorem NOUN ejpam-5960 497 4 2.17 2.17 NUM ejpam-5960 497 5 ( ( PUNCT ejpam-5960 497 6 i i NOUN ejpam-5960 497 7 ) ) PUNCT ejpam-5960 497 8 , , PUNCT ejpam-5960 497 9 we we PRON ejpam-5960 497 10 have have VERB ejpam-5960 497 11 πœ‡ πœ‡ X ejpam-5960 497 12 is be AUX ejpam-5960 497 13 π‘π‘„π›Όβˆ’compact π‘π‘„π›Όβˆ’compact ADJ ejpam-5960 497 14 and and CCONJ ejpam-5960 497 15 by by ADP ejpam-5960 497 16 theorem theorem ADJ ejpam-5960 497 17 3.27 3.27 NUM ejpam-5960 497 18 ( ( PUNCT ejpam-5960 497 19 ii ii NOUN ejpam-5960 497 20 ) ) PUNCT ejpam-5960 497 21 we we PRON ejpam-5960 497 22 have have VERB ejpam-5960 497 23 πœ‡ πœ‡ ADP ejpam-5960 497 24 is be AUX ejpam-5960 497 25 a a DET ejpam-5960 497 26 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 497 27 set set NOUN ejpam-5960 497 28 . . PUNCT ejpam-5960 498 1 theorem theorem VERB ejpam-5960 498 2 3.30 3.30 NUM ejpam-5960 498 3 . . PUNCT ejpam-5960 499 1 let let VERB ejpam-5960 499 2 ( ( PUNCT ejpam-5960 499 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 499 4 , , PUNCT ejpam-5960 499 5 𝜏 𝜏 NOUN ejpam-5960 499 6 ) ) PUNCT ejpam-5960 499 7 be be VERB ejpam-5960 499 8 an an DET ejpam-5960 499 9 l l NOUN ejpam-5960 499 10 - - PUNCT ejpam-5960 499 11 ts ts NOUN ejpam-5960 499 12 and and CCONJ ejpam-5960 499 13 πœ‡ πœ‡ ADP ejpam-5960 499 14 be be VERB ejpam-5960 499 15 a a DET ejpam-5960 499 16 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 499 17 set set NOUN ejpam-5960 499 18 in in ADP ejpam-5960 499 19 ( ( PUNCT ejpam-5960 499 20 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 499 21 , , PUNCT ejpam-5960 499 22 𝜏 𝜏 NOUN ejpam-5960 499 23 ) ) PUNCT ejpam-5960 499 24 , , PUNCT ejpam-5960 499 25 then then ADV ejpam-5960 499 26 πœ‡ πœ‡ SCONJ ejpam-5960 499 27 is be AUX ejpam-5960 499 28 a a DET ejpam-5960 499 29 π‘π›Όβˆ’bounded π‘π›Όβˆ’bounde VERB ejpam-5960 499 30 set set NOUN ejpam-5960 499 31 in in ADP ejpam-5960 499 32 ( ( PUNCT ejpam-5960 499 33 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 499 34 , , PUNCT ejpam-5960 499 35 πœπ‘Œ πœπ‘Œ NOUN ejpam-5960 499 36 ) ) PUNCT ejpam-5960 499 37 . . PUNCT ejpam-5960 500 1 proof proof NOUN ejpam-5960 500 2 . . PUNCT ejpam-5960 501 1 let let VERB ejpam-5960 501 2 πœ‡ πœ‡ PART ejpam-5960 501 3 be be AUX ejpam-5960 501 4 a a DET ejpam-5960 501 5 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 501 6 set set NOUN ejpam-5960 501 7 in in ADP ejpam-5960 501 8 ( ( PUNCT ejpam-5960 501 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 501 10 , , PUNCT ejpam-5960 501 11 𝜏 𝜏 NOUN ejpam-5960 501 12 ) ) PUNCT ejpam-5960 501 13 and and CCONJ ejpam-5960 501 14 πœ™ πœ™ DET ejpam-5960 501 15 β‰  β‰  PROPN ejpam-5960 501 16 π‘Œ π‘Œ PROPN ejpam-5960 501 17 βŠ† βŠ† NUM ejpam-5960 501 18 𝑋. 𝑋. PROPN ejpam-5960 501 19 let let VERB ejpam-5960 501 20 ψ ψ X ejpam-5960 501 21 = = PUNCT ejpam-5960 501 22 { { PUNCT ejpam-5960 501 23 πœŒπ‘– πœŒπ‘– X ejpam-5960 501 24 = = PUNCT ejpam-5960 501 25 πœ‚π‘– πœ‚π‘– PROPN ejpam-5960 501 26 ∧ ∧ PROPN ejpam-5960 501 27 1π‘Œ 1π‘Œ NOUN ejpam-5960 501 28 : : PUNCT ejpam-5960 501 29 πœ‚π‘– πœ‚π‘– PROPN ejpam-5960 501 30 ∈ ∈ PROPN ejpam-5960 501 31 πœβ€² πœβ€² PROPN ejpam-5960 501 32 , , PUNCT ejpam-5960 501 33 𝑖 𝑖 SYM ejpam-5960 501 34 ∈ ∈ PROPN ejpam-5960 502 1 𝐼 𝐼 PROPN ejpam-5960 502 2 } } PUNCT ejpam-5960 502 3 βŠ† βŠ† NUM ejpam-5960 502 4 πœβ€² πœβ€² NOUN ejpam-5960 502 5 π‘Œ π‘Œ PROPN ejpam-5960 502 6 be be VERB ejpam-5960 502 7 an an DET ejpam-5960 502 8 𝛼–rf 𝛼–rf NOUN ejpam-5960 502 9 of of ADP ejpam-5960 502 10 1π‘Œ 1π‘Œ NOUN ejpam-5960 502 11 . . PUNCT ejpam-5960 503 1 hence hence ADV ejpam-5960 503 2 ψ ψ ADP ejpam-5960 503 3 βˆ— βˆ— NOUN ejpam-5960 503 4 = = SYM ejpam-5960 503 5 { { PUNCT ejpam-5960 503 6 πœ‚π‘– πœ‚π‘– X ejpam-5960 503 7 : : PUNCT ejpam-5960 503 8 𝑖 𝑖 SYM ejpam-5960 503 9 ∈ ∈ PROPN ejpam-5960 503 10 𝐼 𝐼 PROPN ejpam-5960 503 11 } } PUNCT ejpam-5960 503 12 βŠ† βŠ† NUM ejpam-5960 503 13 πœβ€² πœβ€² NOUN ejpam-5960 503 14 is be AUX ejpam-5960 503 15 an an DET ejpam-5960 503 16 𝛼–rf 𝛼–rf NUM ejpam-5960 503 17 of of ADP ejpam-5960 503 18 1𝑋. 1𝑋. NUM ejpam-5960 503 19 since since SCONJ ejpam-5960 503 20 πœ‡ πœ‡ ADV ejpam-5960 503 21 is be AUX ejpam-5960 503 22 a a DET ejpam-5960 503 23 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 503 24 set set NOUN ejpam-5960 503 25 in in ADP ejpam-5960 503 26 ( ( PUNCT ejpam-5960 503 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 503 28 , , PUNCT ejpam-5960 503 29 𝜏 𝜏 NOUN ejpam-5960 503 30 ) ) PUNCT ejpam-5960 503 31 , , PUNCT ejpam-5960 503 32 then then ADV ejpam-5960 503 33 there there PRON ejpam-5960 503 34 exists exist VERB ejpam-5960 503 35 Οˆβˆ— Οˆβˆ— NOUN ejpam-5960 503 36 β—¦ β—¦ NOUN ejpam-5960 503 37 = = SYM ejpam-5960 503 38 { { PUNCT ejpam-5960 503 39 πœ‚π‘–π‘š πœ‚π‘–π‘š NOUN ejpam-5960 503 40 : : PUNCT ejpam-5960 503 41 π‘š π‘š X ejpam-5960 503 42 = = SYM ejpam-5960 503 43 1 1 NUM ejpam-5960 503 44 , , PUNCT ejpam-5960 503 45 2 2 NUM ejpam-5960 503 46 , , PUNCT ejpam-5960 503 47 . . PUNCT ejpam-5960 503 48 . . PUNCT ejpam-5960 504 1 . . PUNCT ejpam-5960 505 1 , , PUNCT ejpam-5960 505 2 𝑛 𝑛 X ejpam-5960 505 3 } } PUNCT ejpam-5960 505 4 ∈ ∈ NOUN ejpam-5960 505 5 2(ψ 2(ψ NUM ejpam-5960 505 6 βˆ— βˆ— NOUN ejpam-5960 505 7 ) ) PUNCT ejpam-5960 505 8 such such ADJ ejpam-5960 505 9 that that SCONJ ejpam-5960 505 10 Οˆβˆ— Οˆβˆ— PROPN ejpam-5960 505 11 β—¦ β—¦ NOUN ejpam-5960 505 12 is be AUX ejpam-5960 505 13 an an DET ejpam-5960 505 14 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 505 15 of of ADP ejpam-5960 505 16 πœ‡ πœ‡ ADP ejpam-5960 505 17 and and CCONJ ejpam-5960 505 18 so so ADV ejpam-5960 505 19 ψ ψ X ejpam-5960 505 20 β—¦ β—¦ NOUN ejpam-5960 505 21 = = SYM ejpam-5960 505 22 { { PUNCT ejpam-5960 505 23 πœŒπ‘–π‘š πœŒπ‘–π‘š NOUN ejpam-5960 505 24 = = PUNCT ejpam-5960 505 25 πœ‚π‘–π‘š πœ‚π‘–π‘š PROPN ejpam-5960 505 26 ∧ ∧ PROPN ejpam-5960 505 27 1π‘Œ 1π‘Œ NOUN ejpam-5960 505 28 : : PUNCT ejpam-5960 505 29 π‘š π‘š X ejpam-5960 505 30 = = SYM ejpam-5960 505 31 1 1 NUM ejpam-5960 505 32 , , PUNCT ejpam-5960 505 33 2 2 NUM ejpam-5960 505 34 , , PUNCT ejpam-5960 505 35 . . PUNCT ejpam-5960 505 36 . . PUNCT ejpam-5960 506 1 . . PUNCT ejpam-5960 507 1 , , PUNCT ejpam-5960 507 2 𝑛 𝑛 X ejpam-5960 507 3 } } PUNCT ejpam-5960 507 4 ∈ ∈ NOUN ejpam-5960 507 5 2(ψ 2(ψ NUM ejpam-5960 507 6 ) ) PUNCT ejpam-5960 507 7 is be AUX ejpam-5960 507 8 an an DET ejpam-5960 507 9 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 507 10 of of ADP ejpam-5960 507 11 πœ‡. πœ‡. NOUN ejpam-5960 507 12 hence hence ADV ejpam-5960 507 13 πœ‡ πœ‡ ADV ejpam-5960 507 14 is be AUX ejpam-5960 507 15 a a DET ejpam-5960 507 16 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 507 17 set set NOUN ejpam-5960 507 18 in in ADP ejpam-5960 507 19 ( ( PUNCT ejpam-5960 507 20 πΏπ‘Œ πΏπ‘Œ PROPN ejpam-5960 507 21 , , PUNCT ejpam-5960 507 22 πœπ‘Œ πœπ‘Œ NOUN ejpam-5960 507 23 ) ) PUNCT ejpam-5960 507 24 . . PUNCT ejpam-5960 508 1 4 4 X ejpam-5960 508 2 . . X ejpam-5960 508 3 𝛼–nets 𝛼–net NOUN ejpam-5960 508 4 ’ ’ PART ejpam-5960 508 5 characterizations characterization NOUN ejpam-5960 508 6 of of ADP ejpam-5960 508 7 𝑁.𝛼–boundedness 𝑁.𝛼–boundedness X ejpam-5960 508 8 in in ADP ejpam-5960 508 9 this this DET ejpam-5960 508 10 section section NOUN ejpam-5960 508 11 , , PUNCT ejpam-5960 508 12 we we PRON ejpam-5960 508 13 give give VERB ejpam-5960 508 14 several several ADJ ejpam-5960 508 15 characterizations characterization NOUN ejpam-5960 508 16 of of ADP ejpam-5960 508 17 n.𝛼–boundedness n.𝛼–boundedness PUNCT ejpam-5960 508 18 in in ADP ejpam-5960 508 19 terms term NOUN ejpam-5960 508 20 of of ADP ejpam-5960 508 21 both both CCONJ ejpam-5960 508 22 the the DET ejpam-5960 508 23 upper upper ADJ ejpam-5960 508 24 𝛿–limit 𝛿–limit NOUN ejpam-5960 508 25 of of ADP ejpam-5960 508 26 nets net NOUN ejpam-5960 508 27 of of ADP ejpam-5960 508 28 𝐿–subsets 𝐿–subset NOUN ejpam-5960 508 29 and and CCONJ ejpam-5960 508 30 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 508 31 points point NOUN ejpam-5960 508 32 of of ADP ejpam-5960 508 33 constant constant ADJ ejpam-5960 508 34 molecular molecular ADJ ejpam-5960 508 35 𝛼–nets 𝛼–net NOUN ejpam-5960 508 36 . . PUNCT ejpam-5960 509 1 theorem theorem VERB ejpam-5960 509 2 4.1 4.1 NUM ejpam-5960 509 3 . . PUNCT ejpam-5960 510 1 let let VERB ejpam-5960 510 2 ( ( PUNCT ejpam-5960 510 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 510 4 , , PUNCT ejpam-5960 510 5 𝜏 𝜏 NOUN ejpam-5960 510 6 ) ) PUNCT ejpam-5960 510 7 be be VERB ejpam-5960 510 8 an an DET ejpam-5960 510 9 l l NOUN ejpam-5960 510 10 – – PUNCT ejpam-5960 510 11 ts ts NOUN ejpam-5960 510 12 , , PUNCT ejpam-5960 510 13 𝛼 𝛼 PROPN ejpam-5960 510 14 ∈ ∈ PROPN ejpam-5960 510 15 𝑀 𝑀 PROPN ejpam-5960 510 16 ( ( PUNCT ejpam-5960 510 17 𝐿 𝐿 PROPN ejpam-5960 510 18 ) ) PUNCT ejpam-5960 510 19 and and CCONJ ejpam-5960 510 20 πœ‡ πœ‡ X ejpam-5960 510 21 ∈ ∈ X ejpam-5960 510 22 𝐿𝑋. 𝐿𝑋. X ejpam-5960 510 23 then then ADV ejpam-5960 510 24 πœ‡ πœ‡ ADP ejpam-5960 510 25 is be AUX ejpam-5960 510 26 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 510 27 iff iff PROPN ejpam-5960 510 28 for for SCONJ ejpam-5960 510 29 each each DET ejpam-5960 510 30 constant constant ADJ ejpam-5960 510 31 molecular molecular ADJ ejpam-5960 510 32 𝛼–net 𝛼–net NOUN ejpam-5960 510 33 𝑆 𝑆 PROPN ejpam-5960 510 34 contained contain VERB ejpam-5960 510 35 in in ADP ejpam-5960 510 36 πœ‡ πœ‡ ADP ejpam-5960 510 37 has have VERB ejpam-5960 510 38 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 510 39 point point NOUN ejpam-5960 510 40 in in ADP ejpam-5960 510 41 𝑋 𝑋 NOUN ejpam-5960 510 42 with with ADP ejpam-5960 510 43 height height NOUN ejpam-5960 510 44 𝛼. 𝛼. NOUN ejpam-5960 510 45 proof proof NOUN ejpam-5960 510 46 . . PUNCT ejpam-5960 511 1 suppose suppose VERB ejpam-5960 511 2 that that SCONJ ejpam-5960 511 3 πœ‡ πœ‡ ADP ejpam-5960 511 4 is be AUX ejpam-5960 511 5 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 511 6 and and CCONJ ejpam-5960 511 7 let let VERB ejpam-5960 511 8 𝑆 𝑆 PROPN ejpam-5960 511 9 = = SYM ejpam-5960 511 10 { { PUNCT ejpam-5960 511 11 𝑆(𝑛 𝑆(𝑛 NOUN ejpam-5960 511 12 ) ) PUNCT ejpam-5960 511 13 : : PUNCT ejpam-5960 511 14 𝑛 𝑛 PROPN ejpam-5960 511 15 ∈ ∈ PROPN ejpam-5960 511 16 𝐷 𝐷 PROPN ejpam-5960 511 17 } } PUNCT ejpam-5960 511 18 be be AUX ejpam-5960 511 19 a a DET ejpam-5960 511 20 constant constant ADJ ejpam-5960 511 21 molecular molecular ADJ ejpam-5960 511 22 𝛼–net 𝛼–net NUM ejpam-5960 511 23 in in ADP ejpam-5960 511 24 πœ‡ πœ‡ ADP ejpam-5960 511 25 with with ADP ejpam-5960 511 26 height height NOUN ejpam-5960 511 27 𝛼. 𝛼. NOUN ejpam-5960 511 28 if if SCONJ ejpam-5960 511 29 𝑆 𝑆 PROPN ejpam-5960 511 30 does do AUX ejpam-5960 511 31 not not PART ejpam-5960 511 32 have have VERB ejpam-5960 511 33 any any DET ejpam-5960 511 34 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 511 35 point point NOUN ejpam-5960 511 36 in in ADP ejpam-5960 511 37 𝑋 𝑋 NOUN ejpam-5960 511 38 with with ADP ejpam-5960 511 39 height height NOUN ejpam-5960 511 40 𝛼. 𝛼. NOUN ejpam-5960 511 41 then then ADV ejpam-5960 511 42 for for ADP ejpam-5960 511 43 all all PRON ejpam-5960 511 44 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 511 45 ∈ ∈ PROPN ejpam-5960 511 46 𝑀 𝑀 PROPN ejpam-5960 511 47 ( ( PUNCT ejpam-5960 511 48 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 511 49 ) ) PUNCT ejpam-5960 511 50 , , PUNCT ejpam-5960 511 51 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 511 52 is be AUX ejpam-5960 511 53 not not PART ejpam-5960 511 54 a a DET ejpam-5960 511 55 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 511 56 point point NOUN ejpam-5960 511 57 of of ADP ejpam-5960 511 58 𝑆 𝑆 PROPN ejpam-5960 511 59 and and CCONJ ejpam-5960 511 60 so so ADV ejpam-5960 511 61 there there PRON ejpam-5960 511 62 exists exist VERB ejpam-5960 511 63 πœ†π‘₯ πœ†π‘₯ PROPN ejpam-5960 511 64 ∈ ∈ PROPN ejpam-5960 511 65 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 511 66 and and CCONJ ejpam-5960 511 67 𝑛π‘₯ 𝑛π‘₯ PROPN ejpam-5960 511 68 ∈ ∈ PROPN ejpam-5960 511 69 𝐷 𝐷 PROPN ejpam-5960 511 70 such such ADJ ejpam-5960 511 71 that that PRON ejpam-5960 511 72 for for ADP ejpam-5960 511 73 every every DET ejpam-5960 511 74 π‘š π‘š PROPN ejpam-5960 511 75 ∈ ∈ PROPN ejpam-5960 511 76 𝐷 𝐷 NOUN ejpam-5960 511 77 and and CCONJ ejpam-5960 511 78 π‘š π‘š X ejpam-5960 511 79 β‰₯ β‰₯ NUM ejpam-5960 511 80 𝑛π‘₯ 𝑛π‘₯ PROPN ejpam-5960 511 81 , , PUNCT ejpam-5960 511 82 then then ADV ejpam-5960 511 83 𝑆(π‘š 𝑆(π‘š X ejpam-5960 511 84 ) ) PUNCT ejpam-5960 511 85 ∈ ∈ NOUN ejpam-5960 511 86 𝑐𝑙 𝑐𝑙 X ejpam-5960 512 1 ( ( PUNCT ejpam-5960 512 2 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 512 3 ( ( PUNCT ejpam-5960 512 4 πœ†π‘₯ πœ†π‘₯ NOUN ejpam-5960 512 5 ) ) PUNCT ejpam-5960 512 6 ) ) PUNCT ejpam-5960 512 7 . . PUNCT ejpam-5960 513 1 put put VERB ejpam-5960 513 2 ψ ψ X ejpam-5960 513 3 = = PUNCT ejpam-5960 513 4 { { PUNCT ejpam-5960 513 5 πœ†π‘₯ πœ†π‘₯ NOUN ejpam-5960 513 6 : : PUNCT ejpam-5960 513 7 π‘₯ π‘₯ PUNCT ejpam-5960 513 8 ∈ ∈ PROPN ejpam-5960 513 9 𝑋 𝑋 NOUN ejpam-5960 513 10 and and CCONJ ejpam-5960 513 11 𝛼 𝛼 PROPN ejpam-5960 513 12 ∈ ∈ PROPN ejpam-5960 513 13 𝑀 𝑀 PROPN ejpam-5960 513 14 ( ( PUNCT ejpam-5960 513 15 𝐿 𝐿 PROPN ejpam-5960 513 16 ) ) PUNCT ejpam-5960 513 17 } } PUNCT ejpam-5960 513 18 is be AUX ejpam-5960 513 19 an an DET ejpam-5960 513 20 𝛼–rf 𝛼–rf NUM ejpam-5960 513 21 of of ADP ejpam-5960 513 22 1𝑋. 1𝑋. NUM ejpam-5960 513 23 since since SCONJ ejpam-5960 513 24 πœ‡ πœ‡ ADV ejpam-5960 513 25 is be AUX ejpam-5960 513 26 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 513 27 , , PUNCT ejpam-5960 513 28 then then ADV ejpam-5960 513 29 there there PRON ejpam-5960 513 30 exist exist VERB ejpam-5960 513 31 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 513 32 = = PUNCT ejpam-5960 513 33 { { PUNCT ejpam-5960 513 34 𝑐𝑙 𝑐𝑙 X ejpam-5960 513 35 ( ( PUNCT ejpam-5960 513 36 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 513 37 ( ( PUNCT ejpam-5960 513 38 πœ†π‘₯ πœ†π‘₯ NOUN ejpam-5960 513 39 ) ) PUNCT ejpam-5960 513 40 ) ) PUNCT ejpam-5960 513 41 : : PUNCT ejpam-5960 514 1 𝑖 𝑖 X ejpam-5960 514 2 = = SYM ejpam-5960 514 3 1 1 NUM ejpam-5960 514 4 , , PUNCT ejpam-5960 514 5 2 2 NUM ejpam-5960 514 6 , , PUNCT ejpam-5960 514 7 . . PUNCT ejpam-5960 514 8 . . PUNCT ejpam-5960 515 1 . . PUNCT ejpam-5960 516 1 , , PUNCT ejpam-5960 516 2 π‘˜ π‘˜ X ejpam-5960 516 3 } } PUNCT ejpam-5960 516 4 ∈ ∈ PROPN ejpam-5960 516 5 2(ψ 2(ψ NUM ejpam-5960 516 6 ) ) PUNCT ejpam-5960 516 7 such such ADJ ejpam-5960 516 8 that that SCONJ ejpam-5960 516 9 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 516 10 is be AUX ejpam-5960 516 11 an an DET ejpam-5960 516 12 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 516 13 of of ADP ejpam-5960 516 14 πœ‡. πœ‡. NOUN ejpam-5960 516 15 hence hence ADV ejpam-5960 516 16 , , PUNCT ejpam-5960 516 17 for for ADP ejpam-5960 516 18 each each DET ejpam-5960 516 19 𝑖 𝑖 NOUN ejpam-5960 516 20 ≀ ≀ NOUN ejpam-5960 517 1 π‘˜ π‘˜ PRON ejpam-5960 518 1 we we PRON ejpam-5960 518 2 have have VERB ejpam-5960 518 3 𝑛π‘₯𝑖 𝑛π‘₯𝑖 NOUN ejpam-5960 518 4 ∈ ∈ PROPN ejpam-5960 518 5 𝐷 𝐷 NOUN ejpam-5960 518 6 when when SCONJ ejpam-5960 518 7 π‘š π‘š PROPN ejpam-5960 518 8 β‰₯ β‰₯ PRON ejpam-5960 518 9 𝑛π‘₯𝑖 𝑛π‘₯𝑖 NOUN ejpam-5960 518 10 , , PUNCT ejpam-5960 518 11 𝑆(π‘š 𝑆(π‘š SYM ejpam-5960 518 12 ) ) PUNCT ejpam-5960 518 13 ∈ ∈ NOUN ejpam-5960 519 1 𝑐𝑙 𝑐𝑙 X ejpam-5960 519 2 ( ( PUNCT ejpam-5960 519 3 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 519 4 ( ( PUNCT ejpam-5960 519 5 πœ†π‘₯𝑖 πœ†π‘₯𝑖 NOUN ejpam-5960 519 6 ) ) PUNCT ejpam-5960 519 7 ) ) PUNCT ejpam-5960 519 8 . . PUNCT ejpam-5960 520 1 since since SCONJ ejpam-5960 520 2 𝐷 𝐷 PROPN ejpam-5960 520 3 is be AUX ejpam-5960 520 4 a a DET ejpam-5960 520 5 directed direct VERB ejpam-5960 520 6 set set NOUN ejpam-5960 520 7 , , PUNCT ejpam-5960 520 8 then then ADV ejpam-5960 520 9 there there PRON ejpam-5960 520 10 is be VERB ejpam-5960 520 11 π‘›π‘œ π‘›π‘œ PRON ejpam-5960 520 12 ∈ ∈ PROPN ejpam-5960 520 13 𝐷 𝐷 NOUN ejpam-5960 520 14 such such ADJ ejpam-5960 520 15 that that SCONJ ejpam-5960 520 16 π‘›π‘œ π‘›π‘œ PROPN ejpam-5960 520 17 β‰₯ β‰₯ NOUN ejpam-5960 520 18 𝑛π‘₯𝑖 𝑛π‘₯𝑖 NOUN ejpam-5960 520 19 ( ( PUNCT ejpam-5960 520 20 𝑖 𝑖 SYM ejpam-5960 520 21 = = SYM ejpam-5960 520 22 1 1 NUM ejpam-5960 520 23 , , PUNCT ejpam-5960 520 24 2 2 NUM ejpam-5960 520 25 , , PUNCT ejpam-5960 520 26 . . PUNCT ejpam-5960 520 27 . . PUNCT ejpam-5960 520 28 . . PUNCT ejpam-5960 521 1 , , PUNCT ejpam-5960 521 2 π‘˜ π‘˜ X ejpam-5960 521 3 ) ) PUNCT ejpam-5960 521 4 . . PUNCT ejpam-5960 522 1 hence hence ADV ejpam-5960 522 2 𝑆(π‘š 𝑆(π‘š X ejpam-5960 522 3 ) ) PUNCT ejpam-5960 522 4 ∈ ∈ NOUN ejpam-5960 522 5 βˆ§π‘˜ βˆ§π‘˜ PROPN ejpam-5960 522 6 𝑖=1 𝑖=1 X ejpam-5960 523 1 𝑐𝑙 𝑐𝑙 ADV ejpam-5960 523 2 ( ( PUNCT ejpam-5960 523 3 int(πœ†π‘₯𝑖 int(πœ†π‘₯𝑖 NOUN ejpam-5960 523 4 ) ) PUNCT ejpam-5960 523 5 ) ) PUNCT ejpam-5960 524 1 whenever whenever SCONJ ejpam-5960 524 2 π‘š π‘š X ejpam-5960 524 3 β‰₯ β‰₯ PRON ejpam-5960 524 4 π‘›π‘œ. π‘›π‘œ. VERB ejpam-5960 524 5 this this PRON ejpam-5960 524 6 means mean VERB ejpam-5960 524 7 that that SCONJ ejpam-5960 524 8 𝑆(π‘š 𝑆(π‘š PRON ejpam-5960 524 9 ) ) PUNCT ejpam-5960 524 10 not not PART ejpam-5960 524 11 have have VERB ejpam-5960 524 12 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 524 13 in in ADP ejpam-5960 524 14 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 524 15 , , PUNCT ejpam-5960 524 16 and and CCONJ ejpam-5960 524 17 so so ADV ejpam-5960 524 18 Οˆπ‘œ Οˆπ‘œ ADV ejpam-5960 524 19 is be AUX ejpam-5960 524 20 not not PART ejpam-5960 524 21 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 524 22 of of ADP ejpam-5960 524 23 πœ‡. πœ‡. NOUN ejpam-5960 524 24 this this PRON ejpam-5960 524 25 contradicts contradict VERB ejpam-5960 524 26 the the DET ejpam-5960 524 27 hypothesis hypothesis NOUN ejpam-5960 524 28 that that PRON ejpam-5960 524 29 πœ‡ πœ‡ ADV ejpam-5960 524 30 is be AUX ejpam-5960 524 31 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 524 32 . . PUNCT ejpam-5960 525 1 thus thus ADV ejpam-5960 525 2 𝑆 𝑆 PROPN ejpam-5960 525 3 has have VERB ejpam-5960 525 4 a a DET ejpam-5960 525 5 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 525 6 point point NOUN ejpam-5960 525 7 in in ADP ejpam-5960 525 8 𝑋 𝑋 NOUN ejpam-5960 525 9 with with ADP ejpam-5960 525 10 height height NOUN ejpam-5960 525 11 𝛼. 𝛼. PROPN ejpam-5960 525 12 n. n. PROPN ejpam-5960 525 13 a. a. PROPN ejpam-5960 525 14 alsaedi alsaedi PROPN ejpam-5960 525 15 / / SYM ejpam-5960 525 16 eur eur PROPN ejpam-5960 525 17 . . PUNCT ejpam-5960 526 1 j. j. PROPN ejpam-5960 526 2 pure pure PROPN ejpam-5960 526 3 appl appl PROPN ejpam-5960 526 4 . . PROPN ejpam-5960 526 5 math math PROPN ejpam-5960 526 6 , , PUNCT ejpam-5960 526 7 18 18 NUM ejpam-5960 526 8 ( ( PUNCT ejpam-5960 526 9 4 4 NUM ejpam-5960 526 10 ) ) PUNCT ejpam-5960 526 11 ( ( PUNCT ejpam-5960 526 12 2025 2025 NUM ejpam-5960 526 13 ) ) PUNCT ejpam-5960 526 14 , , PUNCT ejpam-5960 526 15 5960 5960 NUM ejpam-5960 526 16 17 17 NUM ejpam-5960 526 17 of of ADP ejpam-5960 526 18 22 22 NUM ejpam-5960 526 19 conversely conversely ADV ejpam-5960 526 20 , , PUNCT ejpam-5960 526 21 suppose suppose VERB ejpam-5960 526 22 that that SCONJ ejpam-5960 526 23 πœ‡ πœ‡ ADP ejpam-5960 526 24 is be AUX ejpam-5960 526 25 not not PART ejpam-5960 526 26 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 526 27 . . PUNCT ejpam-5960 527 1 then then ADV ejpam-5960 527 2 there there PRON ejpam-5960 527 3 exist exist VERB ejpam-5960 527 4 𝛼 𝛼 PRON ejpam-5960 527 5 ∈ ∈ PROPN ejpam-5960 527 6 𝑀 𝑀 PROPN ejpam-5960 527 7 ( ( PUNCT ejpam-5960 527 8 𝐿 𝐿 PROPN ejpam-5960 527 9 ) ) PUNCT ejpam-5960 527 10 and and CCONJ ejpam-5960 527 11 a a DET ejpam-5960 527 12 family family NOUN ejpam-5960 527 13 ψ ψ X ejpam-5960 527 14 which which PRON ejpam-5960 527 15 is be AUX ejpam-5960 527 16 an an DET ejpam-5960 527 17 𝛼–rf 𝛼–rf NOUN ejpam-5960 527 18 of of ADP ejpam-5960 527 19 1𝑋 1𝑋 NOUN ejpam-5960 527 20 , , PUNCT ejpam-5960 527 21 but but CCONJ ejpam-5960 527 22 for for ADP ejpam-5960 527 23 any any DET ejpam-5960 527 24 family family NOUN ejpam-5960 527 25 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 527 26 ∈ ∈ PROPN ejpam-5960 527 27 2(ψ 2(ψ NUM ejpam-5960 527 28 ) ) PUNCT ejpam-5960 527 29 , , PUNCT ejpam-5960 527 30 we we PRON ejpam-5960 527 31 have have AUX ejpam-5960 527 32 Οˆπ‘œ Οˆπ‘œ PART ejpam-5960 527 33 is be AUX ejpam-5960 527 34 not not PART ejpam-5960 527 35 an an DET ejpam-5960 527 36 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 527 37 of of ADP ejpam-5960 527 38 πœ‡. πœ‡. NOUN ejpam-5960 528 1 then then ADV ejpam-5960 528 2 there there PRON ejpam-5960 528 3 is be VERB ejpam-5960 528 4 a a DET ejpam-5960 528 5 point point NOUN ejpam-5960 528 6 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 528 7 ∈ ∈ PROPN ejpam-5960 528 8 πœ‡ πœ‡ ADP ejpam-5960 528 9 with with ADP ejpam-5960 528 10 height height NOUN ejpam-5960 528 11 𝛼 𝛼 PRON ejpam-5960 528 12 such such ADJ ejpam-5960 528 13 that that SCONJ ejpam-5960 528 14 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 528 15 ≀ ≀ ADJ ejpam-5960 528 16 βˆ§Οˆπ‘œ βˆ§Οˆπ‘œ NOUN ejpam-5960 528 17 and and CCONJ ejpam-5960 528 18 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 528 19 is be AUX ejpam-5960 528 20 denoted denote VERB ejpam-5960 528 21 by by ADP ejpam-5960 528 22 ( ( PUNCT ejpam-5960 528 23 π‘₯(Οˆπ‘œ))𝛼. π‘₯(Οˆπ‘œ))𝛼. VERB ejpam-5960 528 24 since since SCONJ ejpam-5960 528 25 2(ψ 2(ψ NUM ejpam-5960 528 26 ) ) PUNCT ejpam-5960 528 27 is be AUX ejpam-5960 528 28 a a DET ejpam-5960 528 29 directed direct VERB ejpam-5960 528 30 set set NOUN ejpam-5960 528 31 with with ADP ejpam-5960 528 32 relation