id sid tid token lemma pos iajs-2814 1 1 155 155 NUM iajs-2814 1 2 this this DET iajs-2814 1 3 work work NOUN iajs-2814 1 4 is be AUX iajs-2814 1 5 licensed license VERB iajs-2814 1 6 under under ADP iajs-2814 1 7 a a DET iajs-2814 1 8 creative creative ADJ iajs-2814 1 9 commons common NOUN iajs-2814 1 10 attribution attribution NOUN iajs-2814 1 11 4.0 4.0 NUM iajs-2814 1 12 international international ADJ iajs-2814 1 13 license license NOUN iajs-2814 1 14 . . PUNCT iajs-2814 2 1 some some DET iajs-2814 2 2 properties property NOUN iajs-2814 2 3 for for ADP iajs-2814 2 4 the the DET iajs-2814 2 5 restriction restriction NOUN iajs-2814 2 6 of of ADP iajs-2814 2 7 π“Ÿβˆ— π“Ÿβˆ— PROPN iajs-2814 2 8 – – PUNCT iajs-2814 2 9 𝐟𝐒𝐞π₯𝐝 𝐟𝐒𝐞π₯𝐝 NOUN iajs-2814 2 10 of of ADP iajs-2814 2 11 sets set NOUN iajs-2814 2 12 abstract abstract VERB iajs-2814 2 13 the the DET iajs-2814 2 14 restriction restriction NOUN iajs-2814 2 15 concept concept NOUN iajs-2814 2 16 is be AUX iajs-2814 2 17 a a DET iajs-2814 2 18 basic basic ADJ iajs-2814 2 19 feature feature NOUN iajs-2814 2 20 in in ADP iajs-2814 2 21 the the DET iajs-2814 2 22 field field NOUN iajs-2814 2 23 of of ADP iajs-2814 2 24 measure measure NOUN iajs-2814 2 25 theory theory NOUN iajs-2814 2 26 and and CCONJ iajs-2814 2 27 has have VERB iajs-2814 2 28 many many ADJ iajs-2814 2 29 important important ADJ iajs-2814 2 30 properties property NOUN iajs-2814 2 31 . . PUNCT iajs-2814 3 1 this this DET iajs-2814 3 2 article article NOUN iajs-2814 3 3 introduces introduce VERB iajs-2814 3 4 the the DET iajs-2814 3 5 notion notion NOUN iajs-2814 3 6 of of ADP iajs-2814 3 7 restriction restriction NOUN iajs-2814 3 8 of of ADP iajs-2814 3 9 a a DET iajs-2814 3 10 non non ADJ iajs-2814 3 11 - - ADJ iajs-2814 3 12 empty empty ADJ iajs-2814 3 13 class class NOUN iajs-2814 3 14 of of ADP iajs-2814 3 15 subset subset NOUN iajs-2814 3 16 of of ADP iajs-2814 3 17 the the DET iajs-2814 3 18 power power NOUN iajs-2814 3 19 set set VERB iajs-2814 3 20 on on ADP iajs-2814 3 21 a a DET iajs-2814 3 22 nonempty nonempty ADJ iajs-2814 3 23 subset subset NOUN iajs-2814 3 24 of of ADP iajs-2814 3 25 a a DET iajs-2814 3 26 universal universal ADJ iajs-2814 3 27 set set NOUN iajs-2814 3 28 . . PUNCT iajs-2814 4 1 characterization characterization NOUN iajs-2814 4 2 and and CCONJ iajs-2814 4 3 examples example NOUN iajs-2814 4 4 of of ADP iajs-2814 4 5 the the DET iajs-2814 4 6 proposed propose VERB iajs-2814 4 7 concept concept NOUN iajs-2814 4 8 are be AUX iajs-2814 4 9 given give VERB iajs-2814 4 10 , , PUNCT iajs-2814 4 11 and and CCONJ iajs-2814 4 12 several several ADJ iajs-2814 4 13 properties property NOUN iajs-2814 4 14 of of ADP iajs-2814 4 15 restriction restriction NOUN iajs-2814 4 16 are be AUX iajs-2814 4 17 investigated investigate VERB iajs-2814 4 18 . . PUNCT iajs-2814 5 1 furthermore furthermore ADV iajs-2814 5 2 , , PUNCT iajs-2814 5 3 the the DET iajs-2814 5 4 relation relation NOUN iajs-2814 5 5 between between ADP iajs-2814 5 6 the the DET iajs-2814 5 7 p*–field p*–field PROPN iajs-2814 5 8 and and CCONJ iajs-2814 5 9 the the DET iajs-2814 5 10 restriction restriction NOUN iajs-2814 5 11 of of ADP iajs-2814 5 12 the the DET iajs-2814 5 13 p*–field p*–field PROPN iajs-2814 5 14 is be AUX iajs-2814 5 15 studied study VERB iajs-2814 5 16 , , PUNCT iajs-2814 5 17 explaining explain VERB iajs-2814 5 18 that that SCONJ iajs-2814 5 19 the the DET iajs-2814 5 20 restriction restriction NOUN iajs-2814 5 21 of of ADP iajs-2814 5 22 the the DET iajs-2814 5 23 p*–field p*–field PROPN iajs-2814 5 24 is be AUX iajs-2814 5 25 a a DET iajs-2814 5 26 p*–field p*–field NOUN iajs-2814 5 27 too too ADV iajs-2814 5 28 . . PUNCT iajs-2814 6 1 in in ADP iajs-2814 6 2 addition addition NOUN iajs-2814 6 3 , , PUNCT iajs-2814 6 4 it it PRON iajs-2814 6 5 has have AUX iajs-2814 6 6 been be AUX iajs-2814 6 7 shown show VERB iajs-2814 6 8 that that SCONJ iajs-2814 6 9 the the DET iajs-2814 6 10 restriction restriction NOUN iajs-2814 6 11 of of ADP iajs-2814 6 12 the the DET iajs-2814 6 13 p*–field p*–field PROPN iajs-2814 6 14 is be AUX iajs-2814 6 15 not not PART iajs-2814 6 16 necessarily necessarily ADV iajs-2814 6 17 contained contain VERB iajs-2814 6 18 in in ADP iajs-2814 6 19 the the DET iajs-2814 6 20 p*–field p*–field NOUN iajs-2814 6 21 , , PUNCT iajs-2814 6 22 and and CCONJ iajs-2814 6 23 the the DET iajs-2814 6 24 converse converse NOUN iajs-2814 6 25 is be AUX iajs-2814 6 26 true true ADJ iajs-2814 6 27 . . PUNCT iajs-2814 7 1 we we PRON iajs-2814 7 2 provide provide VERB iajs-2814 7 3 a a DET iajs-2814 7 4 necessary necessary ADJ iajs-2814 7 5 condition condition NOUN iajs-2814 7 6 for for ADP iajs-2814 7 7 the the DET iajs-2814 7 8 p*–field p*–field PROPN iajs-2814 7 9 to to PART iajs-2814 7 10 obtain obtain VERB iajs-2814 7 11 that that SCONJ iajs-2814 7 12 the the DET iajs-2814 7 13 restriction restriction NOUN iajs-2814 7 14 of of ADP iajs-2814 7 15 the the DET iajs-2814 7 16 p*–field p*–field PROPN iajs-2814 7 17 is be AUX iajs-2814 7 18 included include VERB iajs-2814 7 19 in in ADP iajs-2814 7 20 the the DET iajs-2814 7 21 p*–field p*–field NOUN iajs-2814 7 22 . . PUNCT iajs-2814 8 1 finally finally ADV iajs-2814 8 2 , , PUNCT iajs-2814 8 3 this this DET iajs-2814 8 4 article article NOUN iajs-2814 8 5 aims aim VERB iajs-2814 8 6 to to PART iajs-2814 8 7 study study VERB iajs-2814 8 8 the the DET iajs-2814 8 9 restriction restriction NOUN iajs-2814 8 10 notion notion NOUN iajs-2814 8 11 and and CCONJ iajs-2814 8 12 give give VERB iajs-2814 8 13 some some DET iajs-2814 8 14 propositions proposition NOUN iajs-2814 8 15 , , PUNCT iajs-2814 8 16 lemmas lemmas ADJ iajs-2814 8 17 , , PUNCT iajs-2814 8 18 and and CCONJ iajs-2814 8 19 theorems theorem NOUN iajs-2814 8 20 related relate VERB iajs-2814 8 21 to to ADP iajs-2814 8 22 the the DET iajs-2814 8 23 proposed propose VERB iajs-2814 8 24 concept