relation NOUN ejpam-5960 528 33 ≀ ≀ NOUN ejpam-5960 528 34 , , PUNCT ejpam-5960 528 35 then then ADV ejpam-5960 528 36 𝑆 𝑆 PROPN ejpam-5960 528 37 = = SYM ejpam-5960 528 38 { { PUNCT ejpam-5960 528 39 π‘₯((Οˆπ‘œ))𝛼 π‘₯((Οˆπ‘œ))𝛼 ADJ ejpam-5960 528 40 : : PUNCT ejpam-5960 528 41 Οˆπ‘œ Οˆπ‘œ PART ejpam-5960 528 42 ∈ ∈ PROPN ejpam-5960 528 43 2(ψ 2(ψ NUM ejpam-5960 528 44 ) ) PUNCT ejpam-5960 528 45 } } PUNCT ejpam-5960 528 46 is be AUX ejpam-5960 528 47 a a DET ejpam-5960 528 48 constant constant ADJ ejpam-5960 528 49 molecular molecular ADJ ejpam-5960 528 50 𝛼–net 𝛼–net NUM ejpam-5960 528 51 in in ADP ejpam-5960 528 52 πœ‡. πœ‡. NOUN ejpam-5960 528 53 take take VERB ejpam-5960 528 54 an an DET ejpam-5960 528 55 arbitrary arbitrary ADJ ejpam-5960 528 56 point point NOUN ejpam-5960 528 57 𝑦𝛼 𝑦𝛼 NOUN ejpam-5960 528 58 in in ADP ejpam-5960 528 59 𝑋 𝑋 NOUN ejpam-5960 528 60 with with ADP ejpam-5960 528 61 height height NOUN ejpam-5960 528 62 𝛼 𝛼 NOUN ejpam-5960 528 63 , , PUNCT ejpam-5960 528 64 since since SCONJ ejpam-5960 528 65 ψ ψ NOUN ejpam-5960 528 66 is be AUX ejpam-5960 528 67 an an DET ejpam-5960 528 68 𝛼–rf 𝛼–rf NOUN ejpam-5960 528 69 of of ADP ejpam-5960 528 70 1𝑋 1𝑋 NOUN ejpam-5960 528 71 , , PUNCT ejpam-5960 528 72 then then ADV ejpam-5960 528 73 there there PRON ejpam-5960 528 74 is be VERB ejpam-5960 528 75 πœ† πœ† DET ejpam-5960 528 76 ∈ ∈ PROPN ejpam-5960 528 77 ψ ψ NOUN ejpam-5960 528 78 such such ADJ ejpam-5960 528 79 that that SCONJ ejpam-5960 528 80 πœ† πœ† DET ejpam-5960 528 81 ∈ ∈ PROPN ejpam-5960 528 82 𝑅𝑦𝛼 𝑅𝑦𝛼 PROPN ejpam-5960 528 83 . . PUNCT ejpam-5960 529 1 hence hence ADV ejpam-5960 529 2 , , PUNCT ejpam-5960 529 3 for for SCONJ ejpam-5960 529 4 each each PRON ejpam-5960 529 5 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 529 6 ∈ ∈ PROPN ejpam-5960 529 7 2(ψ 2(ψ NUM ejpam-5960 529 8 ) ) PUNCT ejpam-5960 529 9 such such ADJ ejpam-5960 529 10 that that SCONJ ejpam-5960 529 11 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 529 12 ( ( PUNCT ejpam-5960 529 13 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 529 14 ( ( PUNCT ejpam-5960 529 15 πœ† πœ† NOUN ejpam-5960 529 16 ) ) PUNCT ejpam-5960 529 17 ) ) PUNCT ejpam-5960 529 18 ∈ ∈ PROPN ejpam-5960 529 19 Οˆπ‘œ Οˆπ‘œ X ejpam-5960 529 20 , , PUNCT ejpam-5960 529 21 there there PRON ejpam-5960 529 22 is be VERB ejpam-5960 529 23 ( ( PUNCT ejpam-5960 529 24 π‘₯(Οˆπ‘œ))𝛼 π‘₯(Οˆπ‘œ))𝛼 PROPN ejpam-5960 529 25 ≀ ≀ NUM ejpam-5960 529 26 βˆ§Οˆπ‘œ βˆ§Οˆπ‘œ PROPN ejpam-5960 529 27 ≀ ≀ PUNCT ejpam-5960 529 28 𝑐𝑙 𝑐𝑙 ADP ejpam-5960 529 29 ( ( PUNCT ejpam-5960 529 30 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 529 31 ( ( PUNCT ejpam-5960 529 32 πœ† πœ† NOUN ejpam-5960 529 33 ) ) PUNCT ejpam-5960 529 34 ) ) PUNCT ejpam-5960 529 35 , , PUNCT ejpam-5960 529 36 i.e. i.e. X ejpam-5960 529 37 , , PUNCT ejpam-5960 529 38 𝑆 𝑆 PROPN ejpam-5960 529 39 ≀ ≀ PROPN ejpam-5960 529 40 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 529 41 ( ( PUNCT ejpam-5960 529 42 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 529 43 ( ( PUNCT ejpam-5960 529 44 πœ† πœ† NOUN ejpam-5960 529 45 ) ) PUNCT ejpam-5960 529 46 ) ) PUNCT ejpam-5960 529 47 . . PUNCT ejpam-5960 530 1 this this PRON ejpam-5960 530 2 shows show VERB ejpam-5960 530 3 that that SCONJ ejpam-5960 530 4 𝑦𝛼 𝑦𝛼 PROPN ejpam-5960 530 5 is be AUX ejpam-5960 530 6 not not PART ejpam-5960 530 7 a a DET ejpam-5960 530 8 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 530 9 point point NOUN ejpam-5960 530 10 of of ADP ejpam-5960 530 11 𝑆 𝑆 PROPN ejpam-5960 530 12 in in ADP ejpam-5960 530 13 𝑋 𝑋 PROPN ejpam-5960 530 14 with with ADP ejpam-5960 530 15 height height NOUN ejpam-5960 530 16 𝛼 𝛼 PROPN ejpam-5960 530 17 , , PUNCT ejpam-5960 530 18 which which PRON ejpam-5960 530 19 contradicts contradict VERB ejpam-5960 530 20 the the DET ejpam-5960 530 21 hypothesis hypothesis NOUN ejpam-5960 530 22 . . PUNCT ejpam-5960 531 1 thus thus ADV ejpam-5960 531 2 πœ‡ πœ‡ PRON ejpam-5960 531 3 is be AUX ejpam-5960 531 4 a a DET ejpam-5960 531 5 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 531 6 . . PUNCT ejpam-5960 532 1 theorem theorem VERB ejpam-5960 532 2 4.2 4.2 NUM ejpam-5960 532 3 . . PUNCT ejpam-5960 533 1 ( ( PUNCT ejpam-5960 533 2 alexander alexander PROPN ejpam-5960 533 3 ’s ’s PART ejpam-5960 533 4 subbase subbase PROPN ejpam-5960 533 5 lemma lemma PROPN ejpam-5960 533 6 ) ) PUNCT ejpam-5960 533 7 suppose suppose VERB ejpam-5960 533 8 that that SCONJ ejpam-5960 533 9 ( ( PUNCT ejpam-5960 533 10 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 533 11 , , PUNCT ejpam-5960 533 12 𝜏 𝜏 NOUN ejpam-5960 533 13 ) ) PUNCT ejpam-5960 533 14 is be AUX ejpam-5960 533 15 a a DET ejpam-5960 533 16 l l NOUN ejpam-5960 533 17 – – PUNCT ejpam-5960 533 18 ts ts NOUN ejpam-5960 533 19 , , PUNCT ejpam-5960 533 20 𝛼 𝛼 PROPN ejpam-5960 533 21 ∈ ∈ PROPN ejpam-5960 533 22 𝑀 𝑀 PROPN ejpam-5960 533 23 ( ( PUNCT ejpam-5960 533 24 𝐿 𝐿 PROPN ejpam-5960 533 25 ) ) PUNCT ejpam-5960 533 26 , , PUNCT ejpam-5960 533 27 πœ‡ πœ‡ SCONJ ejpam-5960 533 28 ∈ ∈ ADP ejpam-5960 533 29 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 533 30 and and CCONJ ejpam-5960 533 31 πœ‰ πœ‰ PROPN ejpam-5960 533 32 is be AUX ejpam-5960 533 33 a a DET ejpam-5960 533 34 subbase subbase NOUN ejpam-5960 533 35 of of ADP ejpam-5960 533 36 πœβ€². πœβ€². NOUN ejpam-5960 533 37 if if SCONJ ejpam-5960 533 38 for for ADP ejpam-5960 533 39 each each DET ejpam-5960 533 40 𝛼–rf 𝛼–rf NOUN ejpam-5960 533 41 ψ ψ NOUN ejpam-5960 533 42 of of ADP ejpam-5960 533 43 1𝑋 1𝑋 PROPN ejpam-5960 533 44 consisting consist VERB ejpam-5960 533 45 of of ADP ejpam-5960 533 46 elements element NOUN ejpam-5960 533 47 of of ADP ejpam-5960 533 48 πœ‰ πœ‰ NOUN ejpam-5960 533 49 , , PUNCT ejpam-5960 533 50 there there PRON ejpam-5960 533 51 is be VERB ejpam-5960 533 52 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 533 53 ∈ ∈ PROPN ejpam-5960 533 54 2(ψ 2(ψ NUM ejpam-5960 533 55 ) ) PUNCT ejpam-5960 533 56 which which PRON ejpam-5960 533 57 is be AUX ejpam-5960 533 58 an an DET ejpam-5960 533 59 𝛼–rcrf 𝛼–rcrf PROPN ejpam-5960 533 60 of of ADP ejpam-5960 533 61 πœ‡ πœ‡ ADP ejpam-5960 533 62 , , PUNCT ejpam-5960 533 63 then then ADV ejpam-5960 533 64 πœ‡ πœ‡ SCONJ ejpam-5960 533 65 is be AUX ejpam-5960 533 66 a a DET ejpam-5960 533 67 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 533 68 set set NOUN ejpam-5960 533 69 . . PUNCT ejpam-5960 534 1 proof proof NOUN ejpam-5960 534 2 . . PUNCT ejpam-5960 535 1 it it PRON ejpam-5960 535 2 is be AUX ejpam-5960 535 3 similar similar ADJ ejpam-5960 535 4 to to AUX ejpam-5960 535 5 theorem theorem VERB ejpam-5960 535 6 5.1 5.1 NUM ejpam-5960 535 7 in in ADP ejpam-5960 535 8 [ [ X ejpam-5960 535 9 15 15 NUM ejpam-5960 535 10 ] ] PUNCT ejpam-5960 535 11 . . PUNCT ejpam-5960 536 1 theorem theorem VERB ejpam-5960 536 2 4.3 4.3 NUM ejpam-5960 536 3 . . PUNCT ejpam-5960 537 1 let let VERB ejpam-5960 537 2 { { PUNCT ejpam-5960 537 3 ( ( PUNCT ejpam-5960 537 4 𝐿𝑋𝑖 𝐿𝑋𝑖 NOUN ejpam-5960 537 5 , , PUNCT ejpam-5960 537 6 πœπ‘– πœπ‘– PROPN ejpam-5960 537 7 ) ) PUNCT ejpam-5960 537 8 : : PUNCT ejpam-5960 538 1 𝑖 𝑖 PUNCT ejpam-5960 538 2 ∈ ∈ PROPN ejpam-5960 539 1 𝐼 𝐼 PROPN ejpam-5960 539 2 } } PUNCT ejpam-5960 539 3 be be VERB ejpam-5960 539 4 a a DET ejpam-5960 539 5 𝐿–ts 𝐿–ts NOUN ejpam-5960 539 6 ’s ’s NOUN ejpam-5960 539 7 and and CCONJ ejpam-5960 539 8 πœ‡π‘– πœ‡π‘– PART ejpam-5960 539 9 be be AUX ejpam-5960 539 10 a a DET ejpam-5960 539 11 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 539 12 set set NOUN ejpam-5960 539 13 in in ADP ejpam-5960 539 14 ( ( PUNCT ejpam-5960 539 15 𝐿𝑋𝑖 𝐿𝑋𝑖 NOUN ejpam-5960 539 16 , , PUNCT ejpam-5960 539 17 πœπ‘– πœπ‘– NOUN ejpam-5960 539 18 ) ) PUNCT ejpam-5960 539 19 for for ADP ejpam-5960 539 20 each each DET ejpam-5960 539 21 𝑖 𝑖 SYM ejpam-5960 539 22 ∈ ∈ PROPN ejpam-5960 539 23 𝐼 𝐼 PROPN ejpam-5960 539 24 , , PUNCT ejpam-5960 539 25 then then ADV ejpam-5960 539 26 the the DET ejpam-5960 539 27 product product NOUN ejpam-5960 539 28 set set VERB ejpam-5960 539 29 πœ‡ πœ‡ ADP ejpam-5960 539 30 = = SYM ejpam-5960 539 31 ∏ ∏ X ejpam-5960 539 32 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 539 33 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 539 34 is be AUX ejpam-5960 539 35 a a DET ejpam-5960 539 36 𝑁𝛼–bounded 𝑁𝛼–bounded PROPN ejpam-5960 539 37 in in ADP ejpam-5960 539 38 the the DET ejpam-5960 539 39 product product NOUN ejpam-5960 539 40 space space NOUN ejpam-5960 539 41 . . PUNCT ejpam-5960 540 1 proof proof NOUN ejpam-5960 540 2 . . PUNCT ejpam-5960 541 1 let let VERB ejpam-5960 541 2 𝛼 𝛼 PRON ejpam-5960 541 3 ∈ ∈ PROPN ejpam-5960 541 4 𝑀 𝑀 PROPN ejpam-5960 541 5 ( ( PUNCT ejpam-5960 541 6 𝐿 𝐿 PROPN ejpam-5960 541 7 ) ) PUNCT ejpam-5960 541 8 and and CCONJ ejpam-5960 541 9 let let VERB ejpam-5960 541 10 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 541 11 ∈ ∈ PROPN ejpam-5960 541 12 𝐿𝑋𝑖 𝐿𝑋𝑖 NOUN ejpam-5960 541 13 be be AUX ejpam-5960 541 14 a a DET ejpam-5960 541 15 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 541 16 set set NOUN ejpam-5960 541 17 in in ADP ejpam-5960 541 18 ( ( PUNCT ejpam-5960 541 19 𝐿𝑋𝑖 𝐿𝑋𝑖 NOUN ejpam-5960 541 20 , , PUNCT ejpam-5960 541 21 πœπ‘– πœπ‘– NOUN ejpam-5960 541 22 ) ) PUNCT ejpam-5960 541 23 for for ADP ejpam-5960 541 24 each each DET ejpam-5960 541 25 𝑖 𝑖 SYM ejpam-5960 541 26 ∈ ∈ PROPN ejpam-5960 541 27 𝐼. 𝐼. PROPN ejpam-5960 541 28 let let VERB ejpam-5960 541 29 the the DET ejpam-5960 541 30 set set NOUN ejpam-5960 541 31 πœ‰ πœ‰ VERB ejpam-5960 541 32 = = PUNCT ejpam-5960 541 33 { { PUNCT ejpam-5960 541 34 π‘ƒβˆ’1 π‘ƒβˆ’1 NOUN ejpam-5960 541 35 𝑖 𝑖 X ejpam-5960 541 36 ( ( PUNCT ejpam-5960 541 37 πœ†π‘– πœ†π‘– NOUN ejpam-5960 541 38 ) ) PUNCT ejpam-5960 541 39 : : PUNCT ejpam-5960 542 1 𝑖 𝑖 PUNCT ejpam-5960 542 2 ∈ ∈ PROPN ejpam-5960 542 3 𝐼 𝐼 PROPN ejpam-5960 542 4 , , PUNCT ejpam-5960 542 5 πœ†π‘– πœ†π‘– PROPN ejpam-5960 542 6 ∈ ∈ PROPN ejpam-5960 542 7 πœβ€² πœβ€² PROPN ejpam-5960 542 8 𝑖 𝑖 AUX ejpam-5960 542 9 } } PUNCT ejpam-5960 542 10 be be AUX ejpam-5960 542 11 a a DET ejpam-5960 542 12 subbase subbase NOUN ejpam-5960 542 13 of of ADP ejpam-5960 542 14 the the DET ejpam-5960 542 15 family family NOUN ejpam-5960 542 16 of of ADP ejpam-5960 542 17 all all DET ejpam-5960 542 18 closed closed ADJ ejpam-5960 542 19 sets set NOUN ejpam-5960 542 20 of of ADP ejpam-5960 542 21 the the DET ejpam-5960 542 22 product product NOUN ejpam-5960 542 23 space space NOUN ejpam-5960 542 24 . . PUNCT ejpam-5960 543 1 to to PART ejpam-5960 543 2 show show VERB ejpam-5960 543 3 that that SCONJ ejpam-5960 543 4 ∏ ∏ PROPN ejpam-5960 543 5 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 543 6 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 543 7 is be AUX ejpam-5960 543 8 a a DET ejpam-5960 543 9 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 543 10 set set NOUN ejpam-5960 543 11 , , PUNCT ejpam-5960 543 12 we we PRON ejpam-5960 543 13 only only ADV ejpam-5960 543 14 need need VERB ejpam-5960 543 15 to to PART ejpam-5960 543 16 verify verify VERB ejpam-5960 543 17 that that PRON ejpam-5960 543 18 for for ADP ejpam-5960 543 19 each each DET ejpam-5960 543 20 𝛼–rf 𝛼–rf NOUN ejpam-5960 543 21 , , PUNCT ejpam-5960 543 22 ψ ψ VERB ejpam-5960 543 23 βŠ† βŠ† NUM ejpam-5960 543 24 πœ‰ πœ‰ NOUN ejpam-5960 543 25 of of ADP ejpam-5960 543 26 the the DET ejpam-5960 543 27 set set NOUN ejpam-5960 543 28 ∏ ∏ PROPN ejpam-5960 543 29 π‘–βˆˆπΌ π‘–βˆˆπΌ PRON ejpam-5960 543 30 𝑋𝑖 𝑋𝑖 PROPN ejpam-5960 543 31 there there PRON ejpam-5960 543 32 exists exist VERB ejpam-5960 543 33 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 543 34 ∈ ∈ PROPN ejpam-5960 543 35 2(ψ 2(ψ NUM ejpam-5960 543 36 ) ) PUNCT ejpam-5960 543 37 such such ADJ ejpam-5960 543 38 that that SCONJ ejpam-5960 543 39 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 543 40 is be AUX ejpam-5960 543 41 an an DET ejpam-5960 543 42 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 543 43 of of ADP ejpam-5960 543 44 the the DET ejpam-5960 543 45 set set NOUN ejpam-5960 543 46 ∏ ∏ PROPN ejpam-5960 544 1 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 544 2 πœ‡π‘–. πœ‡π‘–. NOUN ejpam-5960 544 3 let let VERB ejpam-5960 544 4 ψ ψ X ejpam-5960 544 5 = = SYM ejpam-5960 545 1 βˆ¨π‘›βˆˆn{π‘ƒβˆ’1 βˆ¨π‘›βˆˆn{π‘ƒβˆ’1 VERB ejpam-5960 545 2 𝑖𝑛 𝑖𝑛 INTJ ejpam-5960 546 1 ( ( PUNCT ejpam-5960 546 2 πœ† πœ† X ejpam-5960 546 3 ) ) PUNCT ejpam-5960 546 4 : : PUNCT ejpam-5960 547 1 πœ† πœ† X ejpam-5960 547 2 ∈ ∈ PROPN ejpam-5960 548 1 β„œπ‘–π‘› β„œπ‘–π‘› PROPN ejpam-5960 548 2 , , PUNCT ejpam-5960 548 3 β„œπ‘–π‘› β„œπ‘–π‘› PROPN ejpam-5960 548 4 βŠ† βŠ† NUM ejpam-5960 548 5 πœβ€² πœβ€² NOUN ejpam-5960 548 6 𝑖𝑛 𝑖𝑛 NOUN ejpam-5960 548 7 } } PUNCT ejpam-5960 548 8 . . PUNCT ejpam-5960 549 1 now now ADV ejpam-5960 549 2 , , PUNCT ejpam-5960 549 3 we we PRON ejpam-5960 549 4 consider consider VERB ejpam-5960 549 5 the the DET ejpam-5960 549 6 following follow VERB ejpam-5960 549 7 two two NUM ejpam-5960 549 8 cases case NOUN ejpam-5960 549 9 : : PUNCT ejpam-5960 549 10 ( ( PUNCT ejpam-5960 549 11 i i NOUN ejpam-5960 549 12 ) ) PUNCT ejpam-5960 549 13 there there PRON ejpam-5960 549 14 exists exist VERB ejpam-5960 549 15 π‘–π‘œ π‘–π‘œ ADP ejpam-5960 549 16 ∈ ∈ PROPN ejpam-5960 549 17 𝐼 𝐼 ADP ejpam-5960 549 18 such such DET ejpam-5960 549 19 that that SCONJ ejpam-5960 549 20 no no DET ejpam-5960 549 21 molecular molecular NOUN ejpam-5960 549 22 with with ADP ejpam-5960 549 23 height height NOUN ejpam-5960 549 24 𝛼 𝛼 NOUN ejpam-5960 549 25 is be AUX ejpam-5960 549 26 contained contain VERB ejpam-5960 549 27 in in ADP ejpam-5960 549 28 πœ‡π‘–π‘œ πœ‡π‘–π‘œ NOUN ejpam-5960 549 29 . . PUNCT ejpam-5960 550 1 then then ADV ejpam-5960 550 2 by by ADP ejpam-5960 550 3 the the DET ejpam-5960 550 4 definition definition NOUN ejpam-5960 550 5 of of ADP ejpam-5960 550 6 a a DET ejpam-5960 550 7 product product NOUN ejpam-5960 550 8 set set VERB ejpam-5960 550 9 ∏ ∏ PROPN ejpam-5960 550 10 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 550 11 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 550 12 it it PRON ejpam-5960 550 13 follows follow VERB ejpam-5960 550 14 immediately immediately ADV ejpam-5960 550 15 that that SCONJ ejpam-5960 550 16 no no DET ejpam-5960 550 17 point point NOUN ejpam-5960 550 18 no no DET ejpam-5960 550 19 molecular molecular ADJ ejpam-5960 550 20 with with ADP ejpam-5960 550 21 height height NOUN ejpam-5960 550 22 𝛼 𝛼 NOUN ejpam-5960 550 23 is be AUX ejpam-5960 550 24 contained contain VERB ejpam-5960 550 25 in in ADP ejpam-5960 550 26 ∏ ∏ PROPN ejpam-5960 550 27 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 550 28 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 550 29 and and CCONJ ejpam-5960 550 30 hence hence ADV ejpam-5960 550 31 for for ADP ejpam-5960 550 32 each each PRON ejpam-5960 550 33 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 550 34 ∈ ∈ PROPN ejpam-5960 550 35 2(ψ 2(ψ NUM ejpam-5960 550 36 ) ) PUNCT ejpam-5960 550 37 , , PUNCT ejpam-5960 550 38 we we PRON ejpam-5960 550 39 have have AUX ejpam-5960 550 40 Οˆπ‘œ Οˆπ‘œ PART ejpam-5960 550 41 is be AUX ejpam-5960 550 42 an an DET ejpam-5960 550 43 𝛼–rcrf 𝛼–rcrf PROPN ejpam-5960 550 44 of of ADP ejpam-5960 550 45 ∏ ∏ PROPN ejpam-5960 550 46 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 550 47 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 550 48 and and CCONJ ejpam-5960 550 49 so so ADV ejpam-5960 550 50 ∏ ∏ PROPN ejpam-5960 550 51 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 550 52 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 550 53 is be AUX ejpam-5960 550 54 a a DET ejpam-5960 550 55 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 550 56 set set NOUN ejpam-5960 550 57 . . PUNCT ejpam-5960 551 1 ( ( PUNCT ejpam-5960 551 2 ii ii NOUN ejpam-5960 551 3 ) ) PUNCT ejpam-5960 551 4 for for ADP ejpam-5960 551 5 every every DET ejpam-5960 551 6 𝑖 𝑖 SYM ejpam-5960 551 7 ∈ ∈ PROPN ejpam-5960 551 8 𝐼 𝐼 PROPN ejpam-5960 551 9 , , PUNCT ejpam-5960 551 10 𝑋𝑖 𝑋𝑖 PROPN ejpam-5960 551 11 contains contain VERB ejpam-5960 551 12 a a DET ejpam-5960 551 13 molecular molecular NOUN ejpam-5960 551 14 with with ADP ejpam-5960 551 15 height height NOUN ejpam-5960 551 16 𝛼 𝛼 PROPN ejpam-5960 551 17 , , PUNCT ejpam-5960 551 18 π‘₯𝑖𝛼 π‘₯𝑖𝛼 NOUN ejpam-5960 551 19 say say VERB ejpam-5960 551 20 . . PUNCT ejpam-5960 552 1 then then ADV ejpam-5960 552 2 there there PRON ejpam-5960 552 3 must must AUX ejpam-5960 552 4 be be AUX ejpam-5960 552 5 some some DET ejpam-5960 552 6 𝑛 𝑛 PRON ejpam-5960 552 7 ∈ ∈ NOUN ejpam-5960 552 8 n n PRON ejpam-5960 552 9 such such ADJ ejpam-5960 552 10 that that SCONJ ejpam-5960 552 11 β„œπ‘–π‘› β„œπ‘–π‘› PROPN ejpam-5960 552 12 is be AUX ejpam-5960 552 13 an an DET ejpam-5960 552 14 𝛼–rf 𝛼–rf NOUN ejpam-5960 552 15 of of ADP ejpam-5960 552 16 𝑋𝑖𝑛 𝑋𝑖𝑛 PROPN ejpam-5960 552 17 . . PUNCT ejpam-5960 553 1 in in ADP ejpam-5960 553 2 fact fact NOUN ejpam-5960 553 3 , , PUNCT ejpam-5960 553 4 for for ADP ejpam-5960 553 5 each each DET ejpam-5960 553 6 𝑛 𝑛 PRON ejpam-5960 553 7 ∈ ∈ PROPN ejpam-5960 553 8 n n CCONJ ejpam-5960 553 9 , , PUNCT ejpam-5960 553 10 β„œπ‘–π‘› β„œπ‘–π‘› PROPN ejpam-5960 553 11 is be AUX ejpam-5960 553 12 not not PART ejpam-5960 553 13 an an DET ejpam-5960 553 14 𝛼–rf 𝛼–rf NOUN ejpam-5960 553 15 of of ADP ejpam-5960 553 16 𝑋𝑖𝑛 𝑋𝑖𝑛 PROPN ejpam-5960 553 17 , , PUNCT ejpam-5960 553 18 then then ADV ejpam-5960 553 19 there there PRON ejpam-5960 553 20 exists exist VERB ejpam-5960 553 21 𝑦𝑖𝑛 𝑦𝑖𝑛 PROPN ejpam-5960 553 22 ∈ ∈ PROPN ejpam-5960 553 23 𝑋𝑖𝑛 𝑋𝑖𝑛 PROPN ejpam-5960 553 24 with with ADP ejpam-5960 553 25 𝑦 𝑦 PRON ejpam-5960 553 26 𝑖𝑛 𝑖𝑛 PRON ejpam-5960 553 27 𝛼 𝛼 PROPN ejpam-5960 553 28 ∈ ∈ PROPN ejpam-5960 553 29 1𝑋𝑖𝑛 1𝑋𝑖𝑛 NUM ejpam-5960 553 30 ∧ ∧ PROPN ejpam-5960 553 31 ( ( PUNCT ejpam-5960 553 32 βˆ§β„œπ‘–π‘› βˆ§β„œπ‘–π‘› PROPN ejpam-5960 553 33 ) ) PUNCT ejpam-5960 553 34 . . PUNCT ejpam-5960 554 1 for for ADP ejpam-5960 554 2 each each DET ejpam-5960 554 3 𝑖 𝑖 PROPN ejpam-5960 554 4 βˆ‰ βˆ‰ X ejpam-5960 554 5 { { PUNCT ejpam-5960 554 6 𝑖𝑛 𝑖𝑛 NOUN ejpam-5960 554 7 : : PUNCT ejpam-5960 554 8 𝑛 𝑛 PROPN ejpam-5960 554 9 ∈ ∈ PROPN ejpam-5960 554 10 n n CCONJ ejpam-5960 554 11 } } PUNCT ejpam-5960 554 12 take take VERB ejpam-5960 554 13 𝑦𝑖 𝑦𝑖 NOUN ejpam-5960 554 14 = = PUNCT ejpam-5960 554 15 π‘₯𝑖. π‘₯𝑖. NOUN ejpam-5960 554 16 let let VERB ejpam-5960 554 17 𝑦 𝑦 PRON ejpam-5960 554 18 ∈ ∈ NOUN ejpam-5960 554 19 ∏ ∏ PROPN ejpam-5960 554 20 π‘–βˆˆπΌ π‘–βˆˆπΌ X ejpam-5960 554 21 𝑋𝑖 𝑋𝑖 PROPN ejpam-5960 554 22 with with ADP ejpam-5960 554 23 projections projection NOUN ejpam-5960 554 24 𝑦𝑖 𝑦𝑖 INTJ ejpam-5960 554 25 and and CCONJ ejpam-5960 554 26 𝑖 𝑖 ADP ejpam-5960 554 27 ∈ ∈ PROPN ejpam-5960 554 28 𝐼. 𝐼. PROPN ejpam-5960 554 29 on on ADP ejpam-5960 554 30 the the DET ejpam-5960 554 31 other other ADJ ejpam-5960 554 32 hand hand NOUN ejpam-5960 554 33 , , PUNCT ejpam-5960 554 34 for for ADP ejpam-5960 554 35 each each DET ejpam-5960 554 36 πœ‚ πœ‚ PROPN ejpam-5960 554 37 ∈ ∈ PROPN ejpam-5960 554 38 ψ ψ NOUN ejpam-5960 554 39 , , PUNCT ejpam-5960 554 40 𝑦𝛼 𝑦𝛼 PROPN ejpam-5960 554 41 ∈ ∈ PROPN ejpam-5960 554 42 πœ‚. πœ‚. NOUN ejpam-5960 554 43 to to PART ejpam-5960 554 44 see see VERB ejpam-5960 554 45 this this PRON ejpam-5960 554 46 , , PUNCT ejpam-5960 554 47 let let VERB ejpam-5960 554 48 πœ‚ πœ‚ NOUN ejpam-5960 555 1 = = SYM ejpam-5960 555 2 π‘ƒβˆ’1 π‘ƒβˆ’1 ADJ ejpam-5960 555 3 𝑖𝑛 𝑖𝑛 NOUN ejpam-5960 555 4 ( ( PUNCT ejpam-5960 555 5 πœ† πœ† NOUN ejpam-5960 555 6 ) ) PUNCT ejpam-5960 555 7 , , PUNCT ejpam-5960 555 8 where where SCONJ ejpam-5960 555 9 πœ† πœ† DET ejpam-5960 555 10 ∈ ∈ PROPN ejpam-5960 555 11 β„œπ‘–π‘› β„œπ‘–π‘› PROPN ejpam-5960 555 12 , , PUNCT ejpam-5960 555 13 then then ADV ejpam-5960 555 14 πœ‚(𝑦 πœ‚(𝑦 NOUN ejpam-5960 555 15 ) ) PUNCT ejpam-5960 555 16 = = PUNCT ejpam-5960 556 1 π‘ƒβˆ’1 π‘ƒβˆ’1 X ejpam-5960 556 2 𝑖𝑛 𝑖𝑛 X ejpam-5960 556 3 ( ( PUNCT ejpam-5960 556 4 πœ† πœ† NOUN ejpam-5960 556 5 ) ) PUNCT ejpam-5960 556 6 ( ( PUNCT ejpam-5960 556 7 𝑦 𝑦 NOUN ejpam-5960 556 8 ) ) PUNCT ejpam-5960 556 9 = = SYM ejpam-5960 557 1 πœ†(𝑃𝑖𝑛 πœ†(𝑃𝑖𝑛 PROPN ejpam-5960 557 2 ( ( PUNCT ejpam-5960 557 3 𝑦 𝑦 NOUN ejpam-5960 557 4 ) ) PUNCT ejpam-5960 557 5 ) ) PUNCT ejpam-5960 557 6 = = PUNCT ejpam-5960 558 1 πœ†(𝑦𝑖𝑛 πœ†(𝑦𝑖𝑛 X ejpam-5960 558 2 ) ) PUNCT ejpam-5960 558 3 β‰₯ β‰₯ PROPN ejpam-5960 558 4 𝛼. 𝛼. NOUN ejpam-5960 558 5 since since SCONJ ejpam-5960 558 6 π‘₯ π‘₯ PROPN ejpam-5960 558 7 𝑖𝑛 𝑖𝑛 PRON ejpam-5960 558 8 𝛼 𝛼 NOUN ejpam-5960 558 9 ∈ ∈ PROPN ejpam-5960 558 10 πœ† πœ† ADP ejpam-5960 558 11 and and CCONJ ejpam-5960 558 12 𝑦 𝑦 NOUN ejpam-5960 558 13 = = PUNCT ejpam-5960 558 14 ( ( PUNCT ejpam-5960 558 15 𝑦𝑖1 𝑦𝑖1 ADV ejpam-5960 558 16 , , PUNCT ejpam-5960 558 17 𝑦𝑖2 𝑦𝑖2 NOUN ejpam-5960 558 18 , , PUNCT ejpam-5960 558 19 ... ... PUNCT ejpam-5960 558 20 , , PUNCT ejpam-5960 558 21 𝑦𝑖𝑛 𝑦𝑖𝑛 PROPN ejpam-5960 558 22 ) ) PUNCT ejpam-5960 558 23 , , PUNCT ejpam-5960 558 24 hence hence ADV ejpam-5960 558 25 𝑦𝛼 𝑦𝛼 PROPN ejpam-5960 558 26 ∈ ∈ PROPN ejpam-5960 558 27 πœ‚. πœ‚. NOUN ejpam-5960 558 28 however however ADV ejpam-5960 558 29 , , PUNCT ejpam-5960 558 30 this this PRON ejpam-5960 558 31 is be AUX ejpam-5960 558 32 impossible impossible ADJ ejpam-5960 558 33 since since SCONJ ejpam-5960 558 34 ψ ψ NOUN ejpam-5960 558 35 is be AUX ejpam-5960 558 36 an an DET ejpam-5960 558 37 𝛼–rf 𝛼–rf NUM ejpam-5960 558 38 of of ADP ejpam-5960 558 39 ∏ ∏ PROPN ejpam-5960 558 40 π‘–βˆˆπΌ π‘–βˆˆπΌ DET ejpam-5960 558 41 𝑋𝑖. 𝑋𝑖. PROPN ejpam-5960 558 42 suppose suppose VERB ejpam-5960 558 43 that that SCONJ ejpam-5960 558 44 β„œπ‘–π‘› β„œπ‘–π‘› PROPN ejpam-5960 558 45 is be AUX ejpam-5960 558 46 an an DET ejpam-5960 558 47 𝛼–rf 𝛼–rf NUM ejpam-5960 558 48 of of ADP ejpam-5960 558 49 ∏ ∏ PROPN ejpam-5960 558 50 π‘–βˆˆπΌ π‘–βˆˆπΌ X ejpam-5960 558 51 𝑋𝑖. 