concept NOUN iajs-2814 8 25 . . PUNCT iajs-2814 9 1 keywords keyword NOUN iajs-2814 9 2 : : PUNCT iajs-2814 9 3 Οƒ Οƒ PROPN iajs-2814 9 4 βˆ’field βˆ’field PROPN iajs-2814 9 5 , , PUNCT iajs-2814 9 6 Οƒ Οƒ PROPN iajs-2814 9 7 – – PUNCT iajs-2814 9 8 ring ring NOUN iajs-2814 9 9 , , PUNCT iajs-2814 9 10 field field NOUN iajs-2814 9 11 , , PUNCT iajs-2814 9 12 smallest small ADJ iajs-2814 9 13 Οƒ Οƒ NUM iajs-2814 9 14 βˆ’field βˆ’field NOUN iajs-2814 9 15 and and CCONJ iajs-2814 9 16 restriction restriction NOUN iajs-2814 9 17 . . PUNCT iajs-2814 10 1 1 1 X iajs-2814 10 2 . . X iajs-2814 10 3 introduction introduction NOUN iajs-2814 10 4 in in ADP iajs-2814 10 5 the the DET iajs-2814 10 6 real real ADJ iajs-2814 10 7 analysis analysis NOUN iajs-2814 10 8 and and CCONJ iajs-2814 10 9 probability probability NOUN iajs-2814 10 10 , , PUNCT iajs-2814 10 11 the the DET iajs-2814 10 12 Οƒ Οƒ PROPN iajs-2814 10 13 – – PUNCT iajs-2814 10 14 field field NOUN iajs-2814 10 15 concept concept NOUN iajs-2814 10 16 is be AUX iajs-2814 10 17 the the DET iajs-2814 10 18 class class NOUN iajs-2814 10 19 β„³ β„³ NOUN iajs-2814 10 20 for for ADP iajs-2814 10 21 a a DET iajs-2814 10 22 subset subset NOUN iajs-2814 10 23 of of ADP iajs-2814 10 24 a a DET iajs-2814 10 25 universal universal ADJ iajs-2814 10 26 set set VERB iajs-2814 10 27 𝒰 𝒰 PROPN iajs-2814 10 28 such such ADJ iajs-2814 10 29 that that PRON iajs-2814 10 30 𝒰ϡℳ 𝒰ϡℳ NOUN iajs-2814 11 1 and and CCONJ iajs-2814 11 2 it it PRON iajs-2814 11 3 is be AUX iajs-2814 11 4 closed close VERB iajs-2814 11 5 under under ADP iajs-2814 11 6 the the DET iajs-2814 11 7 complement complement NOUN iajs-2814 11 8 , , PUNCT iajs-2814 11 9 countable countable ADJ iajs-2814 11 10 union union NOUN iajs-2814 12 1 [ [ X iajs-2814 12 2 1 1 NUM iajs-2814 12 3 ] ] PUNCT iajs-2814 12 4 and and CCONJ iajs-2814 12 5 [ [ X iajs-2814 12 6 2 2 NUM iajs-2814 12 7 ] ] PUNCT iajs-2814 12 8 . . PUNCT iajs-2814 13 1 the the DET iajs-2814 13 2 main main ADJ iajs-2814 13 3 reason reason NOUN iajs-2814 13 4 for for ADP iajs-2814 13 5 Οƒ Οƒ PROPN iajs-2814 13 6 – – PUNCT iajs-2814 13 7 field field NOUN iajs-2814 13 8 is be AUX iajs-2814 13 9 the the DET iajs-2814 13 10 idea idea NOUN iajs-2814 13 11 of of ADP iajs-2814 13 12 measure measure NOUN iajs-2814 13 13 , , PUNCT iajs-2814 13 14 which which PRON iajs-2814 13 15 is be AUX iajs-2814 13 16 substantial substantial ADJ iajs-2814 13 17 in in ADP iajs-2814 13 18 the the DET iajs-2814 13 19 real real ADJ iajs-2814 13 20 analysis analysis NOUN iajs-2814 13 21 as as SCONJ iajs-2814 13 22 the the DET iajs-2814 13 23 basis basis NOUN iajs-2814 13 24 of of ADP iajs-2814 13 25 lebesgue lebesgue NOUN iajs-2814 13 26 integrals integral NOUN iajs-2814 13 27 , , PUNCT iajs-2814 13 28 where where SCONJ iajs-2814 13 29 it it PRON iajs-2814 13 30 exponent exponent VERB iajs-2814 13 31 as as ADP iajs-2814 13 32 a a DET iajs-2814 13 33 family family NOUN iajs-2814 13 34 