𝑋𝑖. PROPN ejpam-5960 558 52 by by ADP ejpam-5960 558 53 the the DET ejpam-5960 558 54 𝑁𝛼–boundedness 𝑁𝛼–boundedness PROPN ejpam-5960 558 55 of of ADP ejpam-5960 558 56 πœ‡π‘–π‘› πœ‡π‘–π‘› NOUN ejpam-5960 558 57 , , PUNCT ejpam-5960 558 58 there there PRON ejpam-5960 558 59 exists exist VERB ejpam-5960 558 60 β„œβˆ— β„œβˆ— PROPN ejpam-5960 558 61 𝑖𝑛 𝑖𝑛 PRON ejpam-5960 558 62 ∈ ∈ PROPN ejpam-5960 558 63 2(β„œπ‘–π‘› 2(β„œπ‘–π‘› NUM ejpam-5960 558 64 ) ) PUNCT ejpam-5960 558 65 such such ADJ ejpam-5960 558 66 that that SCONJ ejpam-5960 558 67 β„œβˆ— β„œβˆ— PROPN ejpam-5960 558 68 𝑖𝑛 𝑖𝑛 NOUN ejpam-5960 558 69 is be AUX ejpam-5960 558 70 an an DET ejpam-5960 558 71 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 558 72 of of ADP ejpam-5960 558 73 πœ‡π‘–π‘› πœ‡π‘–π‘› NOUN ejpam-5960 558 74 . . PUNCT ejpam-5960 559 1 consider consider VERB ejpam-5960 559 2 the the DET ejpam-5960 559 3 Οˆπ‘œ Οˆπ‘œ X ejpam-5960 559 4 = = PUNCT ejpam-5960 559 5 { { PUNCT ejpam-5960 559 6 π‘ƒβˆ’1 π‘ƒβˆ’1 NOUN ejpam-5960 559 7 𝑖𝑛 𝑖𝑛 X ejpam-5960 559 8 ( ( PUNCT ejpam-5960 559 9 𝑐𝑙 𝑐𝑙 INTJ ejpam-5960 559 10 ( ( PUNCT ejpam-5960 559 11 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 559 12 ( ( PUNCT ejpam-5960 559 13 πœ† πœ† NOUN ejpam-5960 559 14 ) ) PUNCT ejpam-5960 559 15 ) ) PUNCT ejpam-5960 559 16 ) ) PUNCT ejpam-5960 560 1 : : PUNCT ejpam-5960 560 2 πœ† πœ† X ejpam-5960 560 3 ∈ ∈ X ejpam-5960 560 4 β„œβˆ— β„œβˆ— PROPN ejpam-5960 560 5 𝑖𝑛 𝑖𝑛 X ejpam-5960 560 6 } } PUNCT ejpam-5960 560 7 which which PRON ejpam-5960 560 8 is be AUX ejpam-5960 560 9 a a DET ejpam-5960 560 10 finite finite NOUN ejpam-5960 560 11 subset subset NOUN ejpam-5960 560 12 of of ADP ejpam-5960 560 13 ψ ψ X ejpam-5960 560 14 , , PUNCT ejpam-5960 560 15 i.e. i.e. X ejpam-5960 560 16 , , PUNCT ejpam-5960 560 17 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 560 18 ∈ ∈ PROPN ejpam-5960 560 19 2(ψ 2(ψ NUM ejpam-5960 560 20 ) ) PUNCT ejpam-5960 560 21 , , PUNCT ejpam-5960 560 22 if if SCONJ ejpam-5960 560 23 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 560 24 ∈ ∈ PROPN ejpam-5960 560 25 ∏ ∏ PROPN ejpam-5960 560 26 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 560 27 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 560 28 , , PUNCT ejpam-5960 560 29 then then ADV ejpam-5960 560 30 πœ‡π‘–π‘› πœ‡π‘–π‘› NOUN ejpam-5960 560 31 ( ( PUNCT ejpam-5960 560 32 𝑃𝑖𝑛 𝑃𝑖𝑛 PROPN ejpam-5960 560 33 ( ( PUNCT ejpam-5960 560 34 π‘₯ π‘₯ NOUN ejpam-5960 560 35 ) ) PUNCT ejpam-5960 560 36 ) ) PUNCT ejpam-5960 561 1 = = NUM ejpam-5960 561 2 πœ‡π‘–π‘› πœ‡π‘–π‘› NOUN ejpam-5960 561 3 ( ( PUNCT ejpam-5960 561 4 π‘₯𝑖𝑛 π‘₯𝑖𝑛 NOUN ejpam-5960 561 5 ) ) PUNCT ejpam-5960 561 6 β‰₯ β‰₯ NOUN ejpam-5960 561 7 𝛼 𝛼 NOUN ejpam-5960 562 1 so so ADV ejpam-5960 562 2 there there PRON ejpam-5960 562 3 is be VERB ejpam-5960 562 4 πœ† πœ† DET ejpam-5960 562 5 ∈ ∈ PROPN ejpam-5960 562 6 β„œβˆ— β„œβˆ— PROPN ejpam-5960 562 7 𝑖𝑛 𝑖𝑛 NOUN ejpam-5960 562 8 with with ADP ejpam-5960 562 9 πœ† πœ† DET ejpam-5960 562 10 ∈ ∈ PROPN ejpam-5960 562 11 𝑅 𝑅 PROPN ejpam-5960 562 12 π‘₯ π‘₯ PROPN ejpam-5960 562 13 𝑖𝑛 𝑖𝑛 NOUN ejpam-5960 562 14 𝛼 𝛼 INTJ ejpam-5960 562 15 , , PUNCT ejpam-5960 562 16 i.e. i.e. X ejpam-5960 562 17 , , PUNCT ejpam-5960 562 18 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 562 19 ( ( PUNCT ejpam-5960 562 20 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 562 21 ( ( PUNCT ejpam-5960 562 22 πœ†(π‘₯𝑖𝑛 πœ†(π‘₯𝑖𝑛 NOUN ejpam-5960 562 23 ) ) PUNCT ejpam-5960 562 24 ) ) PUNCT ejpam-5960 562 25 ) ) PUNCT ejpam-5960 563 1 = = SYM ejpam-5960 563 2 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 563 3 ( ( PUNCT ejpam-5960 563 4 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 563 5 ( ( PUNCT ejpam-5960 563 6 πœ†(𝑃𝑖𝑛 πœ†(𝑃𝑖𝑛 PROPN ejpam-5960 563 7 ( ( PUNCT ejpam-5960 563 8 π‘₯ π‘₯ NOUN ejpam-5960 563 9 ) ) PUNCT ejpam-5960 563 10 ) ) PUNCT ejpam-5960 563 11 ) ) PUNCT ejpam-5960 563 12 ) ) PUNCT ejpam-5960 563 13 ≱ ≱ PROPN ejpam-5960 563 14 𝛼 𝛼 VERB ejpam-5960 563 15 , , PUNCT ejpam-5960 563 16 then then ADV ejpam-5960 563 17 π‘ƒβˆ’1 π‘ƒβˆ’1 ADJ ejpam-5960 563 18 𝑖𝑛 𝑖𝑛 X ejpam-5960 563 19 ( ( PUNCT ejpam-5960 563 20 𝑐𝑙 𝑐𝑙 INTJ ejpam-5960 563 21 ( ( PUNCT ejpam-5960 563 22 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 563 23 ( ( PUNCT ejpam-5960 563 24 πœ† πœ† NOUN ejpam-5960 563 25 ) ) PUNCT ejpam-5960 563 26 ) ) PUNCT ejpam-5960 563 27 ) ) PUNCT ejpam-5960 564 1 ( ( PUNCT ejpam-5960 564 2 π‘₯ π‘₯ X ejpam-5960 564 3 ) ) PUNCT ejpam-5960 564 4 ≱ ≱ PROPN ejpam-5960 564 5 𝛼 𝛼 PROPN ejpam-5960 564 6 and and CCONJ ejpam-5960 564 7 therefore therefore ADV ejpam-5960 564 8 π‘ƒβˆ’1 π‘ƒβˆ’1 X ejpam-5960 564 9 𝑖𝑛 𝑖𝑛 X ejpam-5960 564 10 ( ( PUNCT ejpam-5960 564 11 𝑐𝑙 𝑐𝑙 INTJ ejpam-5960 564 12 ( ( PUNCT ejpam-5960 564 13 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 564 14 ( ( PUNCT ejpam-5960 564 15 πœ† πœ† NOUN ejpam-5960 564 16 ) ) PUNCT ejpam-5960 564 17 ) ) PUNCT ejpam-5960 564 18 ) ) PUNCT ejpam-5960 565 1 ∈ ∈ PROPN ejpam-5960 565 2 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 565 3 . . PUNCT ejpam-5960 566 1 this this PRON ejpam-5960 566 2 shows show VERB ejpam-5960 566 3 that that SCONJ ejpam-5960 566 4 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 566 5 ∈ ∈ PROPN ejpam-5960 566 6 2(ψ 2(ψ NUM ejpam-5960 566 7 ) ) PUNCT ejpam-5960 566 8 is be AUX ejpam-5960 566 9 an an DET ejpam-5960 566 10 𝛼–rcrf 𝛼–rcrf PROPN ejpam-5960 566 11 of of ADP ejpam-5960 566 12 ∏ ∏ PROPN ejpam-5960 566 13 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 566 14 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 566 15 and and CCONJ ejpam-5960 566 16 so so ADV ejpam-5960 566 17 ∏ ∏ PROPN ejpam-5960 566 18 π‘–βˆˆπΌ π‘–βˆˆπΌ PROPN ejpam-5960 566 19 πœ‡π‘– πœ‡π‘– ADP ejpam-5960 566 20 is be AUX ejpam-5960 566 21 a a DET ejpam-5960 566 22 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 566 23 set set NOUN ejpam-5960 566 24 . . PUNCT ejpam-5960 567 1 theorem theorem VERB ejpam-5960 567 2 4.4 4.4 NUM ejpam-5960 567 3 . . PUNCT ejpam-5960 568 1 let let VERB ejpam-5960 568 2 𝑆 𝑆 PROPN ejpam-5960 568 3 = = SYM ejpam-5960 568 4 { { PUNCT ejpam-5960 568 5 𝑆(𝑛 𝑆(𝑛 NOUN ejpam-5960 568 6 ) ) PUNCT ejpam-5960 568 7 : : PUNCT ejpam-5960 568 8 𝑛 𝑛 PROPN ejpam-5960 568 9 ∈ ∈ PROPN ejpam-5960 568 10 𝐷 𝐷 PROPN ejpam-5960 568 11 } } PUNCT ejpam-5960 568 12 and and CCONJ ejpam-5960 568 13 𝑇 𝑇 PROPN ejpam-5960 568 14 = = SYM ejpam-5960 568 15 { { PUNCT ejpam-5960 568 16 𝑇 𝑇 PROPN ejpam-5960 568 17 ( ( PUNCT ejpam-5960 568 18 𝑛 𝑛 NOUN ejpam-5960 568 19 ) ) PUNCT ejpam-5960 568 20 : : PUNCT ejpam-5960 568 21 𝑛 𝑛 PROPN ejpam-5960 568 22 ∈ ∈ PROPN ejpam-5960 568 23 𝐷 𝐷 PROPN ejpam-5960 568 24 } } PUNCT ejpam-5960 568 25 be be AUX ejpam-5960 568 26 a a DET ejpam-5960 568 27 molecular molecular ADJ ejpam-5960 568 28 nets net NOUN ejpam-5960 568 29 in in ADP ejpam-5960 568 30 a a DET ejpam-5960 568 31 𝐿–ts 𝐿–ts NOUN ejpam-5960 568 32 ( ( PUNCT ejpam-5960 568 33 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 568 34 , , PUNCT ejpam-5960 568 35 𝜏 𝜏 NOUN ejpam-5960 568 36 ) ) PUNCT ejpam-5960 568 37 such such ADJ ejpam-5960 568 38 that that SCONJ ejpam-5960 568 39 𝑇 𝑇 PROPN ejpam-5960 568 40 ( ( PUNCT ejpam-5960 568 41 𝑛 𝑛 PROPN ejpam-5960 568 42 ) ) PUNCT ejpam-5960 568 43 β‰₯ β‰₯ NOUN ejpam-5960 568 44 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 568 45 ) ) PUNCT ejpam-5960 568 46 for for ADP ejpam-5960 568 47 each each DET ejpam-5960 568 48 𝑛 𝑛 PRON ejpam-5960 568 49 ∈ ∈ PROPN ejpam-5960 568 50 𝐷 𝐷 NOUN ejpam-5960 568 51 and and CCONJ ejpam-5960 568 52 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 568 53 ∈ ∈ PROPN ejpam-5960 568 54 𝑀 𝑀 PROPN ejpam-5960 568 55 ( ( PUNCT ejpam-5960 568 56 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 568 57 ) ) PUNCT ejpam-5960 568 58 . . PUNCT ejpam-5960 569 1 then then ADV ejpam-5960 569 2 the the DET ejpam-5960 569 3 following follow VERB ejpam-5960 569 4 results result NOUN ejpam-5960 569 5 are be AUX ejpam-5960 569 6 true true ADJ ejpam-5960 569 7 : : PUNCT ejpam-5960 569 8 n. n. NOUN ejpam-5960 569 9 a. a. NOUN ejpam-5960 569 10 alsaedi alsaedi PROPN ejpam-5960 569 11 / / SYM ejpam-5960 569 12 eur eur PROPN ejpam-5960 569 13 . . PUNCT ejpam-5960 570 1 j. j. PROPN ejpam-5960 570 2 pure pure PROPN ejpam-5960 570 3 appl appl PROPN ejpam-5960 570 4 . . PROPN ejpam-5960 570 5 math math PROPN ejpam-5960 570 6 , , PUNCT ejpam-5960 570 7 18 18 NUM ejpam-5960 570 8 ( ( PUNCT ejpam-5960 570 9 4 4 NUM ejpam-5960 570 10 ) ) PUNCT ejpam-5960 570 11 ( ( PUNCT ejpam-5960 570 12 2025 2025 NUM ejpam-5960 570 13 ) ) PUNCT ejpam-5960 570 14 , , PUNCT ejpam-5960 570 15 5960 5960 NUM ejpam-5960 570 16 18 18 NUM ejpam-5960 570 17 of of ADP ejpam-5960 570 18 22 22 NUM ejpam-5960 570 19 ( ( PUNCT ejpam-5960 570 20 i i NOUN ejpam-5960 570 21 ) ) PUNCT ejpam-5960 570 22 if if SCONJ ejpam-5960 570 23 𝑆 𝑆 PROPN ejpam-5960 570 24 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ X ejpam-5960 570 25 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 570 26 , , PUNCT ejpam-5960 570 27 then then ADV ejpam-5960 570 28 𝑇 𝑇 PROPN ejpam-5960 570 29 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ PROPN ejpam-5960 570 30 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 570 31 ( ( PUNCT ejpam-5960 570 32 ii ii NOUN ejpam-5960 570 33 ) ) PUNCT ejpam-5960 570 34 if if SCONJ ejpam-5960 570 35 𝑆 𝑆 PROPN ejpam-5960 570 36 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 570 37 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 570 38 , , PUNCT ejpam-5960 570 39 then then ADV ejpam-5960 570 40 𝑇 𝑇 PROPN ejpam-5960 570 41 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 570 42 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 570 43 proof proof NOUN ejpam-5960 570 44 . . PUNCT ejpam-5960 571 1 ( ( PUNCT ejpam-5960 571 2 i i NOUN ejpam-5960 571 3 ) ) PUNCT ejpam-5960 571 4 let let VERB ejpam-5960 571 5 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 571 6 ∈ ∈ PROPN ejpam-5960 571 7 𝑀 𝑀 PROPN ejpam-5960 571 8 ( ( PUNCT ejpam-5960 571 9 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 571 10 ) ) PUNCT ejpam-5960 571 11 such such ADJ ejpam-5960 571 12 that that SCONJ ejpam-5960 571 13 𝑆 𝑆 PROPN ejpam-5960 571 14 π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ π‘π›Όπ΅βˆ’βˆ’βˆ’βˆ’β†’ X ejpam-5960 571 15 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 571 16 , , PUNCT ejpam-5960 571 17 then then ADV ejpam-5960 571 18 for for ADP ejpam-5960 571 19 each each DET ejpam-5960 571 20 πœ† πœ† PRON ejpam-5960 571 21 ∈ ∈ PROPN ejpam-5960 571 22 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 571 23 , , PUNCT ejpam-5960 571 24 there there PRON ejpam-5960 571 25 exists exist VERB ejpam-5960 571 26 𝑛 𝑛 DET ejpam-5960 571 27 ∈ ∈ PROPN ejpam-5960 571 28 𝐷 𝐷 NOUN ejpam-5960 571 29 such such ADJ ejpam-5960 571 30 that that PRON ejpam-5960 571 31 for for ADP ejpam-5960 571 32 each each DET ejpam-5960 571 33 π‘š π‘š PROPN ejpam-5960 571 34 ∈ ∈ PROPN ejpam-5960 571 35 𝐷 𝐷 NOUN ejpam-5960 571 36 and and CCONJ ejpam-5960 571 37 π‘š π‘š ADP ejpam-5960 571 38 β‰₯ β‰₯ NOUN ejpam-5960 571 39 𝑛 𝑛 PRON ejpam-5960 571 40 then then ADV ejpam-5960 571 41 𝑆(π‘š 𝑆(π‘š X ejpam-5960 571 42 ) ) PUNCT ejpam-5960 571 43 βˆ‰ βˆ‰ PROPN ejpam-5960 571 44 πœ†. πœ†. PROPN ejpam-5960 571 45 since since SCONJ ejpam-5960 571 46 𝑇 𝑇 PROPN ejpam-5960 571 47 ( ( PUNCT ejpam-5960 571 48 𝑛 𝑛 PROPN ejpam-5960 571 49 ) ) PUNCT ejpam-5960 571 50 β‰₯ β‰₯ NOUN ejpam-5960 571 51 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 571 52 ) ) PUNCT ejpam-5960 571 53 > > X ejpam-5960 572 1 πœ† πœ† X ejpam-5960 572 2 , , PUNCT ejpam-5960 572 3 and and CCONJ ejpam-5960 572 4 so so ADV ejpam-5960 572 5 for for ADP ejpam-5960 572 6 each each DET ejpam-5960 572 7 πœ† πœ† PRON ejpam-5960 572 8 ∈ ∈ PROPN ejpam-5960 572 9 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 572 10 there there PRON ejpam-5960 572 11 exists exist VERB ejpam-5960 572 12 𝑛 𝑛 DET ejpam-5960 572 13 ∈ ∈ PROPN ejpam-5960 572 14 𝐷 𝐷 NOUN ejpam-5960 572 15 such such ADJ ejpam-5960 572 16 that that PRON ejpam-5960 572 17 for for ADP ejpam-5960 572 18 each each DET ejpam-5960 572 19 π‘š π‘š PROPN ejpam-5960 572 20 ∈ ∈ PROPN ejpam-5960 572 21 𝐷 𝐷 NOUN ejpam-5960 572 22 and and CCONJ ejpam-5960 572 23 π‘š π‘š ADP ejpam-5960 572 24 β‰₯ β‰₯ NOUN ejpam-5960 572 25 𝑛 𝑛 DET ejpam-5960 572 26 then then ADV ejpam-5960 572 27 𝑇 𝑇 PROPN ejpam-5960 572 28 ( ( PUNCT ejpam-5960 572 29 π‘š π‘š NOUN ejpam-5960 572 30 ) ) PUNCT ejpam-5960 572 31 βˆ‰ βˆ‰ PROPN ejpam-5960 572 32 πœ†. πœ†. NOUN ejpam-5960 573 1 this this PRON ejpam-5960 573 2 shows show VERB ejpam-5960 573 3 that that SCONJ ejpam-5960 573 4 𝑇 𝑇 PROPN ejpam-5960 573 5 is be AUX ejpam-5960 573 6 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 573 7 to to PART ejpam-5960 573 8 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 573 9 ( ( PUNCT ejpam-5960 573 10 ii ii NOUN ejpam-5960 573 11 ) ) PUNCT ejpam-5960 573 12 let let VERB ejpam-5960 573 13 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 573 14 ∈ ∈ PROPN ejpam-5960 573 15 𝑀 𝑀 PROPN ejpam-5960 573 16 ( ( PUNCT ejpam-5960 573 17 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 573 18 ) ) PUNCT ejpam-5960 573 19 such such ADJ ejpam-5960 573 20 that that SCONJ ejpam-5960 573 21 𝑆 𝑆 PROPN ejpam-5960 573 22 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PRON ejpam-5960 573 23 π‘₯𝛼 π‘₯𝛼 NOUN ejpam-5960 573 24 , , PUNCT ejpam-5960 573 25 then then ADV ejpam-5960 573 26 for for ADP ejpam-5960 573 27 each each DET ejpam-5960 573 28 πœ† πœ† PRON ejpam-5960 573 29 ∈ ∈ PROPN ejpam-5960 573 30 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 573 31 and and CCONJ ejpam-5960 573 32 each each DET ejpam-5960 573 33 𝑛 𝑛 PRON ejpam-5960 573 34 ∈ ∈ PROPN ejpam-5960 573 35 𝐷 𝐷 PROPN ejpam-5960 573 36 there there PRON ejpam-5960 573 37 exists exist VERB ejpam-5960 573 38 π‘š π‘š PROPN ejpam-5960 573 39 ∈ ∈ PROPN ejpam-5960 573 40 𝐷 𝐷 NOUN ejpam-5960 573 41 such such ADJ ejpam-5960 573 42 that that SCONJ ejpam-5960 573 43 π‘š π‘š PROPN ejpam-5960 573 44 β‰₯ β‰₯ PRON ejpam-5960 573 45 𝑛 𝑛 PRON ejpam-5960 573 46 then then ADV ejpam-5960 573 47 𝑆(π‘š 𝑆(π‘š X ejpam-5960 573 48 ) ) PUNCT ejpam-5960 573 49 βˆ‰ βˆ‰ PROPN ejpam-5960 573 50 πœ†. πœ†. PROPN ejpam-5960 574 1 since since SCONJ ejpam-5960 574 2 𝑇 𝑇 PROPN ejpam-5960 574 3 ( ( PUNCT ejpam-5960 574 4 𝑛 𝑛 PROPN ejpam-5960 574 5 ) ) PUNCT ejpam-5960 574 6 β‰₯ β‰₯ NOUN ejpam-5960 574 7 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 574 8 ) ) PUNCT ejpam-5960 574 9 for for ADP ejpam-5960 574 10 each each DET ejpam-5960 574 11 𝑛 𝑛 PRON ejpam-5960 574 12 ∈ ∈ PROPN ejpam-5960 574 13 𝐷 𝐷 PROPN ejpam-5960 574 14 , , PUNCT ejpam-5960 574 15 then then ADV ejpam-5960 574 16 𝑇 𝑇 PROPN ejpam-5960 574 17 ( ( PUNCT ejpam-5960 574 18 𝑛 𝑛 PROPN ejpam-5960 574 19 ) ) PUNCT ejpam-5960 574 20 β‰₯ β‰₯ NOUN ejpam-5960 574 21 𝑆(𝑛 𝑆(𝑛 NUM ejpam-5960 574 22 ) ) PUNCT ejpam-5960 574 23 > > X ejpam-5960 574 24 πœ†. πœ†. VERB ejpam-5960 574 25 thus thus ADV ejpam-5960 574 26 for for ADP ejpam-5960 574 27 each each DET ejpam-5960 574 28 πœ† πœ† PRON ejpam-5960 574 29 ∈ ∈ PROPN ejpam-5960 574 30 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 574 31 and and CCONJ ejpam-5960 574 32 for for ADP ejpam-5960 574 33 each each DET ejpam-5960 574 34 𝑛 𝑛 PRON ejpam-5960 574 35 ∈ ∈ PROPN ejpam-5960 574 36 𝐷 𝐷 PROPN ejpam-5960 574 37 there there PRON ejpam-5960 574 38 exists exist VERB ejpam-5960 574 39 π‘š π‘š PROPN ejpam-5960 574 40 ∈ ∈ PROPN ejpam-5960 574 41 𝐷 𝐷 NOUN ejpam-5960 574 42 such such ADJ ejpam-5960 574 43 that that SCONJ ejpam-5960 574 44 π‘š π‘š PROPN ejpam-5960 574 45 β‰₯ β‰₯ PRON ejpam-5960 574 46 𝑛 𝑛 PRON ejpam-5960 574 47 then then ADV ejpam-5960 574 48 𝑇 𝑇 PROPN ejpam-5960 574 49 ( ( PUNCT ejpam-5960 574 50 π‘š π‘š NOUN ejpam-5960 574 51 ) ) PUNCT ejpam-5960 575 1 βˆ‰ βˆ‰ PROPN ejpam-5960 576 1 πœ†. πœ†. NOUN ejpam-5960 576 2 this this PRON ejpam-5960 576 3 shows show VERB ejpam-5960 576 4 that that SCONJ ejpam-5960 576 5 𝑇 𝑇 PROPN ejpam-5960 576 6 π‘π›Όπ΅βˆ π‘π›Όπ΅βˆ PROPN ejpam-5960 576 7 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 576 8 theorem theorem VERB ejpam-5960 576 9 4.5 4.5 NUM ejpam-5960 576 10 . . PUNCT ejpam-5960 577 1 let let VERB ejpam-5960 577 2 { { PUNCT ejpam-5960 577 3 πœ‡π‘› πœ‡π‘› NOUN ejpam-5960 577 4 : : PUNCT ejpam-5960 577 5 𝑛 𝑛 PROPN ejpam-5960 577 6 ∈ ∈ PROPN ejpam-5960 577 7 𝐷 𝐷 PROPN ejpam-5960 577 8 } } PUNCT ejpam-5960 577 9 be be AUX ejpam-5960 577 10 a a DET ejpam-5960 577 11 net net NOUN ejpam-5960 577 12 of of ADP ejpam-5960 577 13 closed closed ADJ ejpam-5960 577 14 𝐿-subsets 𝐿-subsets PROPN ejpam-5960 577 15 in in ADP ejpam-5960 577 16 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 577 17 such such ADJ ejpam-5960 577 18 that that DET ejpam-5960 577 19 πœ‡π‘›1 πœ‡π‘›1 NOUN ejpam-5960 577 20 ≀ ≀ NUM ejpam-5960 577 21 πœ‡π‘›2 πœ‡π‘›2 NOUN ejpam-5960 577 22 then then ADV ejpam-5960 577 23 𝛿.lim(πœ‡π‘› 𝛿.lim(πœ‡π‘› NOUN ejpam-5960 577 24 ) ) PUNCT ejpam-5960 577 25 ≀ ≀ NOUN ejpam-5960 578 1 ∧{πœ‡π‘› ∧{πœ‡π‘› PROPN ejpam-5960 578 2 : : PUNCT ejpam-5960 578 3 𝑛 𝑛 PROPN ejpam-5960 578 4 ∈ ∈ PROPN ejpam-5960 578 5 𝐷 𝐷 PROPN ejpam-5960 578 6 } } PUNCT ejpam-5960 578 7 iff iff PROPN ejpam-5960 578 8 𝑛2 𝑛2 NOUN ejpam-5960 578 9 ≀ ≀ PROPN ejpam-5960 578 10 𝑛1 𝑛1 PROPN ejpam-5960 578 11 . . PUNCT ejpam-5960 579 1 proof proof NOUN ejpam-5960 579 2 . . PUNCT ejpam-5960 580 1 let let VERB ejpam-5960 580 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 580 3 ∈ ∈ PROPN ejpam-5960 580 4 𝛿.lim(πœ‡π‘› 𝛿.lim(πœ‡π‘› NOUN ejpam-5960 580 5 ) ) PUNCT ejpam-5960 580 6 and and CCONJ ejpam-5960 580 7 let let VERB ejpam-5960 580 8 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 580 9 βˆ‰ βˆ‰ PROPN ejpam-5960 580 10 ∧{πœ‡π‘› ∧{πœ‡π‘› PROPN ejpam-5960 580 11 : : PUNCT ejpam-5960 580 12 𝑛 𝑛 PROPN ejpam-5960 580 13 ∈ ∈ PROPN ejpam-5960 580 14 𝐷 𝐷 PROPN ejpam-5960 580 15 } } PUNCT ejpam-5960 580 16 hence hence ADV ejpam-5960 580 17 there there PRON ejpam-5960 580 18 is be VERB ejpam-5960 580 19 𝑛 𝑛 PROPN ejpam-5960 580 20 β—¦ β—¦ NOUN ejpam-5960 580 21 ∈ ∈ NOUN ejpam-5960 580 22 𝐷 𝐷 NOUN ejpam-5960 581 1 such such ADJ ejpam-5960 581 2 that that SCONJ ejpam-5960 581 3 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 581 4 βˆ‰ βˆ‰ PROPN ejpam-5960 581 5 πœ‡π‘› πœ‡π‘› PROPN ejpam-5960 581 6 β—¦ β—¦ NOUN ejpam-5960 581 7 . . PUNCT ejpam-5960 582 1 let let VERB ejpam-5960 582 2 𝜌 𝜌 PART ejpam-5960 582 3 = = SYM ejpam-5960 582 4 πœ‡π‘› πœ‡π‘› ADP ejpam-5960 582 5 β—¦ β—¦ NOUN ejpam-5960 582 6 , , PUNCT ejpam-5960 582 7 then then ADV ejpam-5960 582 8 𝜌 𝜌 ADP ejpam-5960 582 9 ∈ ∈ PROPN ejpam-5960 582 10 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 582 11 . . PUNCT ejpam-5960 583 1 since since SCONJ ejpam-5960 583 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 583 3 ∈ ∈ PROPN ejpam-5960 583 4 𝛿.lim(πœ‡π‘› 𝛿.lim(πœ‡π‘› NOUN ejpam-5960 583 5 ) ) PUNCT ejpam-5960 583 6 and and CCONJ ejpam-5960 583 7 𝜌 𝜌 ADP ejpam-5960 583 8 ∈ ∈ PROPN ejpam-5960 583 9 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 583 10 , , PUNCT ejpam-5960 583 11 there there PRON ejpam-5960 583 12 is be VERB ejpam-5960 583 13 𝑛 𝑛 DET ejpam-5960 583 14 ∈ ∈ PROPN ejpam-5960 583 15 𝐷 𝐷 PROPN ejpam-5960 583 16 , , PUNCT ejpam-5960 583 17 𝑛 𝑛 DET ejpam-5960 583 18 β‰₯ β‰₯ NOUN ejpam-5960 583 19 𝑛 𝑛 ADP ejpam-5960 583 20 β—¦ β—¦ NOUN ejpam-5960 583 21 such such ADJ ejpam-5960 583 22 that that SCONJ ejpam-5960 583 23 πœ‡π‘› πœ‡π‘› INTJ ejpam-5960 583 24 β‰° β‰° PROPN ejpam-5960 583 25 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 583 26 ( ( PUNCT ejpam-5960 583 27 int(𝜌 int(𝜌 NOUN ejpam-5960 583 28 ) ) PUNCT ejpam-5960 583 29 ) ) PUNCT ejpam-5960 583 30 , , PUNCT ejpam-5960 583 31 which which PRON ejpam-5960 583 32 contradicts contradict VERB ejpam-5960 583 33 the the DET ejpam-5960 583 34 hypothesis hypothesis NOUN ejpam-5960 583 35 that that PRON ejpam-5960 583 36 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 583 37 βˆ‰ βˆ‰ PROPN ejpam-5960 583 38 ∧{πœ‡π‘› ∧{πœ‡π‘› PROPN ejpam-5960 583 39 : : PUNCT ejpam-5960 583 40 𝑛 𝑛 PROPN ejpam-5960 583 41 ∈ ∈ PROPN ejpam-5960 583 42 𝐷 𝐷 PROPN ejpam-5960 583 43 } } PUNCT ejpam-5960 583 44 . . PUNCT ejpam-5960 584 1 thus thus ADV ejpam-5960 584 2 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 584 3 ∈ ∈ PROPN ejpam-5960 584 4 ∧{πœ‡π‘› ∧{πœ‡π‘› PROPN ejpam-5960 584 5 : : PUNCT ejpam-5960 584 6 𝑛 𝑛 PROPN ejpam-5960 584 7 ∈ ∈ PROPN ejpam-5960 584 8 𝐷 𝐷 PROPN ejpam-5960 584 9 } } PUNCT ejpam-5960 584 10 . . PUNCT ejpam-5960 585 1 theorem theorem VERB ejpam-5960 585 2 4.6 4.6 NUM ejpam-5960 585 3 . . PUNCT ejpam-5960 586 1 let let VERB ejpam-5960 586 2 ( ( PUNCT ejpam-5960 586 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 586 4 , , PUNCT ejpam-5960 586 5 𝜏 𝜏 NOUN ejpam-5960 586 6 ) ) PUNCT ejpam-5960 586 7 be be VERB ejpam-5960 586 8 an an DET ejpam-5960 586 9 l l NOUN ejpam-5960 586 10 – – PUNCT ejpam-5960 586 11 ts ts NOUN ejpam-5960 586 12 and and CCONJ ejpam-5960 586 13 πœ‡ πœ‡ X ejpam-5960 586 14 ∈ ∈ X ejpam-5960 586 15 𝐿𝑋. 𝐿𝑋. X ejpam-5960 586 16 then then ADV ejpam-5960 586 17 πœ‡ πœ‡ SCONJ ejpam-5960 586 18 is be AUX ejpam-5960 586 19 a a DET ejpam-5960 586 20 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 586 21 set set ADJ ejpam-5960 586 22 iff iff PROPN ejpam-5960 586 23 for for ADP ejpam-5960 586 24 every every DET ejpam-5960 586 25 net net NOUN ejpam-5960 586 26 { { PUNCT ejpam-5960 586 27 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 586 28 : : PUNCT ejpam-5960 586 29 𝑛 𝑛 PROPN ejpam-5960 586 30 ∈ ∈ PROPN ejpam-5960 586 31 𝐷 𝐷 PROPN ejpam-5960 586 32 } } PUNCT ejpam-5960 586 33 of of ADP ejpam-5960 586 34 closed closed ADJ ejpam-5960 586 35 𝐿–subsets 𝐿–subset NOUN ejpam-5960 586 36 in in ADP ejpam-5960 586 37 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 586 38 such such ADJ ejpam-5960 586 39 that that SCONJ ejpam-5960 586 40 𝛿 𝛿 ADJ ejpam-5960 586 41 lim(πœ‚π‘› lim(πœ‚π‘› NOUN ejpam-5960 586 42 ) ) PUNCT ejpam-5960 586 43 ( ( PUNCT ejpam-5960 586 44 π‘₯ π‘₯ NOUN ejpam-5960 586 45 ) ) PUNCT ejpam-5960 586 46 < < X ejpam-5960 586 47 𝛼 𝛼 X ejpam-5960 586 48 , , PUNCT ejpam-5960 586 49 for for ADP ejpam-5960 586 50 each each DET ejpam-5960 586 51 π‘₯ π‘₯ PRON ejpam-5960 586 52 ∈ ∈ PROPN ejpam-5960 586 53 𝑋 𝑋 PROPN ejpam-5960 586 54 , , PUNCT ejpam-5960 586 55 there there PRON ejpam-5960 586 56 is be VERB ejpam-5960 586 57 π‘›π‘œ π‘›π‘œ DET ejpam-5960 586 58 ∈ ∈ PROPN ejpam-5960 586 59 𝐷 𝐷 PROPN ejpam-5960 586 60 for for ADP ejpam-5960 586 61 which which PRON ejpam-5960 586 62 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 586 63 ∧ ∧ PROPN ejpam-5960 586 64 πœ‡ πœ‡ ADP ejpam-5960 586 65 = = NOUN ejpam-5960 586 66 0𝑋 0𝑋 NOUN ejpam-5960 586 67 for for ADP ejpam-5960 586 68 every every DET ejpam-5960 586 69 𝑛 𝑛 PROPN ejpam-5960 586 70 ∈ ∈ PROPN ejpam-5960 586 71 𝐷 𝐷 PROPN ejpam-5960 586 72 , , PUNCT ejpam-5960 586 73 𝑛 𝑛 DET ejpam-5960 586 74 β‰₯ β‰₯ NOUN ejpam-5960 586 75 π‘›π‘œ. π‘›π‘œ. NOUN ejpam-5960 586 76 proof proof NOUN ejpam-5960 586 77 . . PUNCT ejpam-5960 587 1 let let VERB ejpam-5960 587 2 πœ‡ πœ‡ SCONJ ejpam-5960 587 3 ∈ ∈ PROPN ejpam-5960 587 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 587 5 be be AUX ejpam-5960 587 6 a a DET ejpam-5960 587 7 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 587 8 set set NOUN ejpam-5960 587 9 and and CCONJ ejpam-5960 587 10 let let VERB ejpam-5960 587 11 { { PUNCT ejpam-5960 587 12 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 587 13 : : PUNCT ejpam-5960 587 14 𝑛 𝑛 PROPN ejpam-5960 587 15 ∈ ∈ PROPN ejpam-5960 587 16 𝐷 𝐷 PROPN ejpam-5960 587 17 } } PUNCT ejpam-5960 587 18 be be AUX ejpam-5960 587 19 a a DET ejpam-5960 587 20 net net NOUN ejpam-5960 587 21 of of ADP ejpam-5960 587 22 closed closed ADJ ejpam-5960 587 23 𝐿–subsets 𝐿–subset NOUN ejpam-5960 587 24 in in ADP ejpam-5960 587 25 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 587 26 such such ADJ ejpam-5960 587 27 that that DET ejpam-5960 587 28 𝛿.lim(πœ‚π‘› 𝛿.lim(πœ‚π‘› PROPN ejpam-5960 587 29 ) ) PUNCT ejpam-5960 587 30 ( ( PUNCT ejpam-5960 587 31 π‘₯ π‘₯ NOUN ejpam-5960 587 32 ) ) PUNCT ejpam-5960 587 33 < < X ejpam-5960 587 34 𝛼 𝛼 X ejpam-5960 587 35 , , PUNCT ejpam-5960 587 36 for for ADP ejpam-5960 587 37 each each DET ejpam-5960 587 38 π‘₯ π‘₯ PRON ejpam-5960 587 39 ∈ ∈ PROPN ejpam-5960 587 40 𝑋. 𝑋. PROPN ejpam-5960 587 41 then then ADV ejpam-5960 587 42 for for ADP ejpam-5960 587 43 every every DET ejpam-5960 587 44 molecular molecular ADJ ejpam-5960 587 45 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 587 46 ∈ ∈ PROPN ejpam-5960 587 47 𝑀 𝑀 PROPN ejpam-5960 587 48 ( ( PUNCT ejpam-5960 587 49 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 587 50 ) ) PUNCT ejpam-5960 587 51 there there PRON ejpam-5960 587 52 exists exist VERB ejpam-5960 587 53 𝜌π‘₯ 𝜌π‘₯ PROPN ejpam-5960 587 54 ∈ ∈ PROPN ejpam-5960 587 55 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 587 56 and and CCONJ ejpam-5960 587 57 𝑛π‘₯ 𝑛π‘₯ PROPN ejpam-5960 587 58 ∈ ∈ PROPN ejpam-5960 587 59 𝐷 𝐷 PROPN ejpam-5960 588 1 such such ADJ ejpam-5960 588 2 that that DET ejpam-5960 588 3 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 588 4 ≀ ≀ ADJ ejpam-5960 588 5 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 588 6 ( ( PUNCT ejpam-5960 588 7 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 588 8 ( ( PUNCT ejpam-5960 588 9 𝜌π‘₯ 𝜌π‘₯ NOUN ejpam-5960 588 10 ) ) PUNCT ejpam-5960 588 11 ) ) PUNCT ejpam-5960 588 12 for for ADP ejpam-5960 588 13 every every DET ejpam-5960 588 14 𝑛 𝑛 PROPN ejpam-5960 588 15 ∈ ∈ PROPN ejpam-5960 588 16 𝐷 𝐷 PROPN ejpam-5960 588 17 , , PUNCT ejpam-5960 588 18 𝑛 𝑛 DET ejpam-5960 588 19 β‰₯ β‰₯ NOUN ejpam-5960 588 20 𝑛π‘₯. 𝑛π‘₯. ADV ejpam-5960 588 21 since since SCONJ ejpam-5960 588 22 𝜌π‘₯ 𝜌π‘₯ PROPN ejpam-5960 588 23 ∈ ∈ PROPN ejpam-5960 588 24 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 588 25 for for ADP ejpam-5960 588 26 every every DET ejpam-5960 588 27 π‘₯ π‘₯ PROPN ejpam-5960 588 28 ∈ ∈ PROPN ejpam-5960 588 29 𝑋 𝑋 PROPN ejpam-5960 588 30 , , PUNCT ejpam-5960 588 31 then then ADV ejpam-5960 588 32 the the DET ejpam-5960 588 33 family family NOUN ejpam-5960 588 34 ψ ψ X ejpam-5960 588 35 = = PUNCT ejpam-5960 588 36 { { PUNCT ejpam-5960 588 37 𝜌π‘₯ 𝜌π‘₯ NOUN ejpam-5960 588 38 : : PUNCT ejpam-5960 588 39 π‘₯ π‘₯ PUNCT ejpam-5960 588 40 ∈ ∈ PROPN ejpam-5960 588 41 𝑋 𝑋 NOUN ejpam-5960 588 42 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ ejpam-5960 588 43 𝛼 𝛼 PROPN ejpam-5960 588 44 ∈ ∈ PROPN ejpam-5960 588 45 𝑀 𝑀 PROPN ejpam-5960 588 46 ( ( PUNCT ejpam-5960 588 47 𝐿 𝐿 PROPN ejpam-5960 588 48 ) ) PUNCT ejpam-5960 588 49 } } PUNCT ejpam-5960 588 50 is be AUX ejpam-5960 588 51 an an DET ejpam-5960 588 52 𝛼–rf 𝛼–rf NUM ejpam-5960 588 53 of of ADP ejpam-5960 588 54 1𝑋. 1𝑋. NUM ejpam-5960 588 55 since since SCONJ ejpam-5960 588 56 πœ‡ πœ‡ ADV ejpam-5960 588 57 is be AUX ejpam-5960 588 58 a a DET ejpam-5960 588 59 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 588 60 , , PUNCT ejpam-5960 588 61 there there PRON ejpam-5960 588 62 exist exist VERB ejpam-5960 588 63 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 588 64 = = PUNCT ejpam-5960 588 65 { { PUNCT ejpam-5960 588 66 𝑐𝑙 𝑐𝑙 X ejpam-5960 588 67 ( ( PUNCT ejpam-5960 588 68 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 588 69 ( ( PUNCT ejpam-5960 588 70 𝜌π‘₯𝑖 𝜌π‘₯𝑖 ADJ ejpam-5960 588 71 ) ) PUNCT ejpam-5960 588 72 ) ) PUNCT ejpam-5960 588 73 : : PUNCT ejpam-5960 589 1 𝑖 𝑖 X ejpam-5960 589 2 = = SYM ejpam-5960 589 3 1 1 NUM ejpam-5960 589 4 , , PUNCT ejpam-5960 589 5 2 2 NUM ejpam-5960 589 6 , , PUNCT ejpam-5960 589 7 ... ... PUNCT ejpam-5960 589 8 , , PUNCT ejpam-5960 589 9 π‘˜ π‘˜ X ejpam-5960 589 10 } } PUNCT ejpam-5960 589 11 ∈ ∈ PROPN ejpam-5960 589 12 2(ψ 2(ψ NUM ejpam-5960 589 13 ) ) PUNCT ejpam-5960 589 14 such such ADJ ejpam-5960 589 15 that that SCONJ ejpam-5960 589 16 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 589 17 is be AUX ejpam-5960 589 18 an an DET ejpam-5960 589 19 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 589 20 of of ADP ejpam-5960 589 21 πœ‡. πœ‡. NOUN ejpam-5960 589 22 put put VERB ejpam-5960 589 23 𝜌 𝜌 ADP ejpam-5960 589 24 = = PUNCT ejpam-5960 589 25 βˆ§π‘˜ βˆ§π‘˜ PROPN ejpam-5960 589 26 𝑖=1 𝑖=1 PROPN ejpam-5960 589 27 𝜌π‘₯𝑖 𝜌π‘₯𝑖 ADJ ejpam-5960 589 28 , , PUNCT ejpam-5960 589 29 then then ADV ejpam-5960 589 30 𝑐𝑙 𝑐𝑙 INTJ ejpam-5960 589 31 ( ( PUNCT ejpam-5960 589 32 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 589 33 ( ( PUNCT ejpam-5960 589 34 𝜌 𝜌 NOUN ejpam-5960 589 35 ) ) PUNCT ejpam-5960 589 36 ) ) PUNCT ejpam-5960 590 1 ∈ ∈ PROPN ejpam-5960 590 2 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 590 3 . . PUNCT ejpam-5960 591 1 since since SCONJ ejpam-5960 591 2 𝐷 𝐷 PROPN ejpam-5960 591 3 is be AUX ejpam-5960 591 4 a a DET ejpam-5960 591 5 directed direct VERB ejpam-5960 591 6 set set NOUN ejpam-5960 591 7 , , PUNCT ejpam-5960 591 8 there there PRON ejpam-5960 591 9 is be VERB ejpam-5960 591 10 π‘›π‘œ π‘›π‘œ PRON ejpam-5960 591 11 ∈ ∈ PROPN ejpam-5960 591 12 𝐷 𝐷 NOUN ejpam-5960 591 13 such such ADJ ejpam-5960 591 14 that that SCONJ ejpam-5960 591 15 π‘›π‘œ π‘›π‘œ PROPN ejpam-5960 591 16 β‰₯ β‰₯ NOUN ejpam-5960 591 17 𝑛π‘₯𝑖 𝑛π‘₯𝑖 NOUN ejpam-5960 591 18 for for ADP ejpam-5960 591 19 every every PRON ejpam-5960 591 20 𝑖 𝑖 SYM ejpam-5960 591 21 = = SYM ejpam-5960 591 22 1 1 NUM ejpam-5960 591 23 , , PUNCT ejpam-5960 591 24 2 2 NUM ejpam-5960 591 25 , , PUNCT ejpam-5960 591 26 ... ... PUNCT ejpam-5960 591 27 , , PUNCT ejpam-5960 591 28 π‘˜. π‘˜. NOUN ejpam-5960 591 29 then then ADV ejpam-5960 591 30 for for ADP ejpam-5960 591 31 every every DET ejpam-5960 591 32 𝑛 𝑛 PROPN ejpam-5960 591 33 ∈ ∈ PROPN ejpam-5960 591 34 𝐷 𝐷 PROPN ejpam-5960 591 35 , , PUNCT ejpam-5960 591 36 𝑛 𝑛 DET ejpam-5960 591 37 β‰₯ β‰₯ NOUN ejpam-5960 591 38 π‘›π‘œ π‘›π‘œ NOUN ejpam-5960 591 39 , , PUNCT ejpam-5960 591 40 we we PRON ejpam-5960 591 41 have have VERB ejpam-5960 591 42 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 591 43 ≀ ≀ NUM ejpam-5960 591 44 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 591 45 ( ( PUNCT ejpam-5960 591 46 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 591 47 ( ( PUNCT ejpam-5960 591 48 βˆ§π‘˜ βˆ§π‘˜ PROPN ejpam-5960 591 49 𝑖=1 𝑖=1 PROPN ejpam-5960 591 50 𝜌π‘₯𝑖 𝜌π‘₯𝑖 ADJ ejpam-5960 591 51 ) ) PUNCT ejpam-5960 591 52 ) ) PUNCT ejpam-5960 592 1 and and CCONJ ejpam-5960 592 2 so so ADV ejpam-5960 592 3 πœ‚π‘› πœ‚π‘› VERB ejpam-5960 592 4 ≀ ≀ PUNCT ejpam-5960 593 1 𝑐𝑙 𝑐𝑙 PRON ejpam-5960 593 2 ( ( PUNCT ejpam-5960 593 3 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 593 4 ( ( PUNCT ejpam-5960 593 5 𝜌 𝜌 NOUN ejpam-5960 593 6 ) ) PUNCT ejpam-5960 593 7 ) ) PUNCT ejpam-5960 593 8 , , PUNCT ejpam-5960 593 9 whenever whenever SCONJ ejpam-5960 593 10 𝑛 𝑛 PRON ejpam-5960 593 11 β‰₯ β‰₯ NOUN ejpam-5960 593 12 π‘›π‘œ. π‘›π‘œ. NOUN ejpam-5960 593 13 since since SCONJ ejpam-5960 593 14 𝑐𝑙 𝑐𝑙 PROPN ejpam-5960 593 15 ( ( PUNCT ejpam-5960 593 16 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 593 17 ( ( PUNCT ejpam-5960 593 18 𝜌 𝜌 NOUN ejpam-5960 593 19 ) ) PUNCT ejpam-5960 593 20 ) ) PUNCT ejpam-5960 594 1 ∧ ∧ PROPN ejpam-5960 594 2 πœ‡ πœ‡ ADP ejpam-5960 594 3 = = X ejpam-5960 594 4 0𝑋 0𝑋 PROPN ejpam-5960 594 5 , , PUNCT ejpam-5960 594 6 then then ADV ejpam-5960 594 7 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 594 8 ∧ ∧ PROPN ejpam-5960 594 9 πœ‡ πœ‡ ADP ejpam-5960 594 10 = = NOUN ejpam-5960 594 11 0𝑋 0𝑋 NOUN ejpam-5960 594 12 for for ADP ejpam-5960 594 13 every every DET ejpam-5960 594 14 𝑛 𝑛 PROPN ejpam-5960 594 15 ∈ ∈ PROPN ejpam-5960 594 16 𝐷 𝐷 PROPN ejpam-5960 594 17 , , PUNCT ejpam-5960 594 18 𝑛 𝑛 DET ejpam-5960 594 19 β‰₯ β‰₯ NOUN ejpam-5960 594 20 π‘›π‘œ. π‘›π‘œ. VERB ejpam-5960 594 21 conversely conversely ADV ejpam-5960 594 22 , , PUNCT ejpam-5960 594 23 suppose suppose VERB ejpam-5960 594 24 that that SCONJ ejpam-5960 594 25 πœ‡ πœ‡ SCONJ ejpam-5960 594 26 ∈ ∈ PROPN ejpam-5960 594 27 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 594 28 satisfies satisfy VERB ejpam-5960 594 29 the the DET ejpam-5960 594 30 condition condition NOUN ejpam-5960 594 31 of of ADP ejpam-5960 594 32 the the DET ejpam-5960 594 33 theorem theorem NOUN ejpam-5960 594 34 . . PUNCT ejpam-5960 595 1 we we PRON ejpam-5960 595 2 prove prove VERB ejpam-5960 595 3 that that SCONJ ejpam-5960 595 4 πœ‡ πœ‡ ADP ejpam-5960 595 5 is be AUX ejpam-5960 595 6 a a DET ejpam-5960 595 7 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 595 8 set set NOUN ejpam-5960 595 9 . . PUNCT ejpam-5960 596 1 let let VERB ejpam-5960 596 2 ψ ψ PRON ejpam-5960 596 3 βŠ† βŠ† NUM ejpam-5960 596 4 πœβ€² πœβ€² NOUN ejpam-5960 596 5 be be AUX ejpam-5960 596 6 an an DET ejpam-5960 596 7 𝛼–rf 𝛼–rf NUM ejpam-5960 596 8 of of ADP ejpam-5960 596 9 1𝑋. 1𝑋. NUM ejpam-5960 596 10 let let VERB ejpam-5960 596 11 𝐷 𝐷 NOUN ejpam-5960 596 12 = = NOUN ejpam-5960 596 13 2(ψ 2(ψ NUM ejpam-5960 596 14 ) ) PUNCT ejpam-5960 596 15 be be VERB ejpam-5960 596 16 the the DET ejpam-5960 596 17 set set NOUN ejpam-5960 596 18 of of ADP ejpam-5960 596 19 all all DET ejpam-5960 596 20 finite finite ADJ ejpam-5960 596 21 subsets subset NOUN ejpam-5960 596 22 of of ADP ejpam-5960 596 23 ψ ψ PRON ejpam-5960 596 24 directed direct VERB ejpam-5960 596 25 by by ADP ejpam-5960 596 26 inclusion inclusion NOUN ejpam-5960 596 27 , , PUNCT ejpam-5960 596 28 and and CCONJ ejpam-5960 596 29 let let VERB ejpam-5960 596 30 { { PUNCT ejpam-5960 596 31 πœ‚Οˆ πœ‚Οˆ X ejpam-5960 596 32 : : PUNCT ejpam-5960 596 33 ψ ψ X ejpam-5960 596 34 ∈ ∈ PROPN ejpam-5960 596 35 𝐷 𝐷 PROPN ejpam-5960 596 36 } } PUNCT ejpam-5960 596 37 be be AUX ejpam-5960 596 38 a a DET ejpam-5960 596 39 net net NOUN ejpam-5960 596 40 of of ADP ejpam-5960 596 41 closed closed ADJ ejpam-5960 596 42 𝐿–subsets 𝐿–subset NOUN ejpam-5960 596 43 in in ADP ejpam-5960 596 44 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 596 45 such such ADJ ejpam-5960 596 46 that that SCONJ ejpam-5960 596 47 πœ‚Οˆ πœ‚Οˆ ADP ejpam-5960 596 48 = = SYM ejpam-5960 596 49 ∧{𝑐𝑙 ∧{𝑐𝑙 PROPN ejpam-5960 596 50 ( ( PUNCT ejpam-5960 596 51 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 596 52 ( ( PUNCT ejpam-5960 596 53 𝜌 𝜌 NOUN ejpam-5960 596 54 ) ) PUNCT ejpam-5960 596 55 ) ) PUNCT ejpam-5960 596 56 : : PUNCT ejpam-5960 597 1 𝜌 𝜌 X ejpam-5960 597 2 ∈ ∈ PROPN ejpam-5960 597 3 ψ ψ NOUN ejpam-5960 597 4 } } PUNCT ejpam-5960 597 5 . . PUNCT ejpam-5960 598 1 obviously obviously ADV ejpam-5960 598 2 , , PUNCT ejpam-5960 598 3 πœ‚Οˆ1 πœ‚Οˆ1 PROPN ejpam-5960 598 4 ≀ ≀ NUM ejpam-5960 598 5 πœ‚Οˆ2 πœ‚Οˆ2 PROPN ejpam-5960 598 6 iff iff PROPN ejpam-5960 598 7 ψ2 ψ2 VERB ejpam-5960 598 8 βŠ† βŠ† NUM ejpam-5960 598 9 ψ1 ψ1 NOUN ejpam-5960 598 10 . . PUNCT ejpam-5960 599 1 hence hence ADV ejpam-5960 599 2 by by ADP ejpam-5960 599 3 theorem theorem NOUN ejpam-5960 599 4 4.5 4.5 NUM ejpam-5960 599 5 it it PRON ejpam-5960 599 6 follows follow VERB ejpam-5960 599 7 that that PRON ejpam-5960 599 8 𝛿.lim(πœ‚Οˆ 𝛿.lim(πœ‚Οˆ NOUN ejpam-5960 599 9 ) ) PUNCT ejpam-5960 599 10 ( ( PUNCT ejpam-5960 599 11 π‘₯ π‘₯ NOUN ejpam-5960 599 12 ) ) PUNCT ejpam-5960 599 13 ≀ ≀ NOUN ejpam-5960 599 14 ∧{πœ‚Οˆ ∧{πœ‚Οˆ NOUN ejpam-5960 599 15 : : PUNCT ejpam-5960 599 16 ψ ψ X ejpam-5960 599 17 ∈ ∈ PROPN ejpam-5960 599 18 𝐷 𝐷 PROPN ejpam-5960 599 19 } } PUNCT ejpam-5960 599 20 . . PUNCT ejpam-5960 600 1 then then ADV ejpam-5960 600 2 ∧{πœ‚Οˆ ∧{πœ‚Οˆ NOUN ejpam-5960 600 3 : : PUNCT ejpam-5960 600 4 ψ ψ X ejpam-5960 600 5 ∈ ∈ PROPN ejpam-5960 600 6 𝐷}(π‘₯ 𝐷}(π‘₯ NOUN ejpam-5960 600 7 ) ) PUNCT ejpam-5960 600 8 = = SYM ejpam-5960 600 9 ∧(∧{𝑐𝑙 ∧(∧{𝑐𝑙 PROPN ejpam-5960 600 10 ( ( PUNCT ejpam-5960 600 11 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 600 12 ( ( PUNCT ejpam-5960 600 13 𝜌 𝜌 NOUN ejpam-5960 600 14 ) ) PUNCT ejpam-5960 600 15 ) ) PUNCT ejpam-5960 600 16 : : PUNCT ejpam-5960 601 1 𝜌 𝜌 X ejpam-5960 601 2 ∈ ∈ PROPN ejpam-5960 601 3 ψ})(π‘₯ ψ})(π‘₯ PROPN ejpam-5960 601 4 ) ) PUNCT ejpam-5960 601 5 < < X ejpam-5960 601 6 𝛼 𝛼 X ejpam-5960 601 7 for for ADP ejpam-5960 601 8 each each DET ejpam-5960 601 9 π‘₯ π‘₯ PRON ejpam-5960 601 10 ∈ ∈ PROPN ejpam-5960 601 11 𝑋. 𝑋. NOUN ejpam-5960 601 12 thus thus ADV ejpam-5960 601 13 𝛿.lim(πœ‚Οˆ 𝛿.lim(πœ‚Οˆ NOUN ejpam-5960 601 14 ) ) PUNCT ejpam-5960 601 15 ( ( PUNCT ejpam-5960 601 16 π‘₯ π‘₯ NOUN ejpam-5960 601 17 ) ) PUNCT ejpam-5960 601 18 < < X ejpam-5960 601 19 𝛼 𝛼 X ejpam-5960 601 20 for for ADP ejpam-5960 601 21 each each DET ejpam-5960 601 22 π‘₯ π‘₯ PRON ejpam-5960 601 23 ∈ ∈ PROPN ejpam-5960 601 24 𝑋. 𝑋. PROPN ejpam-5960 601 25 by by ADP ejpam-5960 601 26 assumption assumption NOUN ejpam-5960 601 27 , , PUNCT ejpam-5960 601 28 there there PRON ejpam-5960 601 29 exists exist VERB ejpam-5960 601 30 an an DET ejpam-5960 601 31 element element NOUN ejpam-5960 601 32 Οˆπ‘œ Οˆπ‘œ ADP ejpam-5960 601 33 ∈ ∈ PROPN ejpam-5960 601 34 𝐷 𝐷 PROPN ejpam-5960 601 35 for for ADP ejpam-5960 601 36 which which PRON ejpam-5960 601 37 πœ‚Οˆ πœ‚Οˆ NOUN ejpam-5960 601 38 ∧ ∧ PROPN ejpam-5960 601 39 πœ‡ πœ‡ ADP ejpam-5960 601 40 = = NOUN ejpam-5960 601 41 0𝑋 0𝑋 NOUN ejpam-5960 601 42 for for ADP ejpam-5960 601 43 every every DET ejpam-5960 601 44 ψ ψ PROPN ejpam-5960 601 45 ∈ ∈ PROPN ejpam-5960 601 46 𝐷 𝐷 PROPN ejpam-5960 601 47 , , PUNCT ejpam-5960 601 48 ψ ψ X ejpam-5960 601 49 β‰₯ β‰₯ PROPN ejpam-5960 601 50 Οˆπ‘œ. Οˆπ‘œ. VERB ejpam-5960 601 51 by by ADP ejpam-5960 601 52 the the DET ejpam-5960 601 53 above above ADV ejpam-5960 601 54 we we PRON ejpam-5960 601 55 have have VERB ejpam-5960 601 56 πœ‚Οˆπ‘œ πœ‚Οˆπ‘œ NOUN ejpam-5960 601 57 ∧ ∧ PROPN ejpam-5960 601 58 πœ‡ πœ‡ ADP ejpam-5960 601 59 = = X ejpam-5960 601 60 0𝑋 0𝑋 NOUN ejpam-5960 601 61 and and CCONJ ejpam-5960 601 62 so so ADV ejpam-5960 601 63 ( ( PUNCT ejpam-5960 601 64 βˆ€π‘₯𝛼 βˆ€π‘₯𝛼 VERB ejpam-5960 601 65 ∈ ∈ PROPN ejpam-5960 601 66 πœ‡ πœ‡ NOUN ejpam-5960 601 67 ) ) PUNCT ejpam-5960 601 68 ( ( PUNCT ejpam-5960 601 69 βˆƒπ‘π‘™ βˆƒπ‘π‘™ NOUN ejpam-5960 601 70 ( ( PUNCT ejpam-5960 601 71 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 601 72 ( ( PUNCT ejpam-5960 601 73 𝜌 𝜌 NOUN ejpam-5960 601 74 ) ) PUNCT ejpam-5960 601 75 ) ) PUNCT ejpam-5960 602 1 ∈ ∈ PROPN ejpam-5960 602 2 Οˆπ‘œ Οˆπ‘œ PROPN ejpam-5960 602 3 ) ) PUNCT ejpam-5960 602 4 ( ( PUNCT ejpam-5960 602 5 𝑐𝑙 𝑐𝑙 X ejpam-5960 602 6 ( ( PUNCT ejpam-5960 602 7 𝑖𝑛𝑑 𝑖𝑛𝑑 PROPN ejpam-5960 602 8 ( ( PUNCT ejpam-5960 602 9 𝜌 𝜌 NOUN ejpam-5960 602 10 ) ) PUNCT ejpam-5960 602 11 ) ) PUNCT ejpam-5960 603 1 ∈ ∈ PROPN ejpam-5960 603 2 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 603 3 ) ) PUNCT ejpam-5960 603 4 . . PUNCT ejpam-5960 604 1 thus thus ADV ejpam-5960 604 2 Οˆπ‘œ Οˆπ‘œ AUX ejpam-5960 604 3 ∈ ∈ PROPN ejpam-5960 604 4 2(ψ 2(ψ NUM ejpam-5960 604 5 ) ) PUNCT ejpam-5960 604 6 is be AUX ejpam-5960 604 7 an an DET ejpam-5960 604 8 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 604 9 of of ADP ejpam-5960 604 10 πœ‡. πœ‡. NOUN ejpam-5960 604 11 hence hence ADV ejpam-5960 604 12 πœ‡ πœ‡ ADV ejpam-5960 604 13 is be AUX ejpam-5960 604 14 a a DET ejpam-5960 604 15 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 604 16 . . PUNCT ejpam-5960 605 1 theorem theorem VERB ejpam-5960 605 2 4.7 4.7 NUM ejpam-5960 605 3 . . PUNCT ejpam-5960 606 1 an an DET ejpam-5960 606 2 l l NOUN ejpam-5960 606 3 – – PUNCT ejpam-5960 606 4 ts ts X ejpam-5960 606 5 ( ( PUNCT ejpam-5960 606 6 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 606 7 , , PUNCT ejpam-5960 606 8 𝜏 𝜏 NOUN ejpam-5960 606 9 ) ) PUNCT ejpam-5960 606 10 is be AUX ejpam-5960 606 11 a a DET ejpam-5960 606 12 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 606 13 iff iff NOUN ejpam-5960 606 14 for for ADP ejpam-5960 606 15 every every DET ejpam-5960 606 16 a a DET ejpam-5960 606 17 net net NOUN ejpam-5960 606 18 { { PUNCT ejpam-5960 606 19 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 606 20 : : PUNCT ejpam-5960 606 21 𝑛 𝑛 PROPN ejpam-5960 606 22 ∈ ∈ PROPN ejpam-5960 606 23 𝐷 𝐷 PROPN ejpam-5960 606 24 } } PUNCT ejpam-5960 606 25 of of ADP ejpam-5960 606 26 closed closed ADJ ejpam-5960 606 27 𝐿–subsets 𝐿–subset NOUN ejpam-5960 606 28 in in ADP ejpam-5960 606 29 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 606 30 such such ADJ ejpam-5960 606 31 that that DET ejpam-5960 606 32 𝛿lim(πœ‚π‘› 𝛿lim(πœ‚π‘› NOUN ejpam-5960 606 33 ) ) PUNCT ejpam-5960 606 34 ( ( PUNCT ejpam-5960 606 35 π‘₯ π‘₯ NOUN ejpam-5960 606 36 ) ) PUNCT ejpam-5960 606 37 < < X ejpam-5960 606 38 𝛼 𝛼 X ejpam-5960 606 39 for for ADP ejpam-5960 606 40 each each DET ejpam-5960 606 41 π‘₯ π‘₯ PRON ejpam-5960 606 42 ∈ ∈ PROPN ejpam-5960 606 43 𝑋 𝑋 PROPN ejpam-5960 606 44 , , PUNCT ejpam-5960 606 45 there there PRON ejpam-5960 606 46 is be VERB ejpam-5960 606 47 π‘›π‘œ π‘›π‘œ DET ejpam-5960 606 48 ∈ ∈ PROPN ejpam-5960 606 49 𝐷 𝐷 PROPN ejpam-5960 606 50 for for ADP ejpam-5960 606 51 n. n. PROPN ejpam-5960 606 52 a. a. NOUN ejpam-5960 606 53 alsaedi alsaedi PROPN ejpam-5960 606 54 / / SYM ejpam-5960 606 55 eur eur PROPN ejpam-5960 606 56 . . PUNCT ejpam-5960 607 1 j. j. PROPN ejpam-5960 607 2 pure pure PROPN ejpam-5960 607 3 appl appl PROPN ejpam-5960 607 4 . . PROPN ejpam-5960 607 5 math math PROPN ejpam-5960 607 6 , , PUNCT ejpam-5960 607 7 18 18 NUM ejpam-5960 607 8 ( ( PUNCT ejpam-5960 607 9 4 4 NUM ejpam-5960 607 10 ) ) PUNCT ejpam-5960 607 11 ( ( PUNCT ejpam-5960 607 12 2025 2025 NUM ejpam-5960 607 13 ) ) PUNCT ejpam-5960 607 14 , , PUNCT ejpam-5960 607 15 5960 5960 NUM ejpam-5960 607 16 19 19 NUM ejpam-5960 607 17 of of ADP ejpam-5960 607 18 22 22 NUM ejpam-5960 607 19 which which DET ejpam-5960 607 20 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 607 21 = = SYM ejpam-5960 607 22 0𝑋 0𝑋 PROPN ejpam-5960 607 23 for for ADP ejpam-5960 607 24 every every DET ejpam-5960 607 25 𝑛 𝑛 PROPN ejpam-5960 607 26 ∈ ∈ PROPN ejpam-5960 607 27 𝐷 𝐷 PROPN ejpam-5960 607 28 , , PUNCT ejpam-5960 607 29 𝑛 𝑛 DET ejpam-5960 607 30 β‰₯ β‰₯ NOUN ejpam-5960 607 31 π‘›π‘œ. π‘›π‘œ. NOUN ejpam-5960 607 32 proof proof NOUN ejpam-5960 607 33 . . PUNCT ejpam-5960 608 1 let let VERB ejpam-5960 608 2 ( ( PUNCT ejpam-5960 608 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 608 4 , , PUNCT ejpam-5960 608 5 𝜏 𝜏 NOUN ejpam-5960 608 6 ) ) PUNCT ejpam-5960 608 7 be be AUX ejpam-5960 608 8 a a DET ejpam-5960 608 9 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 608 10 . . PUNCT ejpam-5960 609 1 then then ADV ejpam-5960 609 2 1𝑋 1𝑋 PROPN ejpam-5960 609 3 is be AUX ejpam-5960 609 4 a a DET ejpam-5960 609 5 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 609 6 and and CCONJ ejpam-5960 609 7 by by ADP ejpam-5960 609 8 theorem theorem NOUN ejpam-5960 609 9 3.27 3.27 NUM ejpam-5960 609 10 ( ( PUNCT ejpam-5960 609 11 i i NOUN ejpam-5960 609 12 ) ) PUNCT ejpam-5960 609 13 , , PUNCT ejpam-5960 609 14 we we PRON ejpam-5960 609 15 have have AUX ejpam-5960 609 16 1𝑋 1𝑋 PROPN ejpam-5960 609 17 is be AUX ejpam-5960 609 18 a a DET ejpam-5960 609 19 𝑁.𝛼–bounded 𝑁.𝛼–bounded ADJ ejpam-5960 609 20 . . PUNCT ejpam-5960 610 1 let let AUX ejpam-5960 610 2 { { PUNCT ejpam-5960 610 3 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 610 4 : : PUNCT ejpam-5960 610 5 𝑛 𝑛 PROPN ejpam-5960 610 6 ∈ ∈ PROPN ejpam-5960 610 7 𝐷 𝐷 PROPN ejpam-5960 610 8 } } PUNCT ejpam-5960 610 9 be be AUX ejpam-5960 610 10 a a DET ejpam-5960 610 11 net net NOUN ejpam-5960 610 12 of of ADP ejpam-5960 610 13 closed closed ADJ ejpam-5960 610 14 𝐿–subsets 𝐿–subset NOUN ejpam-5960 610 15 in in ADP ejpam-5960 610 16 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 610 17 such such ADJ ejpam-5960 610 18 that that DET ejpam-5960 610 19 𝛿.lim(πœ‚π‘› 𝛿.lim(πœ‚π‘› PROPN ejpam-5960 610 20 ) ) PUNCT ejpam-5960 610 21 ( ( PUNCT ejpam-5960 610 22 π‘₯ π‘₯ NOUN ejpam-5960 610 23 ) ) PUNCT ejpam-5960 610 24 < < X ejpam-5960 610 25 𝛼 𝛼 X ejpam-5960 610 26 for for ADP ejpam-5960 610 27 each each DET ejpam-5960 610 28 π‘₯ π‘₯ DET ejpam-5960 610 29 ∈ ∈ PROPN ejpam-5960 610 30 𝑋. 