of of ADP iajs-2814 13 35 events event NOUN iajs-2814 13 36 which which PRON iajs-2814 13 37 may may AUX iajs-2814 13 38 ibn ibn PROPN iajs-2814 13 39 al al PROPN iajs-2814 13 40 - - PUNCT iajs-2814 13 41 haitham haitham PROPN iajs-2814 13 42 journal journal PROPN iajs-2814 13 43 for for ADP iajs-2814 13 44 pure pure ADJ iajs-2814 13 45 and and CCONJ iajs-2814 13 46 applied apply VERB iajs-2814 13 47 sciences sciences PROPN iajs-2814 13 48 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index PROPN iajs-2814 13 49 : : PUNCT iajs-2814 13 50 journal journal PROPN iajs-2814 13 51 homepage homepage NOUN iajs-2814 13 52 doi doi PROPN iajs-2814 13 53 : : PUNCT iajs-2814 13 54 10.30526/35.3.2814 10.30526/35.3.2814 NUM iajs-2814 13 55 article article NOUN iajs-2814 13 56 history history NOUN iajs-2814 13 57 : : PUNCT iajs-2814 13 58 received receive VERB iajs-2814 13 59 20 20 NUM iajs-2814 13 60 february february NOUN iajs-2814 13 61 2022 2022 NUM iajs-2814 13 62 , , PUNCT iajs-2814 13 63 accepted accept VERB iajs-2814 13 64 17 17 NUM iajs-2814 13 65 may may NOUN iajs-2814 13 66 , , PUNCT iajs-2814 13 67 2022 2022 NUM iajs-2814 13 68 , , PUNCT iajs-2814 13 69 published publish VERB iajs-2814 13 70 in in ADP iajs-2814 13 71 july july PROPN iajs-2814 13 72 2022 2022 NUM iajs-2814 13 73 . . 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PUNCT iajs-2814 15 1 hind.f.abbas35386@st.tu.edu.iq hind.f.abbas35386@st.tu.edu.iq PROPN iajs-2814 15 2 ali ali PROPN iajs-2814 15 3 al al PROPN iajs-2814 15 4 - - PUNCT iajs-2814 15 5 fayadh fayadh PROPN iajs-2814 15 6 department department NOUN iajs-2814 15 7 of of ADP iajs-2814 15 8 mathematics mathematics PROPN iajs-2814 15 9 and and CCONJ iajs-2814 15 10 computer computer NOUN iajs-2814 15 11 applications application NOUN iajs-2814 15 12 / / SYM iajs-2814 15 13 college college NOUN iajs-2814 15 14 of of ADP iajs-2814 15 15 science science PROPN iajs-2814 15 16 / / SYM iajs-2814 15 17 al al PROPN iajs-2814 15 18 – – PUNCT iajs-2814 15 19 nahrain nahrain PROPN iajs-2814 15 20 university/ university/ NUM iajs-2814 15 21 iraq iraq PROPN iajs-2814 15 22 aalfayadh@yahoo.com aalfayadh@yahoo.com X iajs-2814 16 1 hassan hassan PROPN iajs-2814 16 2 h. h. PROPN iajs-2814 16 3 ebrahim ebrahim PROPN iajs-2814 16 4 department department PROPN iajs-2814 16 5 of of ADP iajs-2814 16 6 mathematics mathematics PROPN iajs-2814 16 7 / / SYM iajs-2814 16 8 college college NOUN iajs-2814 16 9 of of ADP iajs-2814 16 10 computer computer NOUN iajs-2814 16 11 science science NOUN iajs-2814 16 12 and and CCONJ iajs-2814 16 13 mathematics mathematics PROPN iajs-2814 16 14 / / SYM iajs-2814 16 15 tikrit tikrit NOUN iajs-2814 16 16 university/ university/ NUM iajs-2814 16 17 iraq iraq PROPN iajs-2814 16 18 hassan1962pl@tu.edu.iq hassan1962pl@tu.edu.iq ADJ iajs-2814 16 19 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-2814 16 20 file:///f:/Ψ§Ω„ΨΉΨ―Ψ―%20Ψ§Ω„Ψ«Ψ§Ω†ΩŠ%202022/:%20http:/jih.uobaghdad.edu.iq