𝑋. NOUN ejpam-5960 610 31 hence hence ADV ejpam-5960 610 32 by by ADP ejpam-5960 610 33 theorem theorem NOUN ejpam-5960 610 34 4.6 4.6 NUM ejpam-5960 610 35 , , PUNCT ejpam-5960 610 36 there there PRON ejpam-5960 610 37 exists exist VERB ejpam-5960 610 38 𝑛 𝑛 ADP ejpam-5960 610 39 β—¦ β—¦ NOUN ejpam-5960 610 40 ∈ ∈ NOUN ejpam-5960 610 41 𝐷 𝐷 NOUN ejpam-5960 610 42 such such ADJ ejpam-5960 610 43 that that DET ejpam-5960 610 44 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 610 45 ∧ ∧ PROPN ejpam-5960 610 46 1𝑋 1𝑋 PROPN ejpam-5960 610 47 = = SYM ejpam-5960 610 48 0𝑋 0𝑋 PROPN ejpam-5960 610 49 for for ADP ejpam-5960 610 50 every every DET ejpam-5960 610 51 𝑛 𝑛 PROPN ejpam-5960 610 52 ∈ ∈ PROPN ejpam-5960 610 53 𝐷 𝐷 PROPN ejpam-5960 610 54 , , PUNCT ejpam-5960 611 1 𝑛 𝑛 DET ejpam-5960 611 2 β‰₯ β‰₯ NOUN ejpam-5960 611 3 𝑛 𝑛 PRON ejpam-5960 611 4 β—¦ β—¦ NOUN ejpam-5960 611 5 and and CCONJ ejpam-5960 611 6 so so ADV ejpam-5960 611 7 πœ‚π‘› πœ‚π‘› NOUN ejpam-5960 611 8 = = SYM ejpam-5960 611 9 0𝑋 0𝑋 PROPN ejpam-5960 611 10 for for ADP ejpam-5960 611 11 every every DET ejpam-5960 611 12 𝑛 𝑛 PROPN ejpam-5960 611 13 ∈ ∈ PROPN ejpam-5960 611 14 𝐷 𝐷 PROPN ejpam-5960 611 15 , , PUNCT ejpam-5960 611 16 𝑛 𝑛 PRON ejpam-5960 611 17 β‰₯ β‰₯ NOUN ejpam-5960 611 18 𝑛 𝑛 PROPN ejpam-5960 611 19 β—¦ β—¦ NOUN ejpam-5960 611 20 . . PUNCT ejpam-5960 612 1 conversely conversely ADV ejpam-5960 612 2 , , PUNCT ejpam-5960 612 3 suppose suppose VERB ejpam-5960 612 4 that that SCONJ ejpam-5960 612 5 1𝑋 1𝑋 PROPN ejpam-5960 612 6 satisfies satisfy VERB ejpam-5960 612 7 the the DET ejpam-5960 612 8 condition condition NOUN ejpam-5960 612 9 . . PUNCT ejpam-5960 613 1 we we PRON ejpam-5960 613 2 prove prove VERB ejpam-5960 613 3 that that SCONJ ejpam-5960 613 4 1𝑋 1𝑋 PROPN ejpam-5960 613 5 is be AUX ejpam-5960 613 6 a a DET ejpam-5960 613 7 𝑁𝑄𝛼 𝑁𝑄𝛼 NOUN ejpam-5960 613 8 – – PUNCT ejpam-5960 613 9 compact compact ADJ ejpam-5960 613 10 . . PUNCT ejpam-5960 614 1 let let VERB ejpam-5960 614 2 ψ ψ PART ejpam-5960 614 3 be be AUX ejpam-5960 614 4 an an DET ejpam-5960 614 5 𝛼–rf 𝛼–rf NOUN ejpam-5960 614 6 of of ADP ejpam-5960 614 7 1𝑋 1𝑋 NOUN ejpam-5960 614 8 and and CCONJ ejpam-5960 614 9 let let VERB ejpam-5960 614 10 𝐷 𝐷 NOUN ejpam-5960 614 11 = = NOUN ejpam-5960 614 12 2(ψ 2(ψ NUM ejpam-5960 614 13 ) ) PUNCT ejpam-5960 614 14 be be VERB ejpam-5960 614 15 the the DET ejpam-5960 614 16 set set NOUN ejpam-5960 614 17 of of ADP ejpam-5960 614 18 all all DET ejpam-5960 614 19 finite finite ADJ ejpam-5960 614 20 subsets subset NOUN ejpam-5960 614 21 of of ADP ejpam-5960 614 22 ψ ψ PRON ejpam-5960 614 23 directed direct VERB ejpam-5960 614 24 by by ADP ejpam-5960 614 25 inclusion inclusion NOUN ejpam-5960 614 26 , , PUNCT ejpam-5960 614 27 and and CCONJ ejpam-5960 614 28 let let VERB ejpam-5960 614 29 { { PUNCT ejpam-5960 614 30 πœ‚Οˆ πœ‚Οˆ X ejpam-5960 614 31 : : PUNCT ejpam-5960 614 32 ψ ψ X ejpam-5960 614 33 ∈ ∈ PROPN ejpam-5960 614 34 𝐷 𝐷 PROPN ejpam-5960 614 35 } } PUNCT ejpam-5960 614 36 be be AUX ejpam-5960 614 37 a a DET ejpam-5960 614 38 net net NOUN ejpam-5960 614 39 of of ADP ejpam-5960 614 40 closed closed ADJ ejpam-5960 614 41 𝐿–subsets 𝐿–subset NOUN ejpam-5960 614 42 in in ADP ejpam-5960 614 43 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 614 44 such such ADJ ejpam-5960 614 45 that that SCONJ ejpam-5960 614 46 πœ‚Οˆ πœ‚Οˆ ADP ejpam-5960 614 47 = = SYM ejpam-5960 614 48 ∧{𝑐𝑙 ∧{𝑐𝑙 PROPN ejpam-5960 614 49 ( ( PUNCT ejpam-5960 614 50 int(𝜌 int(𝜌 NOUN ejpam-5960 614 51 ) ) PUNCT ejpam-5960 614 52 ) ) PUNCT ejpam-5960 614 53 : : PUNCT ejpam-5960 615 1 𝜌 𝜌 X ejpam-5960 615 2 ∈ ∈ PROPN ejpam-5960 615 3 ψ ψ NOUN ejpam-5960 615 4 } } PUNCT ejpam-5960 615 5 . . PUNCT ejpam-5960 616 1 obviously obviously ADV ejpam-5960 616 2 , , PUNCT ejpam-5960 616 3 πœ‚Οˆ1 πœ‚Οˆ1 PROPN ejpam-5960 616 4 ≀ ≀ NUM ejpam-5960 616 5 πœ‚Οˆ2 πœ‚Οˆ2 PROPN ejpam-5960 616 6 iff iff PROPN ejpam-5960 616 7 ψ2 ψ2 VERB ejpam-5960 616 8 βŠ† βŠ† NUM ejpam-5960 616 9 ψ1 ψ1 NOUN ejpam-5960 616 10 . . PUNCT ejpam-5960 617 1 hence hence ADV ejpam-5960 617 2 , , PUNCT ejpam-5960 617 3 by by ADP ejpam-5960 617 4 theorem theorem NOUN ejpam-5960 617 5 4.6 4.6 NUM ejpam-5960 617 6 , , PUNCT ejpam-5960 617 7 it it PRON ejpam-5960 617 8 follows follow VERB ejpam-5960 617 9 that that PRON ejpam-5960 617 10 𝛿.lim(πœ‚Οˆ 𝛿.lim(πœ‚Οˆ NOUN ejpam-5960 617 11 ) ) PUNCT ejpam-5960 617 12 ≀ ≀ NOUN ejpam-5960 617 13 ∧{πœ‚Οˆ ∧{πœ‚Οˆ NOUN ejpam-5960 617 14 : : PUNCT ejpam-5960 617 15 ψ ψ X ejpam-5960 617 16 ∈ ∈ PROPN ejpam-5960 617 17 𝐷 𝐷 PROPN ejpam-5960 617 18 } } PUNCT ejpam-5960 617 19 . . PUNCT ejpam-5960 618 1 then then ADV ejpam-5960 618 2 ∧{πœ‚Οˆ ∧{πœ‚Οˆ NOUN ejpam-5960 618 3 : : PUNCT ejpam-5960 618 4 ψ ψ X ejpam-5960 618 5 ∈ ∈ PROPN ejpam-5960 618 6 𝐷}(π‘₯ 𝐷}(π‘₯ NOUN ejpam-5960 618 7 ) ) PUNCT ejpam-5960 618 8 = = SYM ejpam-5960 619 1 ∧{𝑐𝑙 ∧{𝑐𝑙 PROPN ejpam-5960 619 2 ( ( PUNCT ejpam-5960 619 3 int(𝜌 int(𝜌 NOUN ejpam-5960 619 4 ) ) PUNCT ejpam-5960 619 5 ) ) PUNCT ejpam-5960 619 6 : : PUNCT ejpam-5960 620 1 𝜌 𝜌 X ejpam-5960 620 2 ∈ ∈ PROPN ejpam-5960 620 3 ψ}(π‘₯ ψ}(π‘₯ PROPN ejpam-5960 620 4 ) ) PUNCT ejpam-5960 620 5 < < X ejpam-5960 620 6 𝛼 𝛼 X ejpam-5960 620 7 for for ADP ejpam-5960 620 8 every every DET ejpam-5960 620 9 π‘₯ π‘₯ PROPN ejpam-5960 620 10 ∈ ∈ PROPN ejpam-5960 620 11 𝑋.and 𝑋.and NOUN ejpam-5960 620 12 so so ADV ejpam-5960 620 13 𝛿.lim(πœ‚π‘› 𝛿.lim(πœ‚π‘› PROPN ejpam-5960 620 14 ) ) PUNCT ejpam-5960 620 15 ( ( PUNCT ejpam-5960 620 16 π‘₯ π‘₯ NOUN ejpam-5960 620 17 ) ) PUNCT ejpam-5960 620 18 < < X ejpam-5960 620 19 𝛼 𝛼 X ejpam-5960 620 20 , , PUNCT ejpam-5960 620 21 for for ADP ejpam-5960 620 22 each each DET ejpam-5960 620 23 π‘₯ π‘₯ DET ejpam-5960 620 24 ∈ ∈ PROPN ejpam-5960 620 25 𝑋. 𝑋. PROPN ejpam-5960 620 26 by by ADP ejpam-5960 620 27 assumption assumption NOUN ejpam-5960 620 28 , , PUNCT ejpam-5960 620 29 there there PRON ejpam-5960 620 30 exists exist VERB ejpam-5960 620 31 an an DET ejpam-5960 620 32 element element NOUN ejpam-5960 620 33 ψ ψ NOUN ejpam-5960 620 34 β—¦ β—¦ NOUN ejpam-5960 620 35 ∈ ∈ NOUN ejpam-5960 620 36 𝐷 𝐷 NOUN ejpam-5960 620 37 for for ADP ejpam-5960 620 38 which which PRON ejpam-5960 620 39 πœ‚Οˆ πœ‚Οˆ NOUN ejpam-5960 620 40 = = SYM ejpam-5960 620 41 0𝑋 0𝑋 NOUN ejpam-5960 620 42 for for ADP ejpam-5960 620 43 every every DET ejpam-5960 620 44 ψ ψ PROPN ejpam-5960 620 45 ∈ ∈ PROPN ejpam-5960 620 46 𝐷 𝐷 PROPN ejpam-5960 620 47 , , PUNCT ejpam-5960 620 48 ψ ψ X ejpam-5960 620 49 β‰₯ β‰₯ NOUN ejpam-5960 620 50 ψ ψ X ejpam-5960 620 51 β—¦ β—¦ NOUN ejpam-5960 620 52 . . PUNCT ejpam-5960 621 1 thus thus ADV ejpam-5960 621 2 πœ‚Οˆ πœ‚Οˆ ADP ejpam-5960 621 3 β—¦ β—¦ NOUN ejpam-5960 621 4 = = SYM ejpam-5960 621 5 0𝑋 0𝑋 NOUN ejpam-5960 621 6 and and CCONJ ejpam-5960 621 7 so so ADV ejpam-5960 621 8 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 621 9 βˆ‰ βˆ‰ PROPN ejpam-5960 621 10 πœ‚Οˆ πœ‚Οˆ PROPN ejpam-5960 621 11 β—¦ β—¦ NOUN ejpam-5960 621 12 = = SYM ejpam-5960 621 13 ∧{𝑐𝑙 ∧{𝑐𝑙 NUM ejpam-5960 621 14 ( ( PUNCT ejpam-5960 621 15 int(𝜌 int(𝜌 NOUN ejpam-5960 621 16 ) ) PUNCT ejpam-5960 621 17 ) ) PUNCT ejpam-5960 621 18 : : PUNCT ejpam-5960 621 19 𝜌 𝜌 X ejpam-5960 621 20 ∈ ∈ PROPN ejpam-5960 621 21 ψ ψ SYM ejpam-5960 621 22 β—¦ β—¦ NOUN ejpam-5960 621 23 } } PUNCT ejpam-5960 621 24 for for ADP ejpam-5960 621 25 every every DET ejpam-5960 621 26 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 621 27 ∈ ∈ PROPN ejpam-5960 621 28 𝑀 𝑀 PROPN ejpam-5960 621 29 ( ( PUNCT ejpam-5960 621 30 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 621 31 ) ) PUNCT ejpam-5960 621 32 and and CCONJ ejpam-5960 621 33 hence hence ADV ejpam-5960 621 34 ψ ψ VERB ejpam-5960 621 35 β—¦ β—¦ NOUN ejpam-5960 621 36 ∈ ∈ NOUN ejpam-5960 621 37 2(ψ 2(ψ NUM ejpam-5960 621 38 ) ) PUNCT ejpam-5960 621 39 is be AUX ejpam-5960 621 40 an an DET ejpam-5960 621 41 𝛼–rcrf 𝛼–rcrf PROPN ejpam-5960 621 42 of of ADP ejpam-5960 621 43 1𝑋.hence 1𝑋.hence NUM ejpam-5960 621 44 1𝑋 1𝑋 NOUN ejpam-5960 621 45 is be AUX ejpam-5960 621 46 a a DET ejpam-5960 621 47 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 621 48 . . PUNCT ejpam-5960 622 1 theorem theorem NOUN ejpam-5960 622 2 4.8 4.8 NUM ejpam-5960 622 3 . . PUNCT ejpam-5960 623 1 if if SCONJ ejpam-5960 623 2 ( ( PUNCT ejpam-5960 623 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 623 4 , , PUNCT ejpam-5960 623 5 𝜏 𝜏 NOUN ejpam-5960 623 6 ) ) PUNCT ejpam-5960 623 7 is be AUX ejpam-5960 623 8 fully fully ADV ejpam-5960 623 9 stratified stratify VERB ejpam-5960 623 10 and and CCONJ ejpam-5960 623 11 𝐿𝑇2 𝐿𝑇2 PROPN ejpam-5960 623 12 – – PUNCT ejpam-5960 623 13 space space NOUN ejpam-5960 623 14 , , PUNCT ejpam-5960 623 15 then then ADV ejpam-5960 623 16 πœ‡ πœ‡ ADP ejpam-5960 623 17 ∈ ∈ PROPN ejpam-5960 623 18 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 623 19 is be AUX ejpam-5960 623 20 a a DET ejpam-5960 623 21 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 623 22 set set VERB ejpam-5960 623 23 iff iff PROPN ejpam-5960 623 24 πœ‡ πœ‡ ADP ejpam-5960 623 25 is be AUX ejpam-5960 623 26 𝛿–closed 𝛿–close VERB ejpam-5960 623 27 and and CCONJ ejpam-5960 623 28 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 623 29 . . PUNCT ejpam-5960 624 1 proof proof NOUN ejpam-5960 624 2 . . PUNCT ejpam-5960 625 1 let let VERB ejpam-5960 625 2 πœ‡ πœ‡ SCONJ ejpam-5960 625 3 ∈ ∈ PROPN ejpam-5960 625 4 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 625 5 be be VERB ejpam-5960 625 6 an an DET ejpam-5960 625 7 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 625 8 , , PUNCT ejpam-5960 625 9 then then ADV ejpam-5960 625 10 by by ADP ejpam-5960 625 11 theorem theorem ADJ ejpam-5960 625 12 2.18 2.18 NUM ejpam-5960 625 13 ( ( PUNCT ejpam-5960 625 14 ii ii NOUN ejpam-5960 625 15 ) ) PUNCT ejpam-5960 625 16 , , PUNCT ejpam-5960 625 17 we we PRON ejpam-5960 625 18 have have VERB ejpam-5960 625 19 πœ‡ πœ‡ ADP ejpam-5960 625 20 is be AUX ejpam-5960 625 21 𝛿–closed 𝛿–close VERB ejpam-5960 625 22 and and CCONJ ejpam-5960 625 23 by by ADP ejpam-5960 625 24 theorem theorem ADJ ejpam-5960 625 25 3.27 3.27 NUM ejpam-5960 625 26 ( ( PUNCT ejpam-5960 625 27 ii ii NOUN ejpam-5960 625 28 ) ) PUNCT ejpam-5960 625 29 , , PUNCT ejpam-5960 625 30 we we PRON ejpam-5960 625 31 have have VERB ejpam-5960 625 32 πœ‡ πœ‡ X ejpam-5960 625 33 is be AUX ejpam-5960 625 34 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 625 35 . . PUNCT ejpam-5960 626 1 conversely conversely ADV ejpam-5960 626 2 , , PUNCT ejpam-5960 626 3 let let VERB ejpam-5960 626 4 πœ‡ πœ‡ PART ejpam-5960 626 5 be be AUX ejpam-5960 626 6 a a DET ejpam-5960 626 7 𝛿–closed 𝛿–closed ADJ ejpam-5960 626 8 and and CCONJ ejpam-5960 626 9 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 626 10 set set NOUN ejpam-5960 626 11 and and CCONJ ejpam-5960 626 12 let let VERB ejpam-5960 626 13 𝑆 𝑆 PROPN ejpam-5960 626 14 be be AUX ejpam-5960 626 15 a a DET ejpam-5960 626 16 constant constant ADJ ejpam-5960 626 17 𝛼–molecular 𝛼–molecular ADJ ejpam-5960 626 18 net net NOUN ejpam-5960 626 19 in in ADP ejpam-5960 626 20 πœ‡. πœ‡. NOUN ejpam-5960 626 21 since since SCONJ ejpam-5960 626 22 πœ‡ πœ‡ ADV ejpam-5960 626 23 is be AUX ejpam-5960 626 24 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 626 25 , , PUNCT ejpam-5960 626 26 then then ADV ejpam-5960 626 27 by by ADP ejpam-5960 626 28 theorem theorem NOUN ejpam-5960 626 29 4.1 4.1 NUM ejpam-5960 626 30 , , PUNCT ejpam-5960 626 31 we we PRON ejpam-5960 626 32 have have VERB ejpam-5960 626 33 𝑆 𝑆 PROPN ejpam-5960 626 34 has have VERB ejpam-5960 626 35 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 626 36 point point NOUN ejpam-5960 626 37 , , PUNCT ejpam-5960 626 38 say say VERB ejpam-5960 626 39 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 626 40 in in ADP ejpam-5960 626 41 𝑋 𝑋 NOUN ejpam-5960 626 42 with with ADP ejpam-5960 626 43 height height NOUN ejpam-5960 626 44 𝛼. 𝛼. NOUN ejpam-5960 626 45 by by ADP ejpam-5960 626 46 theorem theorem ADJ ejpam-5960 626 47 2.15 2.15 NUM ejpam-5960 626 48 ( ( PUNCT ejpam-5960 626 49 j j PROPN ejpam-5960 626 50 ) ) PUNCT ejpam-5960 626 51 , , PUNCT ejpam-5960 626 52 then then ADV ejpam-5960 626 53 there there PRON ejpam-5960 626 54 is be VERB ejpam-5960 626 55 a a DET ejpam-5960 626 56 subnet subnet NOUN ejpam-5960 626 57 𝑇 𝑇 PROPN ejpam-5960 626 58 of of ADP ejpam-5960 626 59 𝑆 𝑆 PROPN ejpam-5960 626 60 such such ADJ ejpam-5960 626 61 that that SCONJ ejpam-5960 626 62 𝑇 𝑇 PROPN ejpam-5960 626 63 𝛿–converges 𝛿–converge NOUN ejpam-5960 626 64 to to AUX ejpam-5960 626 65 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 627 1 and and CCONJ ejpam-5960 627 2 so so ADV ejpam-5960 627 3 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 627 4 ∈ ∈ PROPN ejpam-5960 627 5 𝛿𝑐𝑙 𝛿𝑐𝑙 X ejpam-5960 627 6 ( ( PUNCT ejpam-5960 627 7 πœ‡ πœ‡ NOUN ejpam-5960 627 8 ) ) PUNCT ejpam-5960 627 9 by by ADP ejpam-5960 627 10 theorem theorem ADJ ejpam-5960 627 11 3.22 3.22 NUM ejpam-5960 627 12 ( ( PUNCT ejpam-5960 627 13 vi vi NOUN ejpam-5960 627 14 ) ) PUNCT ejpam-5960 627 15 . . PUNCT ejpam-5960 628 1 since since SCONJ ejpam-5960 628 2 πœ‡ πœ‡ ADV ejpam-5960 628 3 is be AUX ejpam-5960 628 4 𝛿–closed 𝛿–close VERB ejpam-5960 628 5 , , PUNCT ejpam-5960 628 6 then then ADV ejpam-5960 628 7 𝛿𝑐𝑙 𝛿𝑐𝑙 X ejpam-5960 628 8 ( ( PUNCT ejpam-5960 628 9 πœ‡ πœ‡ NOUN ejpam-5960 628 10 ) ) PUNCT ejpam-5960 628 11 = = SYM ejpam-5960 628 12 πœ‡ πœ‡ ADP ejpam-5960 629 1 and and CCONJ ejpam-5960 629 2 so so ADV ejpam-5960 629 3 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 629 4 ∈ ∈ PROPN ejpam-5960 629 5 πœ‡ πœ‡ ADP ejpam-5960 629 6 , , PUNCT ejpam-5960 629 7 then then ADV ejpam-5960 629 8 by by ADP ejpam-5960 629 9 theorem theorem NOUN ejpam-5960 629 10 2.19 2.19 NUM ejpam-5960 629 11 , , PUNCT ejpam-5960 629 12 we we PRON ejpam-5960 629 13 have have VERB ejpam-5960 629 14 πœ‡ πœ‡ ADP ejpam-5960 629 15 is be AUX ejpam-5960 629 16 a a DET ejpam-5960 629 17 compact compact ADJ ejpam-5960 629 18 set set NOUN ejpam-5960 629 19 . . PUNCT ejpam-5960 630 1 theorem theorem VERB ejpam-5960 630 2 4.9 4.9 NUM ejpam-5960 630 3 . . PUNCT ejpam-5960 631 1 if if SCONJ ejpam-5960 631 2 ( ( PUNCT ejpam-5960 631 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 631 4 , , PUNCT ejpam-5960 631 5 𝜏 𝜏 NOUN ejpam-5960 631 6 ) ) PUNCT ejpam-5960 631 7 is be AUX ejpam-5960 631 8 𝐿𝑅2 𝐿𝑅2 PROPN ejpam-5960 631 9 – – PUNCT ejpam-5960 631 10 space space NOUN ejpam-5960 631 11 , , PUNCT ejpam-5960 631 12 then then ADV ejpam-5960 631 13 πœ‡ πœ‡ ADP ejpam-5960 631 14 ∈ ∈ PROPN ejpam-5960 631 15 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 631 16 is be AUX ejpam-5960 631 17 a a DET ejpam-5960 631 18 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 631 19 set set NOUN ejpam-5960 631 20 iff iff PROPN ejpam-5960 631 21 𝛿𝑐𝑙 𝛿𝑐𝑙 ADP ejpam-5960 631 22 ( ( PUNCT ejpam-5960 631 23 πœ‡ πœ‡ NOUN ejpam-5960 631 24 ) ) PUNCT ejpam-5960 631 25 is be AUX ejpam-5960 631 26 a a DET ejpam-5960 631 27 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 631 28 set set NOUN ejpam-5960 631 29 . . PUNCT ejpam-5960 632 1 proof proof NOUN ejpam-5960 632 2 . . PUNCT ejpam-5960 633 1 if if SCONJ ejpam-5960 633 2 𝛿.𝑐𝑙 𝛿.𝑐𝑙 ADJ ejpam-5960 633 3 ( ( PUNCT ejpam-5960 633 4 πœ‡ πœ‡ NOUN ejpam-5960 633 5 ) ) PUNCT ejpam-5960 633 6 is be AUX ejpam-5960 633 7 a a DET ejpam-5960 633 8 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 633 9 set set NOUN ejpam-5960 633 10 , , PUNCT ejpam-5960 633 11 then then ADV ejpam-5960 633 12 πœ‡ πœ‡ SCONJ ejpam-5960 633 13 is be AUX ejpam-5960 633 14 a a DET ejpam-5960 633 15 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 633 16 set set NOUN ejpam-5960 633 17 ( ( PUNCT ejpam-5960 633 18 by by ADP ejpam-5960 633 19 theorem theorem NOUN ejpam-5960 633 20 3.7 3.7 NUM ejpam-5960 633 21 ) ) PUNCT ejpam-5960 633 22 . . PUNCT ejpam-5960 634 1 conversely conversely ADV ejpam-5960 634 2 , , PUNCT ejpam-5960 634 3 suppose suppose VERB ejpam-5960 634 4 that that SCONJ ejpam-5960 634 5 πœ‡ πœ‡ ADP ejpam-5960 634 6 is be AUX ejpam-5960 634 7 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 634 8 and and CCONJ ejpam-5960 634 9 ψ ψ X ejpam-5960 634 10 = = X ejpam-5960 634 11 { { PUNCT ejpam-5960 634 12 πœ‚π‘₯ πœ‚π‘₯ INTJ ejpam-5960 634 13 𝑗 𝑗 INTJ ejpam-5960 634 14 : : PUNCT ejpam-5960 634 15 𝑗 𝑗 PROPN ejpam-5960 634 16 ∈ ∈ PROPN ejpam-5960 634 17 𝐽 𝐽 PROPN ejpam-5960 634 18 } } PUNCT ejpam-5960 634 19 is be AUX ejpam-5960 634 20 an an DET ejpam-5960 634 21 𝛼–rf 𝛼–rf NUM ejpam-5960 634 22 of of ADP ejpam-5960 634 23 1𝑋. 1𝑋. NUM ejpam-5960 634 24 then then ADV ejpam-5960 634 25 for for SCONJ ejpam-5960 634 26 each each DET ejpam-5960 634 27 π‘₯ π‘₯ PRON ejpam-5960 634 28 ∈ ∈ PROPN ejpam-5960 634 29 𝑋 𝑋 NOUN ejpam-5960 634 30 there there PRON ejpam-5960 634 31 is be VERB ejpam-5960 634 32 πœ‚π‘₯ πœ‚π‘₯ ADP ejpam-5960 634 33 𝑗 𝑗 PROPN ejpam-5960 634 34 ∈ ∈ NOUN ejpam-5960 634 35 ψ ψ NOUN ejpam-5960 634 36 such such ADJ ejpam-5960 634 37 that that PRON ejpam-5960 634 38 πœ‚π‘₯ πœ‚π‘₯ PROPN ejpam-5960 634 39 𝑗 𝑗 PRON ejpam-5960 634 40 ∈ ∈ PROPN ejpam-5960 634 41 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 634 42 . . PUNCT ejpam-5960 635 1 since since SCONJ ejpam-5960 635 2 ( ( PUNCT ejpam-5960 635 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 635 4 , , PUNCT ejpam-5960 635 5 𝜏 𝜏 NOUN ejpam-5960 635 6 ) ) PUNCT ejpam-5960 635 7 is be AUX ejpam-5960 635 8 𝐿𝑅2 𝐿𝑅2 PROPN ejpam-5960 635 9 – – PUNCT ejpam-5960 635 10 space space NOUN ejpam-5960 635 11 , , PUNCT ejpam-5960 635 12 then then ADV ejpam-5960 635 13 there there PRON ejpam-5960 635 14 is be VERB ejpam-5960 635 15 πœ†π‘₯ πœ†π‘₯ NOUN ejpam-5960 635 16 𝑗 𝑗 PROPN ejpam-5960 635 17 ∈ ∈ PROPN ejpam-5960 635 18 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 635 19 and and CCONJ ejpam-5960 635 20 there there PRON ejpam-5960 635 21 is be VERB ejpam-5960 635 22 𝜌π‘₯ 𝜌π‘₯ ADP ejpam-5960 635 23 𝑗 𝑗 PROPN ejpam-5960 635 24 ∈ ∈ PROPN ejpam-5960 635 25 πœβ€² πœβ€² NOUN ejpam-5960 635 26 such such ADJ ejpam-5960 635 27 that that PRON ejpam-5960 635 28 πœ†π‘₯ πœ†π‘₯ NOUN ejpam-5960 635 29 𝑗 𝑗 PROPN ejpam-5960 635 30 ∨ ∨ NUM ejpam-5960 635 31 𝜌π‘₯ 𝜌π‘₯ NOUN ejpam-5960 635 32 𝑗 𝑗 NOUN ejpam-5960 635 33 = = X ejpam-5960 635 34 1𝑋 1𝑋 PROPN ejpam-5960 635 35 and and CCONJ ejpam-5960 635 36 𝜌π‘₯ 𝜌π‘₯ VERB ejpam-5960 635 37 𝑗 𝑗 INTJ ejpam-5960 635 38 ∧ ∧ NOUN ejpam-5960 636 1 πœ‚π‘₯ πœ‚π‘₯ INTJ ejpam-5960 636 2 𝑗 𝑗 NOUN ejpam-5960 636 3 = = NOUN ejpam-5960 636 4 0𝑋. 0𝑋. NUM ejpam-5960 636 5 then then ADV ejpam-5960 637 1 the the DET ejpam-5960 637 2 family family NOUN ejpam-5960 637 3 { { PUNCT ejpam-5960 637 4 πœ†π‘₯ πœ†π‘₯ NOUN ejpam-5960 637 5 𝑗 𝑗 INTJ ejpam-5960 637 6 : : PUNCT ejpam-5960 637 7 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 637 8 ∈ ∈ PROPN ejpam-5960 637 9 𝑀 𝑀 PROPN ejpam-5960 637 10 ( ( PUNCT ejpam-5960 637 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 637 12 ) ) PUNCT ejpam-5960 637 13 } } PUNCT ejpam-5960 637 14 is be AUX ejpam-5960 637 15 an an DET ejpam-5960 637 16 𝛼–rf 𝛼–rf NUM ejpam-5960 637 17 of of ADP ejpam-5960 637 18 1𝑋. 1𝑋. NUM ejpam-5960 637 19 since since SCONJ ejpam-5960 637 20 πœ‡ πœ‡ ADV ejpam-5960 637 21 is be AUX ejpam-5960 637 22 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 637 23 , , PUNCT ejpam-5960 637 24 then then ADV ejpam-5960 637 25 exists exist VERB ejpam-5960 637 26 finite finite PROPN ejpam-5960 637 27 subset subset NOUN ejpam-5960 637 28 𝐽 𝐽 X ejpam-5960 637 29 β—¦ β—¦ NOUN ejpam-5960 637 30 of of ADP ejpam-5960 637 31 𝐽 𝐽 PRON ejpam-5960 637 32 such such ADJ ejpam-5960 637 33 that that SCONJ ejpam-5960 637 34 { { PUNCT ejpam-5960 637 35 πœ†π‘₯ πœ†π‘₯ NOUN ejpam-5960 637 36 𝑗 𝑗 INTJ ejpam-5960 637 37 : : PUNCT ejpam-5960 637 38 𝑗 𝑗 PROPN ejpam-5960 637 39 ∈ ∈ PROPN ejpam-5960 637 40 𝐽 𝐽 NOUN ejpam-5960 637 41 β—¦ β—¦ NOUN ejpam-5960 637 42 } } PUNCT ejpam-5960 637 43 is be AUX ejpam-5960 637 44 an an DET ejpam-5960 637 45 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 637 46 of of ADP ejpam-5960 637 47 πœ‡. πœ‡. NOUN ejpam-5960 637 48 since since SCONJ ejpam-5960 637 49 πœ†π‘₯ πœ†π‘₯ PROPN ejpam-5960 637 50 𝑗 𝑗 PROPN ejpam-5960 637 51 ∨ ∨ NUM ejpam-5960 637 52 𝜌π‘₯ 𝜌π‘₯ NOUN ejpam-5960 637 53 𝑗 𝑗 NOUN ejpam-5960 637 54 = = X ejpam-5960 637 55 1𝑋 1𝑋 PROPN ejpam-5960 637 56 , , PUNCT ejpam-5960 637 57 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 637 58 βˆ‰ βˆ‰ PROPN ejpam-5960 637 59 πœ†π‘₯ πœ†π‘₯ X ejpam-5960 637 60 𝑗 𝑗 INTJ ejpam-5960 637 61 , , PUNCT ejpam-5960 637 62 then then ADV ejpam-5960 637 63 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 637 64 ∈ ∈ PROPN ejpam-5960 637 65 𝜌π‘₯ 𝜌π‘₯ NOUN ejpam-5960 637 66 𝑗 𝑗 INTJ ejpam-5960 637 67 . . PUNCT ejpam-5960 638 1 since since SCONJ ejpam-5960 638 2 𝜌π‘₯ 𝜌π‘₯ VERB ejpam-5960 638 3 𝑗 𝑗 INTJ ejpam-5960 638 4 ∧ ∧ NOUN ejpam-5960 638 5 πœ‚π‘₯ πœ‚π‘₯ INTJ ejpam-5960 638 6 𝑗 𝑗 NOUN ejpam-5960 638 7 = = ADJ ejpam-5960 638 8 0𝑋 0𝑋 PROPN ejpam-5960 638 9 , , PUNCT ejpam-5960 638 10 then then ADV ejpam-5960 638 11 { { PUNCT ejpam-5960 638 12 πœ‚π‘₯ πœ‚π‘₯ NOUN ejpam-5960 638 13 𝑗 𝑗 VERB ejpam-5960 638 14 : : PUNCT ejpam-5960 638 15 𝑗 𝑗 PROPN ejpam-5960 638 16 ∈ ∈ PROPN ejpam-5960 638 17 𝐽 𝐽 NOUN ejpam-5960 638 18 β—¦ β—¦ NOUN ejpam-5960 638 19 } } PUNCT ejpam-5960 638 20 is be AUX ejpam-5960 638 21 an an DET ejpam-5960 638 22 𝛼–rcrf 𝛼–rcrf PROPN ejpam-5960 638 23 of of ADP ejpam-5960 638 24 𝜌π‘₯ 𝜌π‘₯ NOUN ejpam-5960 638 25 𝑗 𝑗 INTJ ejpam-5960 638 26 . . PUNCT ejpam-5960 639 1 therefore therefore ADV ejpam-5960 639 2 πœ‡ πœ‡ ADP ejpam-5960 639 3 ≀ ≀ NUM ejpam-5960 639 4 𝜌π‘₯ 𝜌π‘₯ NOUN ejpam-5960 639 5 𝑗 𝑗 NOUN ejpam-5960 639 6 for for ADP ejpam-5960 639 7 𝐽 𝐽 PROPN ejpam-5960 639 8 ∈ ∈ PROPN ejpam-5960 639 9 𝐽 𝐽 PROPN ejpam-5960 639 10 β—¦ β—¦ NOUN ejpam-5960 639 11 . . PUNCT ejpam-5960 640 1 since since SCONJ ejpam-5960 640 2 𝜌π‘₯ 𝜌π‘₯ VERB ejpam-5960 640 3 𝑗 𝑗 PROPN ejpam-5960 640 4 ∈ ∈ PROPN ejpam-5960 640 5 πœβ€² πœβ€² PROPN ejpam-5960 640 6 and and CCONJ ejpam-5960 640 7 ( ( PUNCT ejpam-5960 640 8 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 640 9 , , PUNCT ejpam-5960 640 10 𝜏 𝜏 NOUN ejpam-5960 640 11 ) ) PUNCT ejpam-5960 640 12 is be AUX ejpam-5960 640 13 𝐿𝑆𝑅2 𝐿𝑆𝑅2 NOUN ejpam-5960 640 14 – – PUNCT ejpam-5960 640 15 space space NOUN ejpam-5960 640 16 , , PUNCT ejpam-5960 640 17 then then ADV ejpam-5960 640 18 by by ADP ejpam-5960 640 19 theorem theorem NOUN ejpam-5960 640 20 2.21 2.21 NUM ejpam-5960 640 21 , , PUNCT ejpam-5960 640 22 we we PRON ejpam-5960 640 23 have have AUX ejpam-5960 640 24 𝛿.