file:///f:/Ψ§Ω„ΨΉΨ―Ψ―%20Ψ§Ω„Ψ«Ψ§Ω†ΩŠ%202022/:%20http:/jih.uobaghdad.edu.iq PROPN iajs-2814 16 21 / / SYM iajs-2814 16 22 index.php index.php VERB iajs-2814 16 23 / / SYM iajs-2814 16 24 j j NOUN iajs-2814 16 25 / / SYM iajs-2814 16 26 index index NOUN iajs-2814 16 27 mailto:hind.f.abbas35386@st.tu.edu.iq mailto:hind.f.abbas35386@st.tu.edu.iq PROPN iajs-2814 16 28 mailto:hind.f.abbas35386@st.tu.edu.iq mailto:hind.f.abbas35386@st.tu.edu.iq PROPN iajs-2814 16 29 mailto:aalfayadh@yahoo.com mailto:aalfayadh@yahoo.com PROPN iajs-2814 16 30 mailto:hassan1962pl@tu.edu.iq mailto:hassan1962pl@tu.edu.iq NOUN iajs-2814 16 31 ihjpas ihjpa NOUN iajs-2814 16 32 . . PUNCT iajs-2814 17 1 53 53 NUM iajs-2814 17 2 ( ( PUNCT iajs-2814 17 3 3)2022 3)2022 NOUN iajs-2814 17 4 156 156 NUM iajs-2814 17 5 be be AUX iajs-2814 17 6 assigned assign VERB iajs-2814 17 7 probability probability NOUN iajs-2814 17 8 [ [ X iajs-2814 17 9 3 3 NUM iajs-2814 17 10 ] ] PUNCT iajs-2814 17 11 and and CCONJ iajs-2814 17 12 [ [ X iajs-2814 17 13 4 4 NUM iajs-2814 17 14 ] ] PUNCT iajs-2814 17 15 . . PUNCT iajs-2814 18 1 in in ADP iajs-2814 18 2 the the DET iajs-2814 18 3 probability probability NOUN iajs-2814 18 4 theory theory NOUN iajs-2814 18 5 , , PUNCT iajs-2814 18 6 a a DET iajs-2814 18 7 Οƒ Οƒ PROPN iajs-2814 18 8 – – PUNCT iajs-2814 18 9 field field NOUN iajs-2814 18 10 is be AUX iajs-2814 18 11 essential essential ADJ iajs-2814 18 12 in in ADP iajs-2814 18 13 the the DET iajs-2814 18 14 conditional conditional NOUN iajs-2814 18 15 expected expect VERB iajs-2814 18 16 . . PUNCT iajs-2814 19 1 also also ADV iajs-2814 19 2 , , PUNCT iajs-2814 19 3 in in ADP iajs-2814 19 4 statistics statistic NOUN iajs-2814 19 5 , , PUNCT iajs-2814 19 6 sub sub PROPN iajs-2814 19 7 Οƒ Οƒ PROPN iajs-2814 19 8 – – PUNCT iajs-2814 19 9 field field NOUN iajs-2814 19 10 is be AUX iajs-2814 19 11 necessary necessary ADJ iajs-2814 19 12 for for ADP iajs-2814 19 13 an an DET iajs-2814 19 14 official official ADJ iajs-2814 19 15 mathematical mathematical ADJ iajs-2814 19 16 definition definition NOUN iajs-2814 19 17 for for ADP iajs-2814 19 18 sufficient sufficient ADJ iajs-2814 19 19 statistic statistic NOUN iajs-2814 19 20 , , PUNCT iajs-2814 19 21 where where SCONJ iajs-2814 19 22 a a DET iajs-2814 19 23 statistic statistic NOUN iajs-2814 19 24 be be AUX iajs-2814 19 25 a a DET iajs-2814 19 26 map map NOUN iajs-2814 19 27 or or CCONJ iajs-2814 19 28 a a DET iajs-2814 19 29 random random ADJ iajs-2814 19 30 variable variable NOUN iajs-2814 19 31 . . PUNCT iajs-2814 20 1 a a DET iajs-2814 20 2 Οƒ Οƒ PROPN iajs-2814 20 3 – – PUNCT iajs-2814 20 4 ring ring NOUN iajs-2814 20 5 idea idea NOUN iajs-2814 20 6 was be AUX iajs-2814 20 7 studied study VERB iajs-2814 20 8 by by ADP iajs-2814 20 9 [ [ X iajs-2814 20 10 5 5 NUM iajs-2814 20 11 ] ] PUNCT iajs-2814 20 12 as as ADP iajs-2814 20 13 a a DET iajs-2814 20 14 class class NOUN iajs-2814 20 15 β„³ β„³ NOUN iajs-2814 20 16 such such ADJ iajs-2814 20 17 that that DET iajs-2814 20 18 b1\b2Ο΅β„³ b1\b2Ο΅β„³ NOUN iajs-2814 20 19 and and CCONJ iajs-2814 20 20 ⋃ ⋃ PROPN iajs-2814 20 21 bn bn NUM iajs-2814 20 22 ∞ ∞ NUM iajs-2814 20 23 𝑛=1 𝑛=1 NOUN iajs-2814 20 24 Ο΅β„³ Ο΅β„³ NUM iajs-2814 20 25 wheneverb1 wheneverb1 NOUN iajs-2814 20 26 , , PUNCT iajs-2814 20 27 b2 b2 NOUN iajs-2814 20 28 , , PUNCT iajs-2814 20 29 … … PUNCT iajs-2814 21 1 Ο΅β„³ Ο΅β„³ NOUN iajs-2814 21 2 . . PUNCT iajs-2814 22 1 many many ADJ iajs-2814 22 2 authors author NOUN iajs-2814 22 3 were be AUX iajs-2814 22 4 interested interested ADJ iajs-2814 22 5 in in ADP iajs-2814 22 6 studying study VERB iajs-2814 22 7 Οƒ Οƒ PROPN iajs-2814 22 8 – – PUNCT iajs-2814 22 9 field field NOUN iajs-2814 22 10 and and CCONJ iajs-2814 22 11 Οƒ Οƒ PROPN iajs-2814 22 12 – – PUNCT iajs-2814 22 13 ring ring NOUN iajs-2814 22 14 ; ; PUNCT iajs-2814 22 15 for for ADP iajs-2814 22 16 example example NOUN iajs-2814 22 17 , , PUNCT iajs-2814 22 18 see see VERB iajs-2814 22 19 [ [ X iajs-2814 22 20 6 6 NUM iajs-2814 22 21 ] ] PUNCT iajs-2814 22 22 , , PUNCT iajs-2814 22 23 [ [ X iajs-2814 22 24 7 7 NUM iajs-2814 22 25 ] ] PUNCT iajs-2814 22 26 , , PUNCT iajs-2814 22 27 and and CCONJ iajs-2814 22 28 [ [ X iajs-2814 22 29 8 8 NUM iajs-2814 22 30 ] ] PUNCT iajs-2814 22 31 . . PUNCT iajs-2814 23 1 in in ADP iajs-2814 23 2 this this DET iajs-2814 23 3 work work NOUN iajs-2814 23 4 , , PUNCT iajs-2814 23 5 we we PRON iajs-2814 23 6 denote denote VERB iajs-2814 23 7 a a DET iajs-2814 23 8 universal universal ADJ iajs-2814 23 9 set set VERB iajs-2814 23 10 by by ADP iajs-2814 23 11 𝒰. 𝒰. PROPN iajs-2814 23 12 preliminaries preliminary NOUN iajs-2814 23 13 in in ADP iajs-2814 23 14 the the DET iajs-2814 23 15 following following NOUN iajs-2814 23 16 , , PUNCT iajs-2814 23 17 we we PRON iajs-2814 23 18 mention mention VERB iajs-2814 23 19 some some DET iajs-2814 23 20 basic basic ADJ iajs-2814 23 21 definitions definition NOUN iajs-2814 23 22 and and CCONJ iajs-2814 23 23 notations notation NOUN iajs-2814 23 24 in in ADP iajs-2814 23 25 measure measure NOUN iajs-2814 23 26 space space NOUN iajs-2814 23 27 that that PRON iajs-2814 23 28 will will AUX iajs-2814 23 29 be be AUX iajs-2814 23 30 used use VERB iajs-2814 23 31 in in ADP iajs-2814 23 32 this this DET iajs-2814 23 33 paper paper NOUN iajs-2814 23 34 . . PUNCT iajs-2814 24 1 definition definition NOUN iajs-2814 24 2 2.1 2.1 NUM iajs-2814 25 1 [ [ X iajs-2814 25 2 9 9 NUM iajs-2814 25 3 ] ] PUNCT iajs-2814 25 4 . . PUNCT iajs-2814 26 1 suppose suppose VERB iajs-2814 26 2 β„³ β„³ NOUN iajs-2814 26 3 is be AUX iajs-2814 26 4 a a DET iajs-2814 26 5 class class NOUN iajs-2814 26 6 of of ADP iajs-2814 26 7 subsets subset NOUN iajs-2814 26 8 of of ADP iajs-2814 26 9 𝒰 𝒰 PROPN iajs-2814 26 10 . . PUNCT iajs-2814 27 1 then then ADV iajs-2814 27 2 , , PUNCT iajs-2814 27 3 β„³ β„³ PROPN iajs-2814 27 4 is be AUX iajs-2814 27 5 π‘‘β„Žπ‘’ π‘‘β„Žπ‘’ ADJ iajs-2814 27 6 π’«βˆ— π’«βˆ— NOUN iajs-2814 27 7 – – PUNCT iajs-2814 27 8 field field NOUN iajs-2814 27 9 of of ADP iajs-2814 27 10 𝒰 𝒰 PROPN iajs-2814 27 11 if if SCONJ iajs-2814 27 12 : : PUNCT iajs-2814 27 13 1Ο† 1Ο† NUM iajs-2814 27 14 Ο΅ Ο΅ X iajs-2814 27 15 β„³. β„³. PROPN iajs-2814 27 16 2n 2n NUM iajs-2814 27 17 , , PUNCT iajs-2814 27 18 mΟ΅β„³ mΟ΅β„³ PROPN iajs-2814 27 19 ; ; PUNCT iajs-2814 27 20 then then ADV iajs-2814 27 21 , , PUNCT iajs-2814 27 22 nβ‹‚m nβ‹‚m PROPN iajs-2814 27 23 Ο΅ Ο΅ ADP iajs-2814 27 24 β„³. β„³. PROPN iajs-2814 27 25 3m2 3m2 ADV iajs-2814 27 26 , , PUNCT iajs-2814 27 27 … … PUNCT iajs-2814 27 28 Ο΅ Ο΅ X iajs-2814 27 29 β„³ β„³ PROPN iajs-2814 27 30 ; ; PUNCT iajs-2814 27 31 then then ADV iajs-2814 27 32 , , PUNCT iajs-2814 27 33 ⋃ ⋃ PROPN iajs-2814 27 34 mi mi X iajs-2814 27 35 ∞ ∞ PROPN iajs-2814 27 36 i=1 i=1 PROPN iajs-2814 28 1 Ο΅ Ο΅ ADP iajs-2814 28 2 β„³. β„³. PROPN iajs-2814 28 3 example example NOUN iajs-2814 28 4 2.2 2.2 NUM iajs-2814 28 5 [ [ X iajs-2814 28 6 9 9 NUM iajs-2814 28 7 ] ] PUNCT iajs-2814 28 8 . . PUNCT iajs-2814 29 1 let let VERB iajs-2814 29 2 𝒰 𝒰 PROPN iajs-2814 29 3 = = VERB iajs-2814 29 4 { { PUNCT iajs-2814 29 5 1,2,3,4 1,2,3,4 NUM iajs-2814 29 6 } } PUNCT iajs-2814 29 7 . . PUNCT iajs-2814 30 1 consider consider VERB iajs-2814 30 2 β„³ β„³ NOUN iajs-2814 30 3 = = NOUN iajs-2814 30 4 { { PUNCT iajs-2814 30 5 Ο†,{1},{1,2},{1,3},{1,2,3 Ο†,{1},{1,2},{1,3},{1,2,3 NOUN iajs-2814 30 6 } } PUNCT iajs-2814 30 7 } } PUNCT iajs-2814 30 8 . . PUNCT iajs-2814 31 1 then then ADV iajs-2814 31 2 β„³ β„³ PROPN iajs-2814 31 3 is be AUX iajs-2814 31 4 a a DET iajs-2814 31 5 π’«βˆ— π’«βˆ— NOUN iajs-2814 31 6 – – PUNCT iajs-2814 31 7 field field NOUN iajs-2814 31 8 of of ADP iajs-2814 31 9 𝒰. 𝒰. PROPN iajs-2814 31 10 definition definition NOUN iajs-2814 31 11 2.3 2.3 NUM iajs-2814 31 12 [ [ X iajs-2814 31 13 5 5 NUM iajs-2814 31 14 ] ] PUNCT iajs-2814 31 15 . . PUNCT iajs-2814 32 1 the the DET iajs-2814 32 2 family family NOUN iajs-2814 32 3 of of ADP iajs-2814 32 4 all all DET iajs-2814 32 5 subsets subset NOUN iajs-2814 32 6 of of ADP iajs-2814 32 7 𝒰 𝒰 PROPN iajs-2814 32 8 is be AUX iajs-2814 32 9 called call VERB iajs-2814 32 10 a a DET iajs-2814 32 11 power power NOUN iajs-2814 32 12 set set NOUN iajs-2814 32 13 and and CCONJ iajs-2814 32 14 denoted denote VERB iajs-2814 32 15 by by ADP iajs-2814 32 16 p(𝒰 p(𝒰 PROPN iajs-2814 32 17 ) ) PUNCT iajs-2814 32 18 , , PUNCT iajs-2814 32 19 in in ADP iajs-2814 32 20 symbols symbol NOUN iajs-2814 32 21 : : PUNCT iajs-2814 32 22 p(𝒰 p(𝒰 X iajs-2814 32 23 ) ) PUNCT iajs-2814 33 1 = = PRON iajs-2814 34 1 { { PUNCT iajs-2814 34 2 b b X iajs-2814 34 3 ∢ ∢ NOUN iajs-2814 34 4 b b PROPN iajs-2814 34 5 is be AUX iajs-2814 34 6 a a DET iajs-2814 34 7 subset subset NOUN iajs-2814 34 8 of of ADP iajs-2814 34 9 𝒰 𝒰 NOUN iajs-2814 34 10 } } PUNCT iajs-2814 34 11 . . PUNCT iajs-2814 35 1 proposition proposition NOUN iajs-2814 35 2 