𝑐𝑙 𝛿.𝑐𝑙 VERB ejpam-5960 640 25 ( ( PUNCT ejpam-5960 640 26 𝜌π‘₯ 𝜌π‘₯ VERB ejpam-5960 640 27 𝑗 𝑗 NOUN ejpam-5960 640 28 ) ) PUNCT ejpam-5960 641 1 = = SYM ejpam-5960 641 2 𝜌π‘₯ 𝜌π‘₯ VERB ejpam-5960 641 3 𝑗 𝑗 NOUN ejpam-5960 642 1 and and CCONJ ejpam-5960 642 2 so so ADV ejpam-5960 642 3 { { PUNCT ejpam-5960 642 4 πœ‚π‘₯ πœ‚π‘₯ NOUN ejpam-5960 642 5 𝑗 𝑗 X ejpam-5960 642 6 : : PUNCT ejpam-5960 642 7 𝑗 𝑗 PROPN ejpam-5960 642 8 ∈ ∈ PROPN ejpam-5960 642 9 𝐽 𝐽 NOUN ejpam-5960 642 10 β—¦ β—¦ NOUN ejpam-5960 642 11 } } PUNCT ejpam-5960 642 12 is be AUX ejpam-5960 642 13 an an DET ejpam-5960 642 14 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 642 15 of of ADP ejpam-5960 642 16 𝛿.𝑐𝑙 𝛿.𝑐𝑙 PROPN ejpam-5960 642 17 ( ( PUNCT ejpam-5960 642 18 𝜌π‘₯ 𝜌π‘₯ NOUN ejpam-5960 642 19 𝑗 𝑗 NOUN ejpam-5960 642 20 ) ) PUNCT ejpam-5960 642 21 and and CCONJ ejpam-5960 643 1 since since SCONJ ejpam-5960 643 2 𝛿.𝑐𝑙 𝛿.𝑐𝑙 PROPN ejpam-5960 643 3 ( ( PUNCT ejpam-5960 643 4 πœ‡ πœ‡ NOUN ejpam-5960 643 5 ) ) PUNCT ejpam-5960 643 6 ≀ ≀ NOUN ejpam-5960 643 7 𝛿.𝑐𝑙 𝛿.𝑐𝑙 PROPN ejpam-5960 643 8 ( ( PUNCT ejpam-5960 643 9 𝜌π‘₯ 𝜌π‘₯ VERB ejpam-5960 643 10 𝑗 𝑗 NOUN ejpam-5960 643 11 ) ) PUNCT ejpam-5960 643 12 , , PUNCT ejpam-5960 643 13 then then ADV ejpam-5960 643 14 { { PUNCT ejpam-5960 643 15 πœ‚π‘₯ πœ‚π‘₯ NOUN ejpam-5960 643 16 𝑗 𝑗 VERB ejpam-5960 643 17 : : PUNCT ejpam-5960 643 18 𝑗 𝑗 PROPN ejpam-5960 643 19 ∈ ∈ PROPN ejpam-5960 643 20 𝐽 𝐽 NOUN ejpam-5960 643 21 β—¦ β—¦ NOUN ejpam-5960 643 22 } } PUNCT ejpam-5960 643 23 is be AUX ejpam-5960 643 24 an an DET ejpam-5960 643 25 𝛼–rcrf 𝛼–rcrf NOUN ejpam-5960 643 26 of of ADP ejpam-5960 643 27 𝛿.𝑐𝑙 𝛿.𝑐𝑙 PROPN ejpam-5960 643 28 ( ( PUNCT ejpam-5960 643 29 πœ‡ πœ‡ NOUN ejpam-5960 643 30 ) ) PUNCT ejpam-5960 643 31 . . PUNCT ejpam-5960 644 1 hence hence ADV ejpam-5960 644 2 𝛿.𝑐𝑙 𝛿.𝑐𝑙 ADJ ejpam-5960 644 3 ( ( PUNCT ejpam-5960 644 4 πœ‡ πœ‡ NOUN ejpam-5960 644 5 ) ) PUNCT ejpam-5960 644 6 is be AUX ejpam-5960 644 7 a a DET ejpam-5960 644 8 𝑁.𝛼–bounded 𝑁.𝛼–bounde VERB ejpam-5960 644 9 set set NOUN ejpam-5960 644 10 . . PUNCT ejpam-5960 645 1 theorem theorem VERB ejpam-5960 645 2 4.10 4.10 NUM ejpam-5960 645 3 . . PUNCT ejpam-5960 646 1 if if SCONJ ejpam-5960 646 2 ( ( PUNCT ejpam-5960 646 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 646 4 , , PUNCT ejpam-5960 646 5 𝜏 𝜏 NOUN ejpam-5960 646 6 ) ) PUNCT ejpam-5960 646 7 is be AUX ejpam-5960 646 8 a a DET ejpam-5960 646 9 𝐿𝑇3 𝐿𝑇3 PROPN ejpam-5960 646 10 – – PUNCT ejpam-5960 646 11 space space NOUN ejpam-5960 646 12 , , PUNCT ejpam-5960 646 13 then then ADV ejpam-5960 646 14 πœ‡ πœ‡ ADP ejpam-5960 646 15 ∈ ∈ PROPN ejpam-5960 646 16 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 646 17 is be AUX ejpam-5960 646 18 a a DET ejpam-5960 646 19 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 646 20 set set VERB ejpam-5960 646 21 iff iff PROPN ejpam-5960 646 22 πœ‡ πœ‡ ADP ejpam-5960 646 23 is be AUX ejpam-5960 646 24 a a DET ejpam-5960 646 25 𝐿–subset 𝐿–subset NOUN ejpam-5960 646 26 of of ADP ejpam-5960 646 27 a a DET ejpam-5960 646 28 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 646 29 set set VERB ejpam-5960 646 30 . . PUNCT ejpam-5960 647 1 n. n. PROPN ejpam-5960 647 2 a. a. PROPN ejpam-5960 647 3 alsaedi alsaedi PROPN ejpam-5960 647 4 / / SYM ejpam-5960 647 5 eur eur PROPN ejpam-5960 647 6 . . PUNCT ejpam-5960 648 1 j. j. PROPN ejpam-5960 648 2 pure pure PROPN ejpam-5960 648 3 appl appl PROPN ejpam-5960 648 4 . . PROPN ejpam-5960 648 5 math math PROPN ejpam-5960 648 6 , , PUNCT ejpam-5960 648 7 18 18 NUM ejpam-5960 648 8 ( ( PUNCT ejpam-5960 648 9 4 4 NUM ejpam-5960 648 10 ) ) PUNCT ejpam-5960 648 11 ( ( PUNCT ejpam-5960 648 12 2025 2025 NUM ejpam-5960 648 13 ) ) PUNCT ejpam-5960 648 14 , , PUNCT ejpam-5960 648 15 5960 5960 NUM ejpam-5960 648 16 20 20 NUM ejpam-5960 648 17 of of ADP ejpam-5960 648 18 22 22 NUM ejpam-5960 648 19 proof proof NOUN ejpam-5960 648 20 . . PUNCT ejpam-5960 649 1 if if SCONJ ejpam-5960 649 2 πœ‡ πœ‡ PRON ejpam-5960 649 3 is be AUX ejpam-5960 649 4 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 649 5 , , PUNCT ejpam-5960 649 6 then then ADV ejpam-5960 649 7 by by ADP ejpam-5960 649 8 theorem theorem NOUN ejpam-5960 649 9 4.9 4.9 NUM ejpam-5960 649 10 , , PUNCT ejpam-5960 649 11 and and CCONJ ejpam-5960 649 12 corollary corollary ADJ ejpam-5960 649 13 2.22 2.22 NUM ejpam-5960 649 14 , , PUNCT ejpam-5960 649 15 we we PRON ejpam-5960 649 16 have have VERB ejpam-5960 649 17 𝛿𝑐𝑙 𝛿𝑐𝑙 ADP ejpam-5960 649 18 ( ( PUNCT ejpam-5960 649 19 πœ‡ πœ‡ NOUN ejpam-5960 649 20 ) ) PUNCT ejpam-5960 649 21 is be AUX ejpam-5960 649 22 𝛿–closed 𝛿–close VERB ejpam-5960 649 23 and and CCONJ ejpam-5960 649 24 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 649 25 set set NOUN ejpam-5960 649 26 , , PUNCT ejpam-5960 649 27 hence hence ADV ejpam-5960 649 28 by by ADP ejpam-5960 649 29 theorem theorem NOUN ejpam-5960 649 30 4.8 4.8 NUM ejpam-5960 649 31 , , PUNCT ejpam-5960 649 32 we we PRON ejpam-5960 649 33 have have VERB ejpam-5960 649 34 𝛿𝑐𝑙 𝛿𝑐𝑙 ADP ejpam-5960 649 35 ( ( PUNCT ejpam-5960 649 36 πœ‡ πœ‡ NOUN ejpam-5960 649 37 ) ) PUNCT ejpam-5960 649 38 is be AUX ejpam-5960 649 39 a a DET ejpam-5960 649 40 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 649 41 set set VERB ejpam-5960 649 42 . . PUNCT ejpam-5960 650 1 conversely conversely ADV ejpam-5960 650 2 , , PUNCT ejpam-5960 650 3 if if SCONJ ejpam-5960 650 4 πœ‡ πœ‡ ADV ejpam-5960 650 5 is be AUX ejpam-5960 650 6 a a DET ejpam-5960 650 7 𝐿–subset 𝐿–subset NOUN ejpam-5960 650 8 of of ADP ejpam-5960 650 9 𝑁𝑄𝛼–compact 𝑁𝑄𝛼–compact NOUN ejpam-5960 650 10 set set NOUN ejpam-5960 650 11 , , PUNCT ejpam-5960 650 12 then then ADV ejpam-5960 650 13 by by ADP ejpam-5960 650 14 theorem theorem ADJ ejpam-5960 650 15 3.27 3.27 NUM ejpam-5960 650 16 ( ( PUNCT ejpam-5960 650 17 iii iii NOUN ejpam-5960 650 18 ) ) PUNCT ejpam-5960 650 19 , , PUNCT ejpam-5960 650 20 we we PRON ejpam-5960 650 21 have have AUX ejpam-5960 650 22 πœ‡ πœ‡ ADP ejpam-5960 650 23 is be AUX ejpam-5960 650 24 a a DET ejpam-5960 650 25 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 650 26 set set NOUN ejpam-5960 650 27 . . PUNCT ejpam-5960 651 1 theorem theorem PROPN ejpam-5960 651 2 4.11 4.11 NUM ejpam-5960 651 3 . . PUNCT ejpam-5960 652 1 assume assume VERB ejpam-5960 652 2 that that SCONJ ejpam-5960 652 3 𝑆 𝑆 PROPN ejpam-5960 652 4 = = SYM ejpam-5960 652 5 { { PUNCT ejpam-5960 652 6 𝑆(𝑛 𝑆(𝑛 NOUN ejpam-5960 652 7 ) ) PUNCT ejpam-5960 652 8 : : PUNCT ejpam-5960 652 9 𝑛 𝑛 PROPN ejpam-5960 652 10 ∈ ∈ PROPN ejpam-5960 652 11 𝐷 𝐷 PROPN ejpam-5960 652 12 } } PUNCT ejpam-5960 652 13 is be AUX ejpam-5960 652 14 a a DET ejpam-5960 652 15 molecular molecular ADJ ejpam-5960 652 16 net net NOUN ejpam-5960 652 17 in in ADP ejpam-5960 652 18 a a DET ejpam-5960 652 19 l l NOUN ejpam-5960 652 20 – – PUNCT ejpam-5960 652 21 ts ts X ejpam-5960 652 22 ( ( PUNCT ejpam-5960 652 23 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 652 24 , , PUNCT ejpam-5960 652 25 𝜏 𝜏 NOUN ejpam-5960 652 26 ) ) PUNCT ejpam-5960 652 27 and and CCONJ ejpam-5960 652 28 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 652 29 ∈ ∈ PROPN ejpam-5960 652 30 𝑀 𝑀 PROPN ejpam-5960 652 31 ( ( PUNCT ejpam-5960 652 32 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 652 33 ) ) PUNCT ejpam-5960 652 34 . . PUNCT ejpam-5960 653 1 then then ADV ejpam-5960 653 2 the the DET ejpam-5960 653 3 following follow VERB ejpam-5960 653 4 results result NOUN ejpam-5960 653 5 are be AUX ejpam-5960 653 6 true true ADJ ejpam-5960 653 7 : : PUNCT ejpam-5960 653 8 ( ( PUNCT ejpam-5960 653 9 i i NOUN ejpam-5960 653 10 ) ) PUNCT ejpam-5960 653 11 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 653 12 is be AUX ejpam-5960 653 13 a a DET ejpam-5960 653 14 𝑁𝛼𝐡–cluster 𝑁𝛼𝐡–clust ADJ ejpam-5960 653 15 point point NOUN ejpam-5960 653 16 of of ADP ejpam-5960 653 17 𝑆 𝑆 PROPN ejpam-5960 653 18 iff iff PROPN ejpam-5960 653 19 there there PRON ejpam-5960 653 20 exists exist VERB ejpam-5960 653 21 a a DET ejpam-5960 653 22 subnet subnet NOUN ejpam-5960 653 23 𝑇 𝑇 PROPN ejpam-5960 653 24 of of ADP ejpam-5960 653 25 𝑆 𝑆 PROPN ejpam-5960 653 26 such such ADJ ejpam-5960 653 27 that that SCONJ ejpam-5960 653 28 𝑇 𝑇 PROPN ejpam-5960 653 29 is be AUX ejpam-5960 653 30 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 653 31 to to PART ejpam-5960 653 32 π‘₯𝛼. π‘₯𝛼. PROPN ejpam-5960 653 33 ( ( PUNCT ejpam-5960 653 34 ii ii NOUN ejpam-5960 653 35 ) ) PUNCT ejpam-5960 653 36 if if SCONJ ejpam-5960 653 37 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 653 38 is be VERB ejpam-5960 653 39 a a DET ejpam-5960 653 40 𝑁𝛼𝐡–cluster 𝑁𝛼𝐡–clust ADJ ejpam-5960 653 41 point point NOUN ejpam-5960 653 42 of of ADP ejpam-5960 653 43 𝑆 𝑆 PROPN ejpam-5960 653 44 , , PUNCT ejpam-5960 653 45 then then ADV ejpam-5960 653 46 𝑇 𝑇 PROPN ejpam-5960 653 47 is be AUX ejpam-5960 653 48 a a DET ejpam-5960 653 49 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 653 50 to to PART ejpam-5960 653 51 π‘₯𝛼 π‘₯𝛼 ADP ejpam-5960 653 52 for for ADP ejpam-5960 653 53 each each DET ejpam-5960 653 54 subnet subnet NOUN ejpam-5960 653 55 𝑇 𝑇 PROPN ejpam-5960 653 56 of of ADP ejpam-5960 653 57 𝑆. 𝑆. PROPN ejpam-5960 653 58 proof proof NOUN ejpam-5960 653 59 . . PUNCT ejpam-5960 654 1 ( ( PUNCT ejpam-5960 654 2 i i NOUN ejpam-5960 654 3 ) ) PUNCT ejpam-5960 654 4 provided provide VERB ejpam-5960 654 5 that that PRON ejpam-5960 654 6 𝑆 𝑆 PROPN ejpam-5960 654 7 = = SYM ejpam-5960 654 8 { { PUNCT ejpam-5960 654 9 𝑆(𝑛 𝑆(𝑛 NOUN ejpam-5960 654 10 ) ) PUNCT ejpam-5960 654 11 : : PUNCT ejpam-5960 654 12 𝑛 𝑛 PROPN ejpam-5960 654 13 ∈ ∈ PROPN ejpam-5960 654 14 𝐷 𝐷 PROPN ejpam-5960 654 15 } } PUNCT ejpam-5960 654 16 and and CCONJ ejpam-5960 654 17 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 654 18 is be AUX ejpam-5960 654 19 a a DET ejpam-5960 654 20 𝑁𝛼𝐡–cluster 𝑁𝛼𝐡–clust ADJ ejpam-5960 654 21 point point NOUN ejpam-5960 654 22 of of ADP ejpam-5960 654 23 𝑆 𝑆 PROPN ejpam-5960 654 24 , , PUNCT ejpam-5960 654 25 then then ADV ejpam-5960 654 26 for for ADP ejpam-5960 654 27 each each DET ejpam-5960 654 28 πœ† πœ† PRON ejpam-5960 654 29 ∈ ∈ PROPN ejpam-5960 654 30 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 654 31 and and CCONJ ejpam-5960 654 32 each each DET ejpam-5960 654 33 𝑛 𝑛 PRON ejpam-5960 654 34 ∈ ∈ PROPN ejpam-5960 654 35 𝐷 𝐷 PROPN ejpam-5960 654 36 there there PRON ejpam-5960 654 37 is be VERB ejpam-5960 654 38 π‘˜ π‘˜ DET ejpam-5960 654 39 ∈ ∈ PROPN ejpam-5960 654 40 𝐷 𝐷 NOUN ejpam-5960 654 41 such such ADJ ejpam-5960 654 42 that that DET ejpam-5960 654 43 𝑆(π‘˜ 𝑆(π‘˜ ADV ejpam-5960 654 44 ) ) PUNCT ejpam-5960 654 45 βˆ‰ βˆ‰ PROPN ejpam-5960 654 46 πœ† πœ† PROPN ejpam-5960 654 47 and and CCONJ ejpam-5960 654 48 π‘˜ π‘˜ DET ejpam-5960 654 49 β‰₯ β‰₯ NOUN ejpam-5960 654 50 𝑛. 𝑛. NOUN ejpam-5960 654 51 taking take VERB ejpam-5960 654 52 π‘˜ π‘˜ PRON ejpam-5960 654 53 = = SYM ejpam-5960 654 54 𝑔(𝑛 𝑔(𝑛 NOUN ejpam-5960 654 55 , , PUNCT ejpam-5960 654 56 πœ† πœ† NOUN ejpam-5960 654 57 ) ) PUNCT ejpam-5960 654 58 , , PUNCT ejpam-5960 654 59 we we PRON ejpam-5960 654 60 get get VERB ejpam-5960 654 61 a a DET ejpam-5960 654 62 mapping mapping NOUN ejpam-5960 654 63 𝑔 𝑔 NOUN ejpam-5960 654 64 : : PUNCT ejpam-5960 654 65 𝐷 𝐷 PROPN ejpam-5960 654 66 Γ— Γ— NOUN ejpam-5960 654 67 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 654 68 β†’ β†’ SYM ejpam-5960 654 69 𝐷 𝐷 NOUN ejpam-5960 654 70 with with ADP ejpam-5960 654 71 𝑆(𝑔(𝑛 𝑆(𝑔(𝑛 NOUN ejpam-5960 654 72 , , PUNCT ejpam-5960 654 73 πœ† πœ† NOUN ejpam-5960 654 74 ) ) PUNCT ejpam-5960 654 75 ) ) PUNCT ejpam-5960 654 76 βˆ‰ βˆ‰ PROPN ejpam-5960 654 77 πœ†. πœ†. PROPN ejpam-5960 654 78 put put VERB ejpam-5960 654 79 𝐸 𝐸 PROPN ejpam-5960 654 80 = = PUNCT ejpam-5960 654 81 𝐷 𝐷 PROPN ejpam-5960 654 82 Γ— Γ— NOUN ejpam-5960 654 83 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 NOUN ejpam-5960 654 84 and and CCONJ ejpam-5960 654 85 we we PRON ejpam-5960 654 86 define define VERB ejpam-5960 654 87 the the DET ejpam-5960 654 88 relation relation NOUN ejpam-5960 654 89 ≀ ≀ NOUN ejpam-5960 654 90 on on ADP ejpam-5960 654 91 𝐸 𝐸 PROPN ejpam-5960 654 92 as as SCONJ ejpam-5960 654 93 follows follow VERB ejpam-5960 654 94 : : PUNCT ejpam-5960 654 95 ( ( PUNCT ejpam-5960 654 96 𝑛 𝑛 ADJ ejpam-5960 654 97 , , PUNCT ejpam-5960 654 98 πœ†1 πœ†1 NOUN ejpam-5960 654 99 ) ) PUNCT ejpam-5960 654 100 ≀ ≀ NOUN ejpam-5960 654 101 ( ( PUNCT ejpam-5960 654 102 𝑛2 𝑛2 NOUN ejpam-5960 654 103 , , PUNCT ejpam-5960 654 104 πœ†2 πœ†2 PROPN ejpam-5960 654 105 ) ) PUNCT ejpam-5960 654 106 iff iff PROPN ejpam-5960 654 107 𝑛1 𝑛1 ADJ ejpam-5960 654 108 ≀ ≀ PUNCT ejpam-5960 654 109 𝑛2 𝑛2 NOUN ejpam-5960 654 110 and and CCONJ ejpam-5960 654 111 πœ†1 πœ†1 VERB ejpam-5960 654 112 ≀ ≀ NUM ejpam-5960 654 113 πœ†2 πœ†2 NOUN ejpam-5960 654 114 , , PUNCT ejpam-5960 654 115 then then ADV ejpam-5960 654 116 ( ( PUNCT ejpam-5960 654 117 𝐸 𝐸 PROPN ejpam-5960 654 118 , , PUNCT ejpam-5960 654 119 ≀ ≀ NUM ejpam-5960 654 120 ) ) PUNCT ejpam-5960 654 121 is be AUX ejpam-5960 654 122 a a DET ejpam-5960 654 123 directed direct VERB ejpam-5960 654 124 set set NOUN ejpam-5960 654 125 . . PUNCT ejpam-5960 655 1 for for ADP ejpam-5960 655 2 each each DET ejpam-5960 655 3 ( ( PUNCT ejpam-5960 655 4 𝑛 𝑛 PROPN ejpam-5960 655 5 , , PUNCT ejpam-5960 655 6 πœ† πœ† NOUN ejpam-5960 655 7 ) ) PUNCT ejpam-5960 655 8 ∈ ∈ PROPN ejpam-5960 655 9 𝐸 𝐸 PROPN ejpam-5960 655 10 , , PUNCT ejpam-5960 655 11 we we PRON ejpam-5960 655 12 choose choose VERB ejpam-5960 655 13 𝑇 𝑇 PROPN ejpam-5960 655 14 ( ( PUNCT ejpam-5960 655 15 𝑛 𝑛 PROPN ejpam-5960 655 16 , , PUNCT ejpam-5960 655 17 πœ† πœ† NOUN ejpam-5960 655 18 ) ) PUNCT ejpam-5960 655 19 = = SYM ejpam-5960 655 20 𝑆(𝑔(𝑛 𝑆(𝑔(𝑛 NOUN ejpam-5960 655 21 , , PUNCT ejpam-5960 655 22 πœ† πœ† NOUN ejpam-5960 655 23 ) ) PUNCT ejpam-5960 655 24 ) ) PUNCT ejpam-5960 655 25 , , PUNCT ejpam-5960 655 26 then then ADV ejpam-5960 655 27 𝑇 𝑇 PROPN ejpam-5960 655 28 = = SYM ejpam-5960 655 29 { { PUNCT ejpam-5960 655 30 𝑇 𝑇 PROPN ejpam-5960 655 31 ( ( PUNCT ejpam-5960 655 32 𝑛 𝑛 PROPN ejpam-5960 655 33 , , PUNCT ejpam-5960 655 34 πœ† πœ† NOUN ejpam-5960 655 35 ) ) PUNCT ejpam-5960 655 36 : : PUNCT ejpam-5960 655 37 ( ( PUNCT ejpam-5960 655 38 𝑛 𝑛 NOUN ejpam-5960 655 39 , , PUNCT ejpam-5960 655 40 πœ† πœ† NOUN ejpam-5960 655 41 ) ) PUNCT ejpam-5960 655 42 ∈ ∈ PROPN ejpam-5960 655 43 𝐸 𝐸 PROPN ejpam-5960 655 44 } } PUNCT ejpam-5960 655 45 is be AUX ejpam-5960 655 46 a a DET ejpam-5960 655 47 subnet subnet NOUN ejpam-5960 655 48 of of ADP ejpam-5960 655 49 𝑆. 𝑆. PROPN ejpam-5960 655 50 because because SCONJ ejpam-5960 655 51 : : PUNCT ejpam-5960 655 52 ( ( PUNCT ejpam-5960 655 53 * * NOUN ejpam-5960 655 54 ) ) PUNCT ejpam-5960 655 55 there there PRON ejpam-5960 655 56 exists exist VERB ejpam-5960 655 57 mapping map VERB ejpam-5960 655 58 𝑓 𝑓 DET ejpam-5960 655 59 : : PUNCT ejpam-5960 655 60 𝐸 𝐸 PROPN ejpam-5960 655 61 β†’ β†’ SYM ejpam-5960 655 62 𝐷 𝐷 NOUN ejpam-5960 655 63 define define NOUN ejpam-5960 655 64 as as SCONJ ejpam-5960 655 65 follows follow VERB ejpam-5960 655 66 𝑓 𝑓 PRON ejpam-5960 655 67 ( ( PUNCT ejpam-5960 655 68 𝑛 𝑛 PROPN ejpam-5960 655 69 , , PUNCT ejpam-5960 655 70 πœ† πœ† NOUN ejpam-5960 655 71 ) ) PUNCT ejpam-5960 655 72 = = SYM ejpam-5960 655 73 𝑛 𝑛 PROPN ejpam-5960 655 74 and and CCONJ ejpam-5960 655 75 𝑇 𝑇 PROPN ejpam-5960 655 76 = = SYM ejpam-5960 655 77 𝑆 𝑆 PROPN ejpam-5960 655 78 β—¦ β—¦ NOUN ejpam-5960 655 79 𝑓 𝑓 PRON ejpam-5960 655 80 . . PUNCT ejpam-5960 656 1 ( ( PUNCT ejpam-5960 656 2 * * PUNCT ejpam-5960 656 3 * * PUNCT ejpam-5960 656 4 ) ) PUNCT ejpam-5960 656 5 let let VERB ejpam-5960 656 6 𝑛1 𝑛1 PROPN ejpam-5960 656 7 ∈ ∈ PROPN ejpam-5960 656 8 𝐷 𝐷 PROPN ejpam-5960 656 9 , , PUNCT ejpam-5960 656 10 then then ADV ejpam-5960 656 11 there there PRON ejpam-5960 656 12 exists exist VERB ejpam-5960 656 13 ( ( PUNCT ejpam-5960 656 14 𝑛1 𝑛1 NOUN ejpam-5960 656 15 , , PUNCT ejpam-5960 656 16 πœ†1 πœ†1 ADJ ejpam-5960 656 17 ) ) PUNCT ejpam-5960 656 18 ∈ ∈ PROPN ejpam-5960 656 19 𝐸 𝐸 PROPN ejpam-5960 656 20 and and CCONJ ejpam-5960 656 21 ( ( PUNCT ejpam-5960 656 22 𝑛1 𝑛1 NOUN ejpam-5960 656 23 , , PUNCT ejpam-5960 656 24 πœ†1 πœ†1 NOUN ejpam-5960 656 25 ) ) PUNCT ejpam-5960 656 26 ≀ ≀ NOUN ejpam-5960 656 27 ( ( PUNCT ejpam-5960 656 28 𝑛2 𝑛2 NOUN ejpam-5960 656 29 , , PUNCT ejpam-5960 656 30 πœ†2 πœ†2 NOUN ejpam-5960 656 31 ) ) PUNCT ejpam-5960 656 32 ∈ ∈ PROPN ejpam-5960 656 33 𝐸 𝐸 PROPN ejpam-5960 656 34 iff iff VERB ejpam-5960 656 35 𝑛1 𝑛1 ADJ ejpam-5960 656 36 ≀ ≀ PUNCT ejpam-5960 656 37 𝑛2 𝑛2 NOUN ejpam-5960 656 38 and and CCONJ ejpam-5960 656 39 πœ†1 πœ†1 VERB ejpam-5960 656 40 ≀ ≀ NUM ejpam-5960 656 41 πœ†2 πœ†2 NOUN ejpam-5960 656 42 , , PUNCT ejpam-5960 656 43 𝑓 𝑓 PRON ejpam-5960 656 44 ( ( PUNCT ejpam-5960 656 45 𝑛2 𝑛2 NOUN ejpam-5960 656 46 , , PUNCT ejpam-5960 656 47 πœ†2 πœ†2 NOUN ejpam-5960 656 48 ) ) PUNCT ejpam-5960 656 49 = = VERB ejpam-5960 657 1 𝑛2 𝑛2 PROPN ejpam-5960 657 2 β‰₯ β‰₯ NOUN ejpam-5960 657 3 𝑛1 𝑛1 PROPN ejpam-5960 657 4 . . PUNCT ejpam-5960 658 1 now now ADV ejpam-5960 658 2 we we PRON ejpam-5960 658 3 prove prove VERB ejpam-5960 658 4 that that SCONJ ejpam-5960 658 5 𝑇 𝑇 PROPN ejpam-5960 658 6 is be AUX ejpam-5960 658 7 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 658 8 to to PART ejpam-5960 658 9 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 658 10 , , PUNCT ejpam-5960 658 11 let let VERB ejpam-5960 658 12 πœ† πœ† DET ejpam-5960 658 13 ∈ ∈ PROPN ejpam-5960 658 14 𝑁𝛼𝐡𝑅π‘₯𝛼 𝑁𝛼𝐡𝑅π‘₯𝛼 PROPN ejpam-5960 658 15 and and CCONJ ejpam-5960 658 16 𝑛 𝑛 DET ejpam-5960 658 17 ∈ ∈ PROPN ejpam-5960 658 18 𝐷 𝐷 PROPN ejpam-5960 658 19 , , PUNCT ejpam-5960 658 20 so so CCONJ ejpam-5960 658 21 ( ( PUNCT ejpam-5960 658 22 𝑛 𝑛 NOUN ejpam-5960 658 23 , , PUNCT ejpam-5960 658 24 πœ† πœ† NOUN ejpam-5960 658 25 ) ) PUNCT ejpam-5960 658 26 ∈ ∈ PROPN ejpam-5960 658 27 𝐸. 𝐸. PROPN ejpam-5960 658 28 therefore therefore ADV ejpam-5960 658 29 , , PUNCT ejpam-5960 658 30 for for ADP ejpam-5960 658 31 each each DET ejpam-5960 658 32 ( ( PUNCT ejpam-5960 658 33 𝑛 𝑛 PROPN ejpam-5960 658 34 , , PUNCT ejpam-5960 658 35 πœ† πœ† NOUN ejpam-5960 658 36 ) ) PUNCT ejpam-5960 658 37 ∈ ∈ PROPN ejpam-5960 658 38 𝐸 𝐸 PROPN ejpam-5960 658 39 and and CCONJ ejpam-5960 658 40 ( ( PUNCT ejpam-5960 658 41 𝑛 𝑛 NOUN ejpam-5960 658 42 , , PUNCT ejpam-5960 658 43 πœ† πœ† NOUN ejpam-5960 658 44 ) ) PUNCT ejpam-5960 658 45 ≀ ≀ NOUN ejpam-5960 658 46 ( ( PUNCT ejpam-5960 658 47 π‘š π‘š NOUN ejpam-5960 658 48 , , PUNCT ejpam-5960 658 49 πœ‚ πœ‚ NOUN ejpam-5960 658 50 ) ) PUNCT ejpam-5960 658 51 then then ADV ejpam-5960 658 52 𝑇 𝑇 PROPN ejpam-5960 658 53 ( ( PUNCT ejpam-5960 658 54 π‘š π‘š PROPN ejpam-5960 658 55 , , PUNCT ejpam-5960 658 56 πœ‚ πœ‚ NOUN ejpam-5960 658 57 ) ) PUNCT ejpam-5960 658 58 = = SYM ejpam-5960 658 59 𝑆(𝑔(π‘š 𝑆(𝑔(π‘š PROPN ejpam-5960 658 60 , , PUNCT ejpam-5960 658 61 πœ‚ πœ‚ NOUN ejpam-5960 658 62 ) ) PUNCT ejpam-5960 658 63 ) ) PUNCT ejpam-5960 658 64 βˆ‰ βˆ‰ PROPN ejpam-5960 658 65 πœ‚ πœ‚ PROPN ejpam-5960 658 66 and and CCONJ ejpam-5960 658 67 πœ† πœ† DET ejpam-5960 658 68 ≀ ≀ NUM ejpam-5960 658 69 πœ‚ πœ‚ NOUN ejpam-5960 658 70 , , PUNCT ejpam-5960 658 71 so so SCONJ ejpam-5960 658 72 𝑇 𝑇 PROPN ejpam-5960 658 73 ( ( PUNCT ejpam-5960 658 74 π‘š π‘š PROPN ejpam-5960 658 75 , , PUNCT ejpam-5960 658 76 πœ‚ πœ‚ NOUN ejpam-5960 658 77 ) ) PUNCT ejpam-5960 658 78 βˆ‰ βˆ‰ PROPN ejpam-5960 658 79 πœ†. πœ†. PROPN ejpam-5960 658 80 thus thus ADV ejpam-5960 658 81 𝑇 𝑇 PROPN ejpam-5960 658 82 is be AUX ejpam-5960 658 83 𝑁𝛼𝐡–converges 𝑁𝛼𝐡–converge NOUN ejpam-5960 658 84 to to PART ejpam-5960 658 85 π‘₯𝛼. π‘₯𝛼. VERB ejpam-5960 658 86 conversely conversely ADV ejpam-5960 658 87 , , PUNCT ejpam-5960 658 88 it it PRON ejpam-5960 658 89 follows follow VERB ejpam-5960 658 90 directly directly ADV ejpam-5960 658 91 from from ADP ejpam-5960 658 92 definition definition NOUN ejpam-5960 658 93 2.12 2.12 NUM ejpam-5960 658 94 . . PUNCT ejpam-5960 659 1 ( ( PUNCT ejpam-5960 659 2 ii ii NOUN ejpam-5960 659 3 ) ) PUNCT ejpam-5960 659 4 it it PRON ejpam-5960 659 5 follows follow VERB ejpam-5960 659 6 directly directly ADV ejpam-5960 659 7 from from ADP ejpam-5960 659 8 definition definition NOUN ejpam-5960 659 9 2.12 2.12 NUM ejpam-5960 659 10 . . PUNCT ejpam-5960 660 1 theorem theorem VERB ejpam-5960 660 2 4.12 4.12 NUM ejpam-5960 660 3 . . PUNCT ejpam-5960 661 1 let let VERB ejpam-5960 661 2 ( ( PUNCT ejpam-5960 661 3 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 661 4 , , PUNCT ejpam-5960 661 5 𝜏 𝜏 NOUN ejpam-5960 661 6 ) ) PUNCT ejpam-5960 661 7 be be VERB ejpam-5960 661 8 an an DET ejpam-5960 661 9 l l NOUN ejpam-5960 661 10 – – PUNCT ejpam-5960 661 11 ts ts NOUN ejpam-5960 661 12 and and CCONJ ejpam-5960 661 13 πœ‡ πœ‡ X ejpam-5960 661 14 ∈ ∈ X ejpam-5960 661 15 𝐿𝑋. 𝐿𝑋. X ejpam-5960 661 16 then then ADV ejpam-5960 661 17 πœ‡ πœ‡ SCONJ ejpam-5960 661 18 is be AUX ejpam-5960 661 19 a a DET ejpam-5960 661 20 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 661 21 set set NOUN ejpam-5960 661 22 iff iff NOUN ejpam-5960 661 23 every every DET ejpam-5960 661 24 𝛼–filter 𝛼–filter NOUN ejpam-5960 661 25 f f NOUN ejpam-5960 661 26 containing contain VERB ejpam-5960 661 27 πœ‡ πœ‡ PRON ejpam-5960 661 28 as as ADP ejpam-5960 661 29 an an DET ejpam-5960 661 30 element element NOUN ejpam-5960 661 31 has have VERB ejpam-5960 661 32 a a DET ejpam-5960 661 33 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 661 34 point point NOUN ejpam-5960 661 35 in in ADP ejpam-5960 661 36 𝑋 𝑋 NOUN ejpam-5960 661 37 with with ADP ejpam-5960 661 38 height height NOUN ejpam-5960 661 39 𝛼. 𝛼. NOUN ejpam-5960 661 40 proof proof NOUN ejpam-5960 661 41 . . PUNCT ejpam-5960 662 1 suppose suppose VERB ejpam-5960 662 2 that that SCONJ ejpam-5960 662 3 πœ‡ πœ‡ ADP ejpam-5960 662 4 is be AUX ejpam-5960 662 5 a a DET ejpam-5960 662 6 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 662 7 and and CCONJ ejpam-5960 662 8 f f PROPN ejpam-5960 662 9 is be AUX ejpam-5960 662 10 a a DET ejpam-5960 662 11 𝛼–filter 𝛼–filter NOUN ejpam-5960 662 12 containing contain VERB ejpam-5960 662 13 πœ‡ πœ‡ PRON ejpam-5960 662 14 as as ADP ejpam-5960 662 15 an an DET ejpam-5960 662 16 element element NOUN ejpam-5960 662 17 ( ( PUNCT ejpam-5960 662 18 𝛼 𝛼 PROPN ejpam-5960 662 19 ∈ ∈ PROPN ejpam-5960 662 20 𝑀 𝑀 PROPN ejpam-5960 662 21 ( ( PUNCT ejpam-5960 662 22 𝐿 𝐿 PROPN ejpam-5960 662 23 ) ) PUNCT ejpam-5960 662 24 ) ) PUNCT ejpam-5960 662 25 , , PUNCT ejpam-5960 662 26 then then ADV ejpam-5960 662 27 πœ† πœ† PROPN ejpam-5960 662 28 ∧ ∧ PROPN ejpam-5960 662 29 πœ‡ πœ‡ ADP ejpam-5960 662 30 ∈ ∈ PROPN ejpam-5960 662 31 f f PROPN ejpam-5960 662 32 for for ADP ejpam-5960 662 33 each each DET ejpam-5960 662 34 πœ† πœ† PROPN ejpam-5960 662 35 ∈ ∈ PROPN ejpam-5960 662 36 f f X ejpam-5960 662 37 , , PUNCT ejpam-5960 662 38 hence hence ADV ejpam-5960 662 39 ∨ ∨ NUM ejpam-5960 662 40 π‘₯βˆˆπ‘‹ π‘₯βˆˆπ‘‹ PROPN ejpam-5960 662 41 ( ( PUNCT ejpam-5960 662 42 πœ† πœ† X ejpam-5960 662 43 ∧ ∧ PROPN ejpam-5960 662 44 πœ‡ πœ‡ ADP ejpam-5960 662 45 ) ) PUNCT ejpam-5960 662 46 ( ( PUNCT ejpam-5960 662 47 π‘₯ π‘₯ NOUN ejpam-5960 662 48 ) ) PUNCT ejpam-5960 662 49 β‰₯ β‰₯ NOUN ejpam-5960 662 50 𝛼 𝛼 NOUN ejpam-5960 662 51 for for ADP ejpam-5960 662 52 each each DET ejpam-5960 662 53 πœ† πœ† PROPN ejpam-5960 662 54 ∈ ∈ PROPN ejpam-5960 662 55 f f X ejpam-5960 662 56 and and CCONJ ejpam-5960 662 57 for for ADP ejpam-5960 662 58 each each PRON ejpam-5960 662 59 π‘₯𝛼 π‘₯𝛼 PROPN ejpam-5960 662 60 ∈ ∈ PROPN ejpam-5960 662 61 𝑀 𝑀 PROPN ejpam-5960 662 62 ( ( PUNCT ejpam-5960 662 63 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 662 64 ) ) PUNCT ejpam-5960 662 65 there there PRON ejpam-5960 662 66 exists exist VERB ejpam-5960 662 67 a a DET ejpam-5960 662 68 molecule molecule NOUN ejpam-5960 662 69 π‘₯ π‘₯ NOUN ejpam-5960 662 70 ( ( PUNCT ejpam-5960 662 71 πœ†,𝛼 πœ†,𝛼 NOUN ejpam-5960 662 72 ) ) PUNCT ejpam-5960 662 73 ∈ ∈ PROPN ejpam-5960 662 74 πœ† πœ† X ejpam-5960 662 75 ∧ ∧ PROPN ejpam-5960 662 76 πœ‡ πœ‡ ADP ejpam-5960 662 77 with with ADP ejpam-5960 662 78 height height NOUN ejpam-5960 662 79 𝛼. 𝛼. NOUN ejpam-5960 662 80 put put VERB ejpam-5960 662 81 𝑆(f 𝑆(f PROPN ejpam-5960 662 82 ) ) PUNCT ejpam-5960 663 1 = = PRON ejpam-5960 663 2 { { PUNCT ejpam-5960 663 3 π‘₯ π‘₯ X ejpam-5960 663 4 ( ( PUNCT ejpam-5960 663 5 πœ†,𝛼 πœ†,𝛼 NOUN ejpam-5960 663 6 ) ) PUNCT ejpam-5960 663 7 : : PUNCT ejpam-5960 663 8 ( ( PUNCT ejpam-5960 663 9 πœ† πœ† NOUN ejpam-5960 663 10 , , PUNCT ejpam-5960 663 11 𝛼 𝛼 NOUN ejpam-5960 663 12 ) ) PUNCT ejpam-5960 663 13 ∈ ∈ PROPN ejpam-5960 663 14 f f X ejpam-5960 663 15 Γ— Γ— PROPN ejpam-5960 663 16 𝑀 𝑀 PROPN ejpam-5960 663 17 ( ( PUNCT ejpam-5960 663 18 𝐿 𝐿 PROPN ejpam-5960 663 19 ) ) PUNCT ejpam-5960 663 20 } } PUNCT ejpam-5960 663 21 . . PUNCT ejpam-5960 664 1 in in ADP ejpam-5960 664 2 f f PROPN ejpam-5960 664 3 Γ— Γ— PROPN ejpam-5960 664 4 𝑀 𝑀 PROPN ejpam-5960 664 5 ( ( PUNCT ejpam-5960 664 6 𝐿 𝐿 PROPN ejpam-5960 664 7 ) ) PUNCT ejpam-5960 664 8 we we PRON ejpam-5960 664 9 define define VERB ejpam-5960 664 10 the the DET ejpam-5960 664 11 relation relation NOUN ejpam-5960 664 12 that that PRON ejpam-5960 664 13 ( ( PUNCT ejpam-5960 664 14 πœ†1 πœ†1 NOUN ejpam-5960 664 15 , , PUNCT ejpam-5960 664 16 𝛼1 𝛼1 NOUN ejpam-5960 664 17 ) ) PUNCT ejpam-5960 664 18 β‰₯ β‰₯ NOUN ejpam-5960 664 19 ( ( PUNCT ejpam-5960 664 20 πœ†2 πœ†2 NOUN ejpam-5960 664 21 , , PUNCT ejpam-5960 664 22 𝛼2)𝑖 𝛼2)𝑖 VERB ejpam-5960 664 23 𝑓 𝑓 PRON ejpam-5960 664 24 𝑓 𝑓 DET ejpam-5960 664 25 πœ†1 πœ†1 NOUN ejpam-5960 664 26 ≀ ≀ NUM ejpam-5960 664 27 πœ†2 πœ†2 NOUN ejpam-5960 664 28 and and CCONJ ejpam-5960 664 29 𝛼1 𝛼1 PROPN ejpam-5960 664 30 β‰₯ β‰₯ NOUN ejpam-5960 664 31 𝛼2 𝛼2 PROPN ejpam-5960 664 32 . . PUNCT ejpam-5960 665 1 then then ADV ejpam-5960 665 2 , , PUNCT ejpam-5960 665 3 f f PROPN ejpam-5960 665 4 Γ— Γ— PROPN ejpam-5960 665 5 𝑀 𝑀 PROPN ejpam-5960 665 6 ( ( PUNCT ejpam-5960 665 7 𝐿 𝐿 PROPN ejpam-5960 665 8 ) ) PUNCT ejpam-5960 665 9 is be AUX ejpam-5960 665 10 a a DET ejpam-5960 665 11 directed direct VERB ejpam-5960 665 12 set set NOUN ejpam-5960 665 13 with with ADP ejpam-5960 665 14 this this DET ejpam-5960 665 15 relation relation NOUN ejpam-5960 665 16 , , PUNCT ejpam-5960 665 17 and and CCONJ ejpam-5960 665 18 𝑆(f 𝑆(f PROPN ejpam-5960 665 19 ) ) PUNCT ejpam-5960 665 20 is be AUX ejpam-5960 665 21 a a DET ejpam-5960 665 22 constant constant ADJ ejpam-5960 665 23 molecular molecular ADJ ejpam-5960 665 24 𝛼–net 𝛼–net NUM ejpam-5960 665 25 in in ADP ejpam-5960 665 26 πœ‡. πœ‡. NOUN ejpam-5960 665 27 since since SCONJ ejpam-5960 665 28 πœ‡ πœ‡ ADV ejpam-5960 665 29 is be AUX ejpam-5960 665 30 a a DET ejpam-5960 665 31 n.𝛼–bounded n.𝛼–bounde VERB ejpam-5960 665 32 set set NOUN ejpam-5960 665 33 , , PUNCT ejpam-5960 665 34 then then ADV ejpam-5960 665 35 by by ADP ejpam-5960 665 36 theorem theorem NOUN ejpam-5960 665 37 4.1 4.1 NUM ejpam-5960 665 38 , , PUNCT ejpam-5960 665 39 𝑆(f 𝑆(f PART ejpam-5960 665 40 ) ) PUNCT ejpam-5960 665 41 has have AUX ejpam-5960 665 42 a a DET ejpam-5960 665 43 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 665 44 point point NOUN ejpam-5960 665 45 in in ADP ejpam-5960 665 46 𝑋 𝑋 NOUN ejpam-5960 665 47 with with ADP ejpam-5960 665 48 height height NOUN ejpam-5960 665 49 𝛼 𝛼 NOUN ejpam-5960 665 50 , , PUNCT ejpam-5960 665 51 say say INTJ ejpam-5960 665 52 π‘₯𝛼. π‘₯𝛼. X ejpam-5960 665 53 so so ADV ejpam-5960 665 54 , , PUNCT ejpam-5960 665 55 by by ADP ejpam-5960 665 56 theorem theorem NOUN ejpam-5960 665 57 2.28 2.28 NUM ejpam-5960 665 58 , , PUNCT ejpam-5960 665 59 f f PROPN ejpam-5960 665 60 𝛿–cluster 𝛿–cluster PROPN ejpam-5960 665 61 to to ADP ejpam-5960 665 62 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 665 63 as as ADV ejpam-5960 665 64 well well ADV ejpam-5960 665 65 . . PUNCT ejpam-5960 666 1 conversely conversely ADV ejpam-5960 666 2 , , PUNCT ejpam-5960 666 3 suppose suppose VERB ejpam-5960 666 4 that that SCONJ ejpam-5960 666 5 the the DET ejpam-5960 666 6 condition condition NOUN ejpam-5960 666 7 is be AUX ejpam-5960 666 8 satisfied satisfied ADJ ejpam-5960 666 9 and and CCONJ ejpam-5960 666 10 𝑆 𝑆 PROPN ejpam-5960 666 11 = = SYM ejpam-5960 666 12 { { PUNCT ejpam-5960 666 13 𝑆(𝑛 𝑆(𝑛 NOUN ejpam-5960 666 14 ) ) PUNCT ejpam-5960 666 15 : : PUNCT ejpam-5960 666 16 𝑛 𝑛 PROPN ejpam-5960 666 17 ∈ ∈ PROPN ejpam-5960 666 18 𝐷 𝐷 PROPN ejpam-5960 666 19 } } PUNCT ejpam-5960 666 20 is be AUX ejpam-5960 666 21 a a DET ejpam-5960 666 22 constant constant ADJ ejpam-5960 666 23 molecular molecular ADJ ejpam-5960 666 24 𝛼–net 𝛼–net NUM ejpam-5960 666 25 in in ADP ejpam-5960 666 26 πœ‡. πœ‡. NOUN ejpam-5960 666 27 let let VERB ejpam-5960 666 28 πœ†π‘š πœ†π‘š VERB ejpam-5960 666 29 = = SYM ejpam-5960 666 30 ∨(𝑆(π‘š ∨(𝑆(π‘š NOUN ejpam-5960 666 31 ) ) PUNCT ejpam-5960 666 32 ) ) PUNCT ejpam-5960 666 33 for for ADP ejpam-5960 666 34 each each DET ejpam-5960 666 35 π‘š π‘š PROPN ejpam-5960 666 36 ∈ ∈ PROPN ejpam-5960 666 37 𝐷 𝐷 PROPN ejpam-5960 666 38 , , PUNCT ejpam-5960 666 39 𝑛 𝑛 DET ejpam-5960 666 40 β‰₯ β‰₯ NOUN ejpam-5960 666 41 π‘š. π‘š. ADV ejpam-5960 666 42 since since SCONJ ejpam-5960 666 43 𝐷 𝐷 PROPN ejpam-5960 666 44 is be AUX ejpam-5960 666 45 a a DET ejpam-5960 666 46 directed direct VERB ejpam-5960 666 47 set set NOUN ejpam-5960 666 48 , , PUNCT ejpam-5960 666 49 then then ADV ejpam-5960 666 50 the the DET ejpam-5960 666 51 family family NOUN ejpam-5960 666 52 { { PUNCT ejpam-5960 666 53 πœ†π‘š πœ†π‘š NOUN ejpam-5960 666 54 : : PUNCT ejpam-5960 666 55 π‘š π‘š PROPN ejpam-5960 666 56 ∈ ∈ PROPN ejpam-5960 666 57 𝐷 𝐷 PROPN ejpam-5960 666 58 } } PUNCT ejpam-5960 666 59 can can AUX ejpam-5960 666 60 generate generate VERB ejpam-5960 666 61 a a DET ejpam-5960 666 62 filter filter NOUN ejpam-5960 666 63 f f NOUN ejpam-5960 666 64 ( ( PUNCT ejpam-5960 666 65 𝑆 𝑆 PROPN ejpam-5960 666 66 ) ) PUNCT ejpam-5960 666 67 . . PUNCT ejpam-5960 667 1 since since SCONJ ejpam-5960 667 2 𝑆 𝑆 PROPN ejpam-5960 667 3 is be AUX ejpam-5960 667 4 a a DET ejpam-5960 667 5 constant constant ADJ ejpam-5960 667 6 molecular molecular ADJ ejpam-5960 667 7 𝛼–net 𝛼–net NUM ejpam-5960 667 8 , , PUNCT ejpam-5960 667 9 then then ADV ejpam-5960 667 10 for for ADP ejpam-5960 667 11 each each DET ejpam-5960 667 12 𝛼 𝛼 PROPN ejpam-5960 667 13 ∈ ∈ PROPN ejpam-5960 667 14 𝑀 𝑀 PROPN ejpam-5960 667 15 ( ( PUNCT ejpam-5960 667 16 𝐿 𝐿 PROPN ejpam-5960 667 17 ) ) PUNCT ejpam-5960 667 18 ( ( PUNCT ejpam-5960 667 19 βˆƒπ‘› βˆƒπ‘› PROPN ejpam-5960 667 20 ∈ ∈ PROPN ejpam-5960 667 21 𝐷 𝐷 PROPN ejpam-5960 667 22 ) ) PUNCT ejpam-5960 667 23 ( ( PUNCT ejpam-5960 667 24 βˆ€π‘š βˆ€π‘š PROPN ejpam-5960 667 25 ∈ ∈ PROPN ejpam-5960 667 26 𝐷 𝐷 NOUN ejpam-5960 667 27 , , PUNCT ejpam-5960 667 28 π‘š π‘š PROPN ejpam-5960 667 29 β‰₯ β‰₯ NUM ejpam-5960 667 30 𝑛 𝑛 NOUN ejpam-5960 667 31 ) ) PUNCT ejpam-5960 667 32 ( ( PUNCT ejpam-5960 667 33 ∨(𝑆(π‘š ∨(𝑆(π‘š NOUN ejpam-5960 667 34 ) ) PUNCT ejpam-5960 667 35 ) ) PUNCT ejpam-5960 668 1 = = SYM ejpam-5960 668 2 𝛼 𝛼 X ejpam-5960 668 3 ) ) PUNCT ejpam-5960 668 4 , , PUNCT ejpam-5960 668 5 hence∨(πœ†π‘š(π‘₯ hence∨(πœ†π‘š(π‘₯ NUM ejpam-5960 668 6 ) ) PUNCT ejpam-5960 668 7 ) ) PUNCT ejpam-5960 669 1 = = SYM ejpam-5960 669 2 ∨(∨(𝑆(𝑛 ∨(∨(𝑆(𝑛 PROPN ejpam-5960 669 3 ) ) PUNCT ejpam-5960 669 4 ) ) PUNCT ejpam-5960 669 5 ) ) PUNCT ejpam-5960 670 1 = = PUNCT ejpam-5960 670 2 𝛼 𝛼 X ejpam-5960 670 3 , , PUNCT ejpam-5960 670 4 𝑛 𝑛 DET ejpam-5960 670 5 β‰₯ β‰₯ NOUN ejpam-5960 670 6 π‘š π‘š X ejpam-5960 670 7 and and CCONJ ejpam-5960 670 8 so so ADV ejpam-5960 670 9 ∨(πœ†π‘š(π‘₯ ∨(πœ†π‘š(π‘₯ ADJ ejpam-5960 670 10 ) ) PUNCT ejpam-5960 670 11 ) ) PUNCT ejpam-5960 671 1 = = PUNCT ejpam-5960 671 2 𝛼. 𝛼. NOUN ejpam-5960 671 3 since since SCONJ ejpam-5960 671 4 f f PROPN ejpam-5960 671 5 ( ( PUNCT ejpam-5960 671 6 𝑆 𝑆 PROPN ejpam-5960 671 7 ) ) PUNCT ejpam-5960 671 8 is be AUX ejpam-5960 671 9 produced produce VERB ejpam-5960 671 10 by by ADP ejpam-5960 671 11 { { PUNCT ejpam-5960 671 12 πœ†π‘š πœ†π‘š NOUN ejpam-5960 671 13 : : PUNCT ejpam-5960 671 14 π‘š π‘š PROPN ejpam-5960 671 15 ∈ ∈ PROPN ejpam-5960 671 16 𝐷 𝐷 PROPN ejpam-5960 671 17 } } PUNCT ejpam-5960 671 18 , , PUNCT ejpam-5960 671 19 then then ADV ejpam-5960 671 20 for for ADP ejpam-5960 671 21 each each DET ejpam-5960 671 22 πœ† πœ† PROPN ejpam-5960 671 23 ∈ ∈ PROPN ejpam-5960 671 24 f f X ejpam-5960 671 25 ( ( PUNCT ejpam-5960 671 26 𝑆 𝑆 PROPN ejpam-5960 671 27 ) ) PUNCT ejpam-5960 671 28 contains contain VERB ejpam-5960 671 29 some some DET ejpam-5960 671 30 πœ†π‘š πœ†π‘š PROPN ejpam-5960 671 31 and and CCONJ ejpam-5960 671 32 therefore therefore ADV ejpam-5960 671 33 ∨(πœ†(π‘₯ ∨(πœ†(π‘₯ ADJ ejpam-5960 671 34 ) ) PUNCT ejpam-5960 671 35 ) ) PUNCT ejpam-5960 672 1 = = SYM ejpam-5960 672 2 𝛼. 𝛼. NOUN ejpam-5960 672 3 hence hence ADV ejpam-5960 672 4 f f PROPN ejpam-5960 672 5 ( ( PUNCT ejpam-5960 672 6 𝑆 𝑆 PROPN ejpam-5960 672 7 ) ) PUNCT ejpam-5960 672 8 is be AUX ejpam-5960 672 9 an an DET ejpam-5960 672 10 𝛼–filter 𝛼–filter NOUN ejpam-5960 672 11 . . PUNCT ejpam-5960 673 1 by by ADP ejpam-5960 673 2 assumption assumption NOUN ejpam-5960 673 3 , , PUNCT ejpam-5960 673 4 f f PROPN ejpam-5960 673 5 ( ( PUNCT ejpam-5960 673 6 𝑆 𝑆 PROPN ejpam-5960 673 7 ) ) PUNCT ejpam-5960 673 8 has have VERB ejpam-5960 673 9 a a DET ejpam-5960 673 10 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 673 11 point point NOUN ejpam-5960 673 12 in in ADP ejpam-5960 673 13 𝑋 𝑋 NOUN ejpam-5960 673 14 with with ADP ejpam-5960 673 15 height height NOUN ejpam-5960 673 16 𝛼 𝛼 PROPN ejpam-5960 673 17 , , PUNCT ejpam-5960 673 18 say say INTJ ejpam-5960 673 19 π‘₯𝛼. π‘₯𝛼. VERB ejpam-5960 673 20 thus thus ADV ejpam-5960 673 21 , , PUNCT ejpam-5960 673 22 for for ADP ejpam-5960 673 23 each each DET ejpam-5960 673 24 πœ‡ πœ‡ ADP ejpam-5960 673 25 ∈ ∈ PROPN ejpam-5960 673 26 𝑅π‘₯𝛼 𝑅π‘₯𝛼 PROPN ejpam-5960 673 27 and and CCONJ ejpam-5960 673 28 for for ADP ejpam-5960 673 29 each each DET ejpam-5960 673 30 πœ† πœ† PROPN ejpam-5960 673 31 ∈ ∈ PROPN ejpam-5960 673 32 f f X ejpam-5960 673 33 ( ( PUNCT ejpam-5960 673 34 𝑆 𝑆 PROPN ejpam-5960 673 35 ) ) PUNCT ejpam-5960 673 36 . . PUNCT ejpam-5960 674 1 in in ADP ejpam-5960 674 2 particular particular ADJ ejpam-5960 674 3 , , PUNCT ejpam-5960 674 4 πœ†π‘š πœ†π‘š AUX ejpam-5960 674 5 we we PRON ejpam-5960 674 6 have have VERB ejpam-5960 674 7 n. n. PROPN ejpam-5960 674 8 a. a. NOUN ejpam-5960 674 9 alsaedi alsaedi PROPN ejpam-5960 674 10 / / SYM ejpam-5960 674 11 eur eur PROPN ejpam-5960 674 12 . . PUNCT ejpam-5960 675 1 j. j. PROPN ejpam-5960 675 2 pure pure PROPN ejpam-5960 675 3 appl appl PROPN ejpam-5960 675 4 . . PROPN ejpam-5960 675 5 math math PROPN ejpam-5960 675 6 , , PUNCT ejpam-5960 675 7 18 18 NUM ejpam-5960 675 8 ( ( PUNCT ejpam-5960 675 9 4 4 NUM ejpam-5960 675 10 ) ) PUNCT ejpam-5960 675 11 ( ( PUNCT ejpam-5960 675 12 2025 2025 NUM ejpam-5960 675 13 ) ) PUNCT ejpam-5960 675 14 , , PUNCT ejpam-5960 675 15 5960 5960 NUM ejpam-5960 675 16 21 21 NUM ejpam-5960 675 17 of of ADP ejpam-5960 675 18 22 22 NUM ejpam-5960 675 19 πœ†π‘š πœ†π‘š AUX ejpam-5960 675 20 β‰° β‰° PROPN ejpam-5960 675 21 πœ‡ πœ‡ ADV ejpam-5960 675 22 , , PUNCT ejpam-5960 675 23 and and CCONJ ejpam-5960 675 24 by by ADP ejpam-5960 675 25 theorem theorem NOUN ejpam-5960 675 26 2.29 2.29 NUM ejpam-5960 675 27 , , PUNCT ejpam-5960 675 28 we we PRON ejpam-5960 675 29 have have VERB ejpam-5960 675 30 𝑆 𝑆 PROPN ejpam-5960 675 31 has have VERB ejpam-5960 675 32 a a DET ejpam-5960 675 33 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 675 34 point point NOUN ejpam-5960 675 35 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 675 36 and and CCONJ ejpam-5960 675 37 by by ADP ejpam-5960 675 38 theorem theorem NOUN ejpam-5960 675 39 4.1 4.1 NUM ejpam-5960 675 40 , , PUNCT ejpam-5960 675 41 we we PRON ejpam-5960 675 42 have have VERB ejpam-5960 675 43 πœ‡ πœ‡ ADP ejpam-5960 675 44 is be AUX ejpam-5960 675 45 a a DET ejpam-5960 675 46 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 675 47 . . PUNCT ejpam-5960 675 48 theorem theorem VERB ejpam-5960 675 49 4.13 4.13 NUM ejpam-5960 675 50 . . PUNCT ejpam-5960 676 1 if if SCONJ ejpam-5960 676 2 a a DET ejpam-5960 676 3 set set NOUN ejpam-5960 676 4 πœ‡ πœ‡ X ejpam-5960 676 5 in in ADP ejpam-5960 676 6 a a DET ejpam-5960 676 7 l l NOUN ejpam-5960 676 8 – – PUNCT ejpam-5960 676 9 ts ts X ejpam-5960 676 10 ( ( PUNCT ejpam-5960 676 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 676 12 , , PUNCT ejpam-5960 676 13 𝜏 𝜏 NOUN ejpam-5960 676 14 ) ) PUNCT ejpam-5960 676 15 is be AUX ejpam-5960 676 16 a a DET ejpam-5960 676 17 n.𝛼–bounded n.𝛼–bounded NUM ejpam-5960 676 18 , , PUNCT ejpam-5960 676 19 then then ADV ejpam-5960 676 20 every every DET ejpam-5960 676 21 𝛼–ideal 𝛼–ideal ADJ ejpam-5960 676 22 𝐼 𝐼 PROPN ejpam-5960 676 23 in in ADP ejpam-5960 676 24 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 676 25 and and CCONJ ejpam-5960 676 26 πœ‡ πœ‡ ADP ejpam-5960 676 27 βˆ‰ βˆ‰ PROPN ejpam-5960 676 28 𝐼 𝐼 PROPN ejpam-5960 676 29 has have VERB ejpam-5960 676 30 a a DET ejpam-5960 676 31 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 676 32 point point NOUN ejpam-5960 676 33 in in ADP ejpam-5960 676 34 𝑋 𝑋 NOUN ejpam-5960 676 35 with with ADP ejpam-5960 676 36 height height NOUN ejpam-5960 676 37 𝛼. 𝛼. NOUN ejpam-5960 676 38 proof proof NOUN ejpam-5960 676 39 . . PUNCT ejpam-5960 677 1 let let VERB ejpam-5960 677 2 𝐼 𝐼 PRON ejpam-5960 677 3 be be AUX ejpam-5960 677 4 an an DET ejpam-5960 677 5 𝛼–ideal 𝛼–ideal NOUN ejpam-5960 677 6 in in ADP ejpam-5960 677 7 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 677 8 and and CCONJ ejpam-5960 677 9 πœ‡ πœ‡ ADP ejpam-5960 677 10 ∈ ∈ PROPN ejpam-5960 677 11 𝐿𝑋 𝐿𝑋 PROPN ejpam-5960 677 12 be be VERB ejpam-5960 677 13 a a DET ejpam-5960 677 14 𝑁𝛼–bounded 𝑁𝛼–bounde VERB ejpam-5960 677 15 with with ADP ejpam-5960 677 16 πœ‡ πœ‡ ADP ejpam-5960 677 17 βˆ‰ βˆ‰ PROPN ejpam-5960 677 18 𝐼. 𝐼. PROPN ejpam-5960 677 19 then then ADV ejpam-5960 677 20 for for ADP ejpam-5960 677 21 each each DET ejpam-5960 677 22 πœ‚ πœ‚ NOUN ejpam-5960 677 23 ∈ ∈ NOUN ejpam-5960 678 1 𝐼 𝐼 SCONJ ejpam-5960 678 2 we we PRON ejpam-5960 678 3 have have VERB ejpam-5960 678 4 ∨ ∨ PROPN ejpam-5960 678 5 π‘₯βˆˆπ‘‹ π‘₯βˆˆπ‘‹ PROPN ejpam-5960 678 6 πœ‚(π‘₯ πœ‚(π‘₯ PROPN ejpam-5960 678 7 ) ) PUNCT ejpam-5960 678 8 < < X ejpam-5960 679 1 𝛼 𝛼 X ejpam-5960 679 2 , , PUNCT ejpam-5960 679 3 and and CCONJ ejpam-5960 679 4 then then ADV ejpam-5960 679 5 for for ADP ejpam-5960 679 6 each each DET ejpam-5960 679 7 𝛼 𝛼 PROPN ejpam-5960 679 8 ∈ ∈ PROPN ejpam-5960 679 9 𝑀 𝑀 PROPN ejpam-5960 679 10 ( ( PUNCT ejpam-5960 679 11 𝐿 𝐿 PROPN ejpam-5960 679 12 ) ) PUNCT ejpam-5960 679 13 there there PRON ejpam-5960 679 14 exists exist VERB ejpam-5960 679 15 a a DET ejpam-5960 679 16 molecule molecule NOUN ejpam-5960 679 17 𝑆(πœ‚ 𝑆(πœ‚ PROPN ejpam-5960 679 18 , , PUNCT ejpam-5960 679 19 𝛼 𝛼 NOUN ejpam-5960 679 20 ) ) PUNCT ejpam-5960 679 21 = = SYM ejpam-5960 679 22 π‘₯ π‘₯ PROPN ejpam-5960 679 23 ( ( PUNCT ejpam-5960 679 24 πœ‚,𝛼 πœ‚,𝛼 NOUN ejpam-5960 679 25 ) ) PUNCT ejpam-5960 679 26 βˆ‰ βˆ‰ PROPN ejpam-5960 680 1 πœ‚. πœ‚. NOUN ejpam-5960 680 2 put put VERB ejpam-5960 680 3 𝐷 𝐷 PROPN ejpam-5960 680 4 ( ( PUNCT ejpam-5960 680 5 𝐼 𝐼 PROPN ejpam-5960 680 6 ) ) PUNCT ejpam-5960 680 7 = = SYM ejpam-5960 680 8 { { PUNCT ejpam-5960 680 9 ( ( PUNCT ejpam-5960 680 10 πœ‚ πœ‚ NOUN ejpam-5960 680 11 , , PUNCT ejpam-5960 680 12 𝛼 𝛼 NOUN ejpam-5960 680 13 ) ) PUNCT ejpam-5960 680 14 : : PUNCT ejpam-5960 681 1 π‘₯ π‘₯ X ejpam-5960 681 2 ( ( PUNCT ejpam-5960 681 3 πœ‚,𝛼 πœ‚,𝛼 NOUN ejpam-5960 681 4 ) ) PUNCT ejpam-5960 681 5 ∈ ∈ PROPN ejpam-5960 681 6 πœ‡ πœ‡ ADP ejpam-5960 681 7 , , PUNCT ejpam-5960 681 8 πœ‚ πœ‚ PROPN ejpam-5960 681 9 ∈ ∈ PROPN ejpam-5960 681 10 𝐼 𝐼 PROPN ejpam-5960 681 11 and and CCONJ ejpam-5960 681 12 π‘₯ π‘₯ PROPN ejpam-5960 681 13 ( ( PUNCT ejpam-5960 681 14 πœ‚,𝛼 πœ‚,𝛼 NOUN ejpam-5960 681 15 ) ) PUNCT ejpam-5960 681 16 βˆ‰ βˆ‰ PROPN ejpam-5960 681 17 πœ‚}.in πœ‚}.in X ejpam-5960 681 18 𝐷 𝐷 PROPN ejpam-5960 681 19 ( ( PUNCT ejpam-5960 681 20 𝐼 𝐼 PROPN ejpam-5960 681 21 ) ) PUNCT ejpam-5960 681 22 we we PRON ejpam-5960 681 23 define define VERB ejpam-5960 681 24 the the DET ejpam-5960 681 25 relation relation NOUN ejpam-5960 681 26 that that PRON ejpam-5960 681 27 ( ( PUNCT ejpam-5960 681 28 πœ‚1 πœ‚1 NOUN ejpam-5960 681 29 , , PUNCT ejpam-5960 681 30 𝛼1 𝛼1 NOUN ejpam-5960 681 31 ) ) PUNCT ejpam-5960 681 32 β‰₯ β‰₯ NOUN ejpam-5960 681 33 ( ( PUNCT ejpam-5960 681 34 πœ‚2 πœ‚2 ADJ ejpam-5960 681 35 , , PUNCT ejpam-5960 681 36 𝛼2 𝛼2 PROPN ejpam-5960 681 37 ) ) PUNCT ejpam-5960 681 38 iff iff PROPN ejpam-5960 681 39 πœ‚1 πœ‚1 PROPN ejpam-5960 681 40 β‰₯ β‰₯ NOUN ejpam-5960 681 41 πœ‚2 πœ‚2 PROPN ejpam-5960 681 42 . . PUNCT ejpam-5960 682 1 then then ADV ejpam-5960 682 2 ( ( PUNCT ejpam-5960 682 3 𝐷 𝐷 PROPN ejpam-5960 682 4 ( ( PUNCT ejpam-5960 682 5 𝐼 𝐼 PROPN ejpam-5960 682 6 ) ) PUNCT ejpam-5960 682 7 , , PUNCT ejpam-5960 682 8 β‰₯ β‰₯ NUM ejpam-5960 682 9 ) ) PUNCT ejpam-5960 682 10 is be AUX ejpam-5960 682 11 a a DET ejpam-5960 682 12 directed direct VERB ejpam-5960 682 13 set set NOUN ejpam-5960 682 14 with with ADP ejpam-5960 682 15 this this DET ejpam-5960 682 16 relation relation NOUN ejpam-5960 682 17 and and CCONJ ejpam-5960 682 18 𝑆(𝐼 𝑆(𝐼 NOUN ejpam-5960 682 19 ) ) PUNCT ejpam-5960 682 20 = = PRON ejpam-5960 682 21 { { PUNCT ejpam-5960 682 22 𝑆(πœ‚ 𝑆(πœ‚ PROPN ejpam-5960 682 23 , , PUNCT ejpam-5960 682 24 𝛼 𝛼 NOUN ejpam-5960 682 25 ) ) PUNCT ejpam-5960 682 26 = = SYM ejpam-5960 682 27 π‘₯ π‘₯ PROPN ejpam-5960 682 28 ( ( PUNCT ejpam-5960 682 29 πœ‚,𝛼 πœ‚,𝛼 NOUN ejpam-5960 682 30 ) ) PUNCT ejpam-5960 682 31 : : PUNCT ejpam-5960 682 32 ( ( PUNCT ejpam-5960 682 33 πœ‚ πœ‚ NOUN ejpam-5960 682 34 , , PUNCT ejpam-5960 682 35 𝛼 𝛼 ADJ ejpam-5960 682 36 ) ) PUNCT ejpam-5960 682 37 ∈ ∈ PROPN ejpam-5960 682 38 𝐷 𝐷 PROPN ejpam-5960 682 39 ( ( PUNCT ejpam-5960 682 40 𝐼 𝐼 PROPN ejpam-5960 682 41 ) ) PUNCT ejpam-5960 682 42 } } PUNCT ejpam-5960 682 43 is be AUX ejpam-5960 682 44 a a DET ejpam-5960 682 45 constant constant ADJ ejpam-5960 682 46 molecular molecular ADJ ejpam-5960 682 47 𝛼–net 𝛼–net NUM ejpam-5960 682 48 in in ADP ejpam-5960 682 49 πœ‡. πœ‡. NOUN ejpam-5960 682 50 since since SCONJ ejpam-5960 682 51 πœ‡ πœ‡ ADV ejpam-5960 682 52 is be AUX ejpam-5960 682 53 a a DET ejpam-5960 682 54 𝑁𝛼–bounded 𝑁𝛼–bounded NUM ejpam-5960 682 55 set set NOUN ejpam-5960 682 56 , , PUNCT ejpam-5960 682 57 then then ADV ejpam-5960 682 58 by by ADP ejpam-5960 682 59 theorem theorem NOUN ejpam-5960 682 60 4.1 4.1 NUM ejpam-5960 682 61 , , PUNCT ejpam-5960 682 62 𝑆(𝐼 𝑆(𝐼 NOUN ejpam-5960 682 63 ) ) PUNCT ejpam-5960 682 64 has have VERB ejpam-5960 682 65 a a DET ejpam-5960 682 66 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 682 67 point point NOUN ejpam-5960 682 68 in in ADP ejpam-5960 682 69 𝑋 𝑋 NOUN ejpam-5960 682 70 with with ADP ejpam-5960 682 71 height height NOUN ejpam-5960 682 72 𝛼 𝛼 PROPN ejpam-5960 682 73 , , PUNCT ejpam-5960 682 74 say say VERB ejpam-5960 682 75 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 682 76 , , PUNCT ejpam-5960 682 77 by by ADP ejpam-5960 682 78 theorem theorem NOUN ejpam-5960 682 79 2.30 2.30 NUM ejpam-5960 682 80 , , PUNCT ejpam-5960 682 81 we we PRON ejpam-5960 682 82 have have AUX ejpam-5960 682 83 π‘₯𝛼 π‘₯𝛼 ADV ejpam-5960 682 84 is be AUX ejpam-5960 682 85 also also ADV ejpam-5960 682 86 a a DET ejpam-5960 682 87 𝛿–cluster 𝛿–cluster NOUN ejpam-5960 682 88 point point NOUN ejpam-5960 682 89 of of ADP ejpam-5960 682 90 𝐼. 