2.4 2.4 NUM iajs-2814 36 1 [ [ SYM iajs-2814 36 2 9 9 NUM iajs-2814 36 3 ] ] PUNCT iajs-2814 36 4 . . PUNCT iajs-2814 37 1 if if SCONJ iajs-2814 37 2 { { PUNCT iajs-2814 37 3 β„³i}iϡι β„³i}iϡι NOUN iajs-2814 37 4 is be AUX iajs-2814 37 5 a a DET iajs-2814 37 6 family family NOUN iajs-2814 37 7 of of ADP iajs-2814 37 8 π’«βˆ— π’«βˆ— NOUN iajs-2814 37 9 – – PUNCT iajs-2814 37 10 field field NOUN iajs-2814 37 11 of of ADP iajs-2814 37 12 𝒰 𝒰 PROPN iajs-2814 37 13 , , PUNCT iajs-2814 37 14 then then ADV iajs-2814 37 15 so so ADV iajs-2814 37 16 is be AUX iajs-2814 37 17 β‹‚ β‹‚ PROPN iajs-2814 37 18 β„³i β„³i PROPN iajs-2814 37 19 iϡι iϡι VERB iajs-2814 37 20 . . PUNCT iajs-2814 38 1 definition definition NOUN iajs-2814 38 2 2.5 2.5 NUM iajs-2814 39 1 [ [ X iajs-2814 39 2 9 9 NUM iajs-2814 39 3 ] ] PUNCT iajs-2814 39 4 . . PUNCT iajs-2814 40 1 let let VERB iajs-2814 40 2 ℐ ℐ PRON iajs-2814 40 3 βŠ† βŠ† NUM iajs-2814 40 4 p(𝒰 p(𝒰 NOUN iajs-2814 40 5 ) ) PUNCT iajs-2814 40 6 . . PUNCT iajs-2814 41 1 then then ADV iajs-2814 41 2 , , PUNCT iajs-2814 41 3 π’«βˆ—(ℐ π’«βˆ—(ℐ NUM iajs-2814 41 4 ) ) PUNCT iajs-2814 41 5 = = SYM iajs-2814 41 6 β‹‚{β„³i β‹‚{β„³i NOUN iajs-2814 41 7 : : PUNCT iajs-2814 42 1 β„³i β„³i PROPN iajs-2814 42 2 is be AUX iajs-2814 42 3 a a DET iajs-2814 42 4 π’«βˆ— π’«βˆ— NOUN iajs-2814 42 5 – – PUNCT iajs-2814 42 6 field field NOUN iajs-2814 42 7 of of ADP iajs-2814 42 8 𝒰 𝒰 PROPN iajs-2814 42 9 and and CCONJ iajs-2814 42 10 β„³i β„³i PROPN iajs-2814 42 11 βŠ‡ βŠ‡ NOUN iajs-2814 42 12 ℐ ℐ PROPN iajs-2814 42 13 , , PUNCT iajs-2814 42 14 βˆ€i βˆ€i NOUN iajs-2814 42 15 ∈ ∈ NOUN iajs-2814 42 16 ΞΉ ΞΉ AUX iajs-2814 42 17 } } PUNCT iajs-2814 42 18 is be AUX iajs-2814 42 19 called call VERB iajs-2814 42 20 the the DET iajs-2814 42 21 π’«βˆ— π’«βˆ— NOUN iajs-2814 42 22 – – PUNCT iajs-2814 42 23 field field NOUN iajs-2814 42 24 generated generate VERB iajs-2814 42 25 by by ADP iajs-2814 42 26 ℐ. ℐ. PROPN iajs-2814 42 27 proposition proposition NOUN iajs-2814 42 28 2.6 2.6 NUM iajs-2814 42 29 [ [ X iajs-2814 42 30 9 9 NUM iajs-2814 42 31 ] ] PUNCT iajs-2814 42 32 . . PUNCT iajs-2814 43 1 if if SCONJ iajs-2814 43 2 ℐ ℐ PRON iajs-2814 43 3 βŠ† βŠ† NUM iajs-2814 43 4 p(𝒰 p(𝒰 NOUN iajs-2814 43 5 ) ) PUNCT iajs-2814 43 6 , , PUNCT iajs-2814 43 7 then then ADV iajs-2814 43 8 π’«βˆ—(ℐ π’«βˆ—(ℐ NUM iajs-2814 43 9 ) ) PUNCT iajs-2814 43 10 is be AUX iajs-2814 43 11 the the DET iajs-2814 43 12 smallest small ADJ iajs-2814 43 13 π’«βˆ— π’«βˆ— NOUN iajs-2814 43 14 – – PUNCT iajs-2814 43 15 field field NOUN iajs-2814 43 16 of of ADP iajs-2814 43 17 𝒰 𝒰 PROPN iajs-2814 43 18 that that PRON iajs-2814 43 19 contains contain VERB iajs-2814 43 20 ℐ. ℐ. NOUN iajs-2814 43 21 proposition proposition NOUN iajs-2814 43 22 2.7 2.7 NUM iajs-2814 43 23 [ [ X iajs-2814 43 24 5 5 NUM iajs-2814 43 25 ] ] PUNCT iajs-2814 43 26 . . PUNCT iajs-2814 44 1 if if SCONJ iajs-2814 44 2 β„³ β„³ PROPN iajs-2814 44 3 is be AUX iajs-2814 44 4 Οƒ Οƒ