𝐼. PROPN ejpam-5960 682 91 acknowledgements acknowledgement VERB ejpam-5960 682 92 the the DET ejpam-5960 682 93 author author NOUN ejpam-5960 682 94 is be AUX ejpam-5960 682 95 thankful thankful ADJ ejpam-5960 682 96 to to ADP ejpam-5960 682 97 the the DET ejpam-5960 682 98 referees referee NOUN ejpam-5960 682 99 for for ADP ejpam-5960 682 100 their their PRON ejpam-5960 682 101 kind kind ADJ ejpam-5960 682 102 suggestions suggestion NOUN ejpam-5960 682 103 , , PUNCT ejpam-5960 682 104 which which PRON ejpam-5960 682 105 considerably considerably ADV ejpam-5960 682 106 improved improve VERB ejpam-5960 682 107 the the DET ejpam-5960 682 108 presentation presentation NOUN ejpam-5960 682 109 of of ADP ejpam-5960 682 110 the the DET ejpam-5960 682 111 paper paper NOUN ejpam-5960 682 112 . . PUNCT ejpam-5960 683 1 references reference NOUN ejpam-5960 683 2 [ [ X ejpam-5960 683 3 1 1 NUM ejpam-5960 683 4 ] ] PUNCT ejpam-5960 683 5 p. p. NOUN ejpam-5960 683 6 t. t. PROPN ejpam-5960 683 7 lamprinos lamprinos PROPN ejpam-5960 683 8 . . PUNCT ejpam-5960 684 1 a a DET ejpam-5960 684 2 topological topological ADJ ejpam-5960 684 3 notion notion NOUN ejpam-5960 684 4 of of ADP ejpam-5960 684 5 boundedness boundedness PROPN ejpam-5960 684 6 . . PUNCT ejpam-5960 685 1 manuscripta manuscripta PROPN ejpam-5960 685 2 mathematica mathematica PROPN ejpam-5960 685 3 , , PUNCT ejpam-5960 685 4 10:289–296 10:289–296 PROPN ejpam-5960 685 5 , , PUNCT ejpam-5960 685 6 1973 1973 NUM ejpam-5960 685 7 . . PUNCT ejpam-5960 686 1 [ [ X ejpam-5960 686 2 2 2 X ejpam-5960 686 3 ] ] PUNCT ejpam-5960 686 4 s. s. PROPN ejpam-5960 686 5 m. m. PROPN ejpam-5960 686 6 karnik karnik PROPN ejpam-5960 686 7 . . PUNCT ejpam-5960 687 1 on on ADP ejpam-5960 687 2 boundedness boundedness PROPN ejpam-5960 687 3 . . PUNCT ejpam-5960 688 1 mathematica mathematica PROPN ejpam-5960 688 2 japonica japonica PROPN ejpam-5960 688 3 , , PUNCT ejpam-5960 688 4 19:165 19:165 NUM ejpam-5960 688 5 , , PUNCT ejpam-5960 688 6 1974 1974 NUM ejpam-5960 688 7 . . PUNCT ejpam-5960 689 1 [ [ X ejpam-5960 689 2 3 3 X ejpam-5960 689 3 ] ] PUNCT ejpam-5960 689 4 s. s. PROPN ejpam-5960 689 5 t. t. PROPN ejpam-5960 689 6 hu hu PROPN ejpam-5960 689 7 . . PROPN ejpam-5960 689 8 boundedness boundedness PROPN ejpam-5960 689 9 in in ADP ejpam-5960 689 10 topological topological ADJ ejpam-5960 689 11 spaces space NOUN ejpam-5960 689 12 . . PUNCT ejpam-5960 690 1 journal journal PROPN ejpam-5960 690 2 de de PROPN ejpam-5960 690 3 mathématiques mathématiques PROPN ejpam-5960 690 4 pures pure NOUN ejpam-5960 690 5 et et NOUN ejpam-5960 690 6 appliquées appliquée NOUN ejpam-5960 690 7 , , PUNCT ejpam-5960 690 8 28:287–320 28:287–320 NUM ejpam-5960 690 9 , , PUNCT ejpam-5960 690 10 1949 1949 NUM ejpam-5960 690 11 . . PUNCT ejpam-5960 691 1 [ [ X ejpam-5960 691 2 4 4 X ejpam-5960 691 3 ] ] PUNCT ejpam-5960 691 4 p. p. NOUN ejpam-5960 691 5 t. t. PROPN ejpam-5960 691 6 lamprinos lamprinos PROPN ejpam-5960 691 7 . . PUNCT ejpam-5960 692 1 some some DET ejpam-5960 692 2 weaker weak ADJ ejpam-5960 692 3 forms form NOUN ejpam-5960 692 4 of of ADP ejpam-5960 692 5 topological topological ADJ ejpam-5960 692 6 boundedness boundedness NOUN ejpam-5960 692 7 . . PUNCT ejpam-5960 693 1 annales annales PROPN ejpam-5960 693 2 de de ADP ejpam-5960 693 3 la la PROPN ejpam-5960 693 4 société société PROPN ejpam-5960 693 5 scientifique scientifique PROPN ejpam-5960 693 6 de de X ejpam-5960 693 7 bruxelles bruxelle NOUN ejpam-5960 693 8 , , PUNCT ejpam-5960 693 9 90:109–124 90:109–124 NUM ejpam-5960 693 10 , , PUNCT ejpam-5960 693 11 1976 1976 NUM ejpam-5960 693 12 . . PUNCT ejpam-5960 694 1 [ [ X ejpam-5960 694 2 5 5 X ejpam-5960 694 3 ] ] PUNCT ejpam-5960 694 4 c. c. PROPN ejpam-5960 694 5 l. l. PROPN ejpam-5960 694 6 chang chang PROPN ejpam-5960 694 7 . . PUNCT ejpam-5960 695 1 fuzzy fuzzy ADJ ejpam-5960 695 2 topological topological ADJ ejpam-5960 695 3 spaces space NOUN ejpam-5960 695 4 . . PUNCT ejpam-5960 696 1 journal journal PROPN ejpam-5960 696 2 of of ADP ejpam-5960 696 3 mathematical mathematical ADJ ejpam-5960 696 4 analysis analysis NOUN ejpam-5960 696 5 and and CCONJ ejpam-5960 696 6 applications application NOUN ejpam-5960 696 7 , , PUNCT ejpam-5960 696 8 24:182–190 24:182–190 NUM ejpam-5960 696 9 , , PUNCT ejpam-5960 696 10 1968 1968 NUM ejpam-5960 696 11 . . PUNCT ejpam-5960 697 1 [ [ X ejpam-5960 697 2 6 6 NUM ejpam-5960 697 3 ] ] PUNCT ejpam-5960 697 4 t. t. PROPN ejpam-5960 697 5 e. e. PROPN ejpam-5960 697 6 ganter ganter PROPN ejpam-5960 697 7 , , PUNCT ejpam-5960 697 8 r. r. PROPN ejpam-5960 697 9 c. c. PROPN ejpam-5960 697 10 steinlage steinlage PROPN ejpam-5960 697 11 , , PUNCT ejpam-5960 697 12 and and CCONJ ejpam-5960 697 13 r. r. PROPN ejpam-5960 697 14 h. h. PROPN ejpam-5960 697 15 warren warren PROPN ejpam-5960 697 16 . . PUNCT ejpam-5960 698 1 compactness compactness NOUN ejpam-5960 698 2 in in ADP ejpam-5960 698 3 fuzzy fuzzy ADJ ejpam-5960 698 4 topological topological ADJ ejpam-5960 698 5 space space NOUN ejpam-5960 698 6 . . PUNCT ejpam-5960 699 1 journal journal PROPN ejpam-5960 699 2 of of ADP ejpam-5960 699 3 mathematical mathematical ADJ ejpam-5960 699 4 analysis analysis NOUN ejpam-5960 699 5 and and CCONJ ejpam-5960 699 6 applications application NOUN ejpam-5960 699 7 , , PUNCT ejpam-5960 699 8 62:547–562 62:547–562 NUM ejpam-5960 699 9 , , PUNCT ejpam-5960 699 10 1978 1978 NUM ejpam-5960 699 11 . . PUNCT ejpam-5960 700 1 [ [ X ejpam-5960 700 2 7 7 X ejpam-5960 700 3 ] ] X ejpam-5960 700 4 g. g. PROPN ejpam-5960 700 5 meng meng PROPN ejpam-5960 700 6 . . PUNCT ejpam-5960 701 1 on on ADP ejpam-5960 701 2 the the DET ejpam-5960 701 3 sum sum NOUN ejpam-5960 701 4 of of ADP ejpam-5960 701 5 𝑙–fuzzy 𝑙–fuzzy ADJ ejpam-5960 701 6 topological topological ADJ ejpam-5960 701 7 spaces space NOUN ejpam-5960 701 8 . . PUNCT ejpam-5960 702 1 fuzzy fuzzy ADJ ejpam-5960 702 2 sets set NOUN ejpam-5960 702 3 and and CCONJ ejpam-5960 702 4 systems system NOUN ejpam-5960 702 5 , , PUNCT ejpam-5960 702 6 59:65–77 59:65–77 NUM ejpam-5960 702 7 , , PUNCT ejpam-5960 702 8 1993 1993 NUM ejpam-5960 702 9 . . PUNCT ejpam-5960 703 1 [ [ X ejpam-5960 703 2 8 8 NUM ejpam-5960 703 3 ] ] PUNCT ejpam-5960 703 4 z. z. PROPN ejpam-5960 703 5 li li PROPN ejpam-5960 703 6 . . PROPN ejpam-5960 703 7 compactness compactness NOUN ejpam-5960 703 8 in in ADP ejpam-5960 703 9 fuzzy fuzzy ADJ ejpam-5960 703 10 topological topological ADJ ejpam-5960 703 11 spaces space NOUN ejpam-5960 703 12 . . PUNCT ejpam-5960 704 1 kexue kexue PROPN ejpam-5960 704 2 tongbao tongbao VERB ejpam-5960 704 3 , , PUNCT ejpam-5960 704 4 29(5):582–585 29(5):582–585 PROPN ejpam-5960 704 5 , , PUNCT ejpam-5960 704 6 1984 1984 NUM ejpam-5960 704 7 . . PUNCT ejpam-5960 705 1 [ [ X ejpam-5960 705 2 9 9 NUM ejpam-5960 705 3 ] ] X ejpam-5960 705 4 g. g. PROPN ejpam-5960 705 5 j. j. PROPN ejpam-5960 705 6 wang wang PROPN ejpam-5960 705 7 . . PUNCT ejpam-5960 705 8 theory theory NOUN ejpam-5960 705 9 of of ADP ejpam-5960 705 10 topological topological ADJ ejpam-5960 705 11 molecular molecular ADJ ejpam-5960 705 12 lattices lattice NOUN ejpam-5960 705 13 . . PUNCT ejpam-5960 706 1 fuzzy fuzzy ADJ ejpam-5960 706 2 sets set NOUN ejpam-5960 706 3 and and CCONJ ejpam-5960 706 4 systems system NOUN ejpam-5960 706 5 , , PUNCT ejpam-5960 706 6 47(3):351–376 47(3):351–376 PROPN ejpam-5960 706 7 , , PUNCT ejpam-5960 706 8 1992 1992 NUM ejpam-5960 706 9 . . PUNCT ejpam-5960 707 1 [ [ X ejpam-5960 707 2 10 10 NUM ejpam-5960 707 3 ] ] X ejpam-5960 707 4 d. d. PROPN ejpam-5960 707 5 n. n. PROPN ejpam-5960 707 6 georgiou georgiou PROPN ejpam-5960 707 7 and and CCONJ ejpam-5960 707 8 b. b. PROPN ejpam-5960 707 9 k. k. PROPN ejpam-5960 707 10 papadopoulos papadopoulos PROPN ejpam-5960 707 11 . . PUNCT ejpam-5960 708 1 on on ADP ejpam-5960 708 2 nearly nearly ADV ejpam-5960 708 3 compact compact ADJ ejpam-5960 708 4 topological topological ADJ ejpam-5960 708 5 and and CCONJ ejpam-5960 708 6 fuzzy fuzzy ADJ ejpam-5960 708 7 topological topological ADJ ejpam-5960 708 8 spaces space NOUN ejpam-5960 708 9 . . PUNCT ejpam-5960 709 1 research research NOUN ejpam-5960 709 2 communications communication NOUN ejpam-5960 709 3 of of ADP ejpam-5960 709 4 democritus democritus PROPN ejpam-5960 709 5 university university PROPN ejpam-5960 709 6 , , PUNCT ejpam-5960 709 7 pages page NOUN ejpam-5960 709 8 1–18 1–18 NUM ejpam-5960 709 9 , , PUNCT ejpam-5960 709 10 1997 1997 NUM ejpam-5960 709 11 . . PUNCT ejpam-5960 710 1 [ [ X ejpam-5960 710 2 11 11 NUM ejpam-5960 710 3 ] ] PUNCT ejpam-5960 710 4 d. d. PROPN ejpam-5960 710 5 n. n. PROPN ejpam-5960 710 6 georgiou georgiou PROPN ejpam-5960 710 7 and and CCONJ ejpam-5960 710 8 b. b. PROPN ejpam-5960 710 9 k. k. PROPN ejpam-5960 710 10 papadopoulos papadopoulos PROPN ejpam-5960 710 11 . . PROPN ejpam-5960 711 1 boundedness boundedness PROPN ejpam-5960 711 2 and and CCONJ ejpam-5960 711 3 fuzzy fuzzy ADJ ejpam-5960 711 4 sets set NOUN ejpam-5960 711 5 . . PUNCT ejpam-5960 712 1 journal journal NOUN ejpam-5960 712 2 of of ADP ejpam-5960 712 3 fuzzy fuzzy ADJ ejpam-5960 712 4 mathematics mathematic NOUN ejpam-5960 712 5 , , PUNCT ejpam-5960 712 6 6(4):941–955 6(4):941–955 NOUN ejpam-5960 712 7 , , PUNCT ejpam-5960 712 8 1998 1998 NUM ejpam-5960 712 9 . . PUNCT ejpam-5960 713 1 [ [ X ejpam-5960 713 2 12 12 NUM ejpam-5960 713 3 ] ] X ejpam-5960 713 4 d. d. PROPN ejpam-5960 713 5 n. n. PROPN ejpam-5960 713 6 georgiou georgiou PROPN ejpam-5960 713 7 and and CCONJ ejpam-5960 713 8 b. b. PROPN ejpam-5960 713 9 k. k. PROPN ejpam-5960 713 10 papadopoulos papadopoulos PROPN ejpam-5960 713 11 . . PROPN ejpam-5960 714 1 on on ADP ejpam-5960 714 2 fuzzy fuzzy ADJ ejpam-5960 714 3 boundedness boundedness NOUN ejpam-5960 714 4 . . PUNCT ejpam-5960 715 1 panamerican panamerican PROPN ejpam-5960 715 2 mathematical mathematical ADJ ejpam-5960 715 3 journal journal PROPN ejpam-5960 715 4 , , PUNCT ejpam-5960 715 5 10(1):25–43 10(1):25–43 NUM ejpam-5960 715 6 , , PUNCT ejpam-5960 715 7 2000 2000 NUM ejpam-5960 715 8 . . PUNCT ejpam-5960 716 1 n. n. NOUN ejpam-5960 716 2 a. a. PROPN ejpam-5960 716 3 alsaedi alsaedi PROPN ejpam-5960 716 4 / / SYM ejpam-5960 716 5 eur eur PROPN ejpam-5960 716 6 . . PUNCT ejpam-5960 717 1 j. j. PROPN ejpam-5960 717 2 pure pure PROPN ejpam-5960 717 3 appl appl PROPN ejpam-5960 717 4 . . PROPN ejpam-5960 717 5 math math PROPN ejpam-5960 717 6 , , PUNCT ejpam-5960 717 7 18 18 NUM ejpam-5960 717 8 ( ( PUNCT ejpam-5960 717 9 4 4 NUM ejpam-5960 717 10 ) ) PUNCT ejpam-5960 717 11 ( ( PUNCT ejpam-5960 717 12 2025 2025 NUM ejpam-5960 717 13 ) ) PUNCT ejpam-5960 717 14 , , PUNCT ejpam-5960 717 15 5960 5960 NUM ejpam-5960 717 16 22 22 NUM ejpam-5960 717 17 of of ADP ejpam-5960 717 18 22 22 NUM ejpam-5960 717 19 [ [ X ejpam-5960 717 20 13 13 NUM ejpam-5960 717 21 ] ] PUNCT ejpam-5960 717 22 a. a. NOUN ejpam-5960 717 23 a. a. PROPN ejpam-5960 717 24 nouh nouh PROPN ejpam-5960 717 25 . . PUNCT ejpam-5960 718 1 boundedness boundedness PROPN ejpam-5960 718 2 in in ADP ejpam-5960 718 3 𝑙–fuzzy 𝑙–fuzzy ADJ ejpam-5960 718 4 topological topological ADJ ejpam-5960 718 5 space space NOUN ejpam-5960 718 6 . . PUNCT ejpam-5960 719 1 journal journal NOUN ejpam-5960 719 2 of of ADP ejpam-5960 719 3 fuzzy fuzzy ADJ ejpam-5960 719 4 mathematics mathematic NOUN ejpam-5960 719 5 , , PUNCT ejpam-5960 719 6 11(2):1–11 11(2):1–11 NOUN ejpam-5960 719 7 , , PUNCT ejpam-5960 719 8 2003 2003 NUM ejpam-5960 719 9 . . PUNCT ejpam-5960 720 1 [ [ X ejpam-5960 720 2 14 14 NUM ejpam-5960 720 3 ] ] X ejpam-5960 720 4 n. n. PROPN ejpam-5960 720 5 a. a. NOUN ejpam-5960 720 6 alsaedi alsaedi PROPN ejpam-5960 720 7 . . PUNCT ejpam-5960 721 1 nearly nearly ADV ejpam-5960 721 2 πœ”-boundedness πœ”-boundedness NOUN ejpam-5960 721 3 in in ADP ejpam-5960 721 4 𝑙-topological 𝑙-topological ADJ ejpam-5960 721 5 space space NOUN ejpam-5960 721 6 . . PUNCT ejpam-5960 722 1 pure pure ADJ ejpam-5960 722 2 mathematical mathematical ADJ ejpam-5960 722 3 sciences science NOUN ejpam-5960 722 4 , , PUNCT ejpam-5960 722 5 12(1):1–19 12(1):1–19 NUM ejpam-5960 722 6 , , PUNCT ejpam-5960 722 7 2023 2023 NUM ejpam-5960 722 8 . . PUNCT ejpam-5960 723 1 [ [ X ejpam-5960 723 2 15 15 NUM ejpam-5960 723 3 ] ] X ejpam-5960 723 4 d. d. PROPN ejpam-5960 723 5 s. s. PROPN ejpam-5960 723 6 zhao zhao PROPN ejpam-5960 723 7 . . PUNCT ejpam-5960 724 1 the the DET ejpam-5960 724 2 𝑛-compactness 𝑛-compactness NOUN ejpam-5960 724 3 in in ADP ejpam-5960 724 4 fuzzy fuzzy ADJ ejpam-5960 724 5 topological topological ADJ ejpam-5960 724 6 spaces space NOUN ejpam-5960 724 7 . . PUNCT ejpam-5960 725 1 journal journal PROPN ejpam-5960 725 2 of of ADP ejpam-5960 725 3 mathematical mathematical ADJ ejpam-5960 725 4 analysis analysis NOUN ejpam-5960 725 5 and and CCONJ ejpam-5960 725 6 applications application NOUN ejpam-5960 725 7 , , PUNCT ejpam-5960 725 8 128:64–79 128:64–79 NUM ejpam-5960 725 9 , , PUNCT ejpam-5960 725 10 1987 1987 NUM ejpam-5960 725 11 . . PUNCT ejpam-5960 726 1 [ [ X ejpam-5960 726 2 16 16 NUM ejpam-5960 726 3 ] ] PUNCT ejpam-5960 726 4 j. j. PROPN ejpam-5960 726 5 a. a. PROPN ejpam-5960 726 6 goguen goguen PROPN ejpam-5960 726 7 . . PUNCT ejpam-5960 727 1 𝑙-fuzzy 𝑙-fuzzy PROPN ejpam-5960 727 2 sets set VERB ejpam-5960 727 3 . . PUNCT ejpam-5960 728 1 journal journal NOUN ejpam-5960 728 2 of of ADP ejpam-5960 728 3 mathematical mathematical ADJ ejpam-5960 728 4 analysis analysis NOUN ejpam-5960 728 5 and and CCONJ ejpam-5960 728 6 applications application NOUN ejpam-5960 728 7 , , PUNCT ejpam-5960 728 8 18(1):145–174 18(1):145–174 NUM ejpam-5960 728 9 , , PUNCT ejpam-5960 728 10 1967 1967 NUM ejpam-5960 728 11 . . PUNCT ejpam-5960 729 1 [ [ X ejpam-5960 729 2 17 17 NUM ejpam-5960 729 3 ] ] X ejpam-5960 729 4 g. g. PROPN ejpam-5960 729 5 j. j. PROPN ejpam-5960 729 6 wang wang PROPN ejpam-5960 729 7 . . PUNCT ejpam-5960 730 1 generalized generalize VERB ejpam-5960 730 2 topological topological ADJ ejpam-5960 730 3 molecular molecular ADJ ejpam-5960 730 4 lattice lattice NOUN ejpam-5960 730 5 . . PUNCT ejpam-5960 731 1 scientia scientia PROPN ejpam-5960 731 2 sinica sinica PROPN ejpam-5960 731 3 series series PROPN ejpam-5960 731 4 a a PROPN ejpam-5960 731 5 , , PUNCT ejpam-5960 731 6 ( ( PUNCT ejpam-5960 731 7 8):785–789 8):785–789 ADV ejpam-5960 731 8 , , PUNCT ejpam-5960 731 9 1984 1984 NUM ejpam-5960 731 10 . . PUNCT ejpam-5960 732 1 [ [ X ejpam-5960 732 2 18 18 NUM ejpam-5960 732 3 ] ] X ejpam-5960 732 4 y. y. PROPN ejpam-5960 732 5 m. m. PROPN ejpam-5960 732 6 liu liu PROPN ejpam-5960 732 7 . . PUNCT ejpam-5960 732 8 completely completely ADV ejpam-5960 732 9 distributive distributive ADJ ejpam-5960 732 10 law law NOUN ejpam-5960 732 11 and and CCONJ ejpam-5960 732 12 induced induced ADJ ejpam-5960 732 13 spaces space NOUN ejpam-5960 732 14 . . PUNCT ejpam-5960 733 1 in in ADP ejpam-5960 733 2 preprints preprint NOUN ejpam-5960 733 3 of of ADP ejpam-5960 733 4 second second ADJ ejpam-5960 733 5 ifsa ifsa PROPN ejpam-5960 733 6 congress congress PROPN ejpam-5960 733 7 , , PUNCT ejpam-5960 733 8 pages page NOUN ejpam-5960 733 9 460–463 460–463 NUM ejpam-5960 733 10 , , PUNCT ejpam-5960 733 11 1987 1987 NUM ejpam-5960 733 12 . . PUNCT ejpam-5960 734 1 [ [ X ejpam-5960 734 2 19 19 NUM ejpam-5960 734 3 ] ] X ejpam-5960 734 4 r. r. PROPN ejpam-5960 734 5 lowen lowen PROPN ejpam-5960 734 6 . . PUNCT ejpam-5960 735 1 fuzzy fuzzy ADJ ejpam-5960 735 2 topological topological ADJ ejpam-5960 735 3 space space NOUN ejpam-5960 735 4 . . PUNCT ejpam-5960 736 1 journal journal PROPN ejpam-5960 736 2 of of ADP ejpam-5960 736 3 mathematical mathematical ADJ ejpam-5960 736 4 analysis analysis NOUN ejpam-5960 736 5 and and CCONJ ejpam-5960 736 6 applications application NOUN ejpam-5960 736 7 , , PUNCT ejpam-5960 736 8 56:621–633 56:621–633 NUM ejpam-5960 736 9 , , PUNCT ejpam-5960 736 10 1976 1976 NUM ejpam-5960 736 11 . . PUNCT ejpam-5960 737 1 [ [ X ejpam-5960 737 2 20 20 NUM ejpam-5960 737 3 ] ] X ejpam-5960 737 4 g. g. PROPN ejpam-5960 737 5 j. j. PROPN ejpam-5960 737 6 wang wang PROPN ejpam-5960 737 7 . . PUNCT ejpam-5960 738 1 a a DET ejpam-5960 738 2 new new ADJ ejpam-5960 738 3 fuzzy fuzzy ADJ ejpam-5960 738 4 compactness compactness NOUN ejpam-5960 738 5 defined define VERB ejpam-5960 738 6 by by ADP ejpam-5960 738 7 fuzzy fuzzy ADJ ejpam-5960 738 8 nets net NOUN ejpam-5960 738 9 . . PUNCT ejpam-5960 739 1 journal journal NOUN ejpam-5960 739 2 of of ADP ejpam-5960 739 3 mathematical mathematical ADJ ejpam-5960 739 4 analysis analysis NOUN ejpam-5960 739 5 and and CCONJ ejpam-5960 739 6 applications application NOUN ejpam-5960 739 7 , , PUNCT ejpam-5960 739 8 94:1–23 94:1–23 NUM ejpam-5960 739 9 , , PUNCT ejpam-5960 739 10 1983 1983 NUM ejpam-5960 739 11 . . PUNCT ejpam-5960 740 1 [ [ X ejpam-5960 740 2 21 21 NUM ejpam-5960 740 3 ] ] X ejpam-5960 740 4 d. d. PROPN ejpam-5960 740 5 n. n. PROPN ejpam-5960 740 6 georgiou georgiou PROPN ejpam-5960 740 7 and and CCONJ ejpam-5960 740 8 b. b. PROPN ejpam-5960 740 9 k. k. PROPN ejpam-5960 740 10 papadopoulos papadopoulos PROPN ejpam-5960 740 11 . . PROPN ejpam-5960 741 1 on on ADP ejpam-5960 741 2 πœƒ-fuzzy πœƒ-fuzzy NOUN ejpam-5960 741 3 convergences convergence NOUN ejpam-5960 741 4 . . PUNCT ejpam-5960 742 1 fuzzy fuzzy ADJ ejpam-5960 742 2 sets set NOUN ejpam-5960 742 3 and and CCONJ ejpam-5960 742 4 systems system NOUN ejpam-5960 742 5 , , PUNCT ejpam-5960 742 6 116:385–399 116:385–399 NUM ejpam-5960 742 7 , , PUNCT ejpam-5960 742 8 2000 2000 NUM ejpam-5960 742 9 . . PUNCT ejpam-5960 743 1 [ [ X ejpam-5960 743 2 22 22 NUM ejpam-5960 743 3 ] ] PUNCT ejpam-5960 743 4 s. s. PROPN ejpam-5960 743 5 l. l. PROPN ejpam-5960 743 6 chen chen PROPN ejpam-5960 743 7 and and CCONJ ejpam-5960 743 8 j. j. PROPN ejpam-5960 743 9 s. s. PROPN ejpam-5960 743 10 cheng cheng PROPN ejpam-5960 743 11 . . PUNCT ejpam-5960 744 1 on on ADP ejpam-5960 744 2 convergence convergence NOUN ejpam-5960 744 3 of of ADP ejpam-5960 744 4 𝑙-nets 𝑙-net NOUN ejpam-5960 744 5 of of ADP ejpam-5960 744 6 fuzzy fuzzy ADJ ejpam-5960 744 7 sets set NOUN ejpam-5960 744 8 . . PUNCT ejpam-5960 745 1 journal journal NOUN ejpam-5960 745 2 of of ADP ejpam-5960 745 3 fuzzy fuzzy ADJ ejpam-5960 745 4 mathematics mathematic NOUN ejpam-5960 745 5 , , PUNCT ejpam-5960 745 6 2:517–524 2:517–524 NUM ejpam-5960 745 7 , , PUNCT ejpam-5960 745 8 1994 1994 NUM ejpam-5960 745 9 . . PUNCT ejpam-5960 746 1 [ [ X ejpam-5960 746 2 23 23 NUM ejpam-5960 746 3 ] ] PUNCT ejpam-5960 746 4 j. j. PROPN ejpam-5960 746 5 x. x. PROPN ejpam-5960 746 6 fang fang PROPN ejpam-5960 746 7 and and CCONJ ejpam-5960 746 8 b. b. PROPN ejpam-5960 746 9 l. l. PROPN ejpam-5960 746 10 ren ren PROPN ejpam-5960 746 11 . . PUNCT ejpam-5960 747 1 a a DET ejpam-5960 747 2 set set NOUN ejpam-5960 747 3 of of ADP ejpam-5960 747 4 new new ADJ ejpam-5960 747 5 separation separation NOUN ejpam-5960 747 6 axioms axiom NOUN ejpam-5960 747 7 in in ADP ejpam-5960 747 8 𝑙-fuzzy 𝑙-fuzzy PROPN ejpam-5960 747 9 topological topological ADJ ejpam-5960 747 10 spaces space NOUN ejpam-5960 747 11 . . PUNCT ejpam-5960 748 1 fuzzy fuzzy ADJ ejpam-5960 748 2 sets set NOUN ejpam-5960 748 3 and and CCONJ ejpam-5960 748 4 systems system NOUN ejpam-5960 748 5 , , PUNCT ejpam-5960 748 6 96:359–366 96:359–366 PROPN ejpam-5960 748 7 , , PUNCT ejpam-5960 748 8 1998 1998 NUM ejpam-5960 748 9 . . PUNCT ejpam-5960 749 1 [ [ X ejpam-5960 749 2 24 24 NUM ejpam-5960 749 3 ] ] X ejpam-5960 749 4 n. n. PROPN ejpam-5960 749 5 a. a. NOUN ejpam-5960 749 6 alsaedi alsaedi PROPN ejpam-5960 749 7 . . PUNCT ejpam-5960 750 1 𝛼-boundedness 𝛼-boundedness NOUN ejpam-5960 750 2 in in ADP ejpam-5960 750 3 𝑙-topological 𝑙-topological ADJ ejpam-5960 750 4 spaces space NOUN ejpam-5960 750 5 . . PUNCT ejpam-5960 751 1 submitted submit VERB ejpam-5960 751 2 . . PUNCT ejpam-5960 752 1 [ [ X ejpam-5960 752 2 25 25 NUM ejpam-5960 752 3 ] ] PUNCT ejpam-5960 752 4 z. z. PROPN ejpam-5960 752 5 q. q. PROPN ejpam-5960 752 6 yang yang PROPN ejpam-5960 752 7 . . PROPN ejpam-5960 752 8 ideal ideal PROPN ejpam-5960 752 9 in in ADP ejpam-5960 752 10 topological topological ADJ ejpam-5960 752 11 molecular molecular ADJ ejpam-5960 752 12 lattices lattice NOUN ejpam-5960 752 13 . . PUNCT ejpam-5960 753 1 acta acta PROPN ejpam-5960 753 2 mathematica mathematica PROPN ejpam-5960 753 3 sinica sinica PROPN ejpam-5960 753 4 , , PUNCT ejpam-5960 753 5 29(2):276–279 29(2):276–279 PROPN ejpam-5960 753 6 , , PUNCT ejpam-5960 753 7 1986 1986 NUM ejpam-5960 753 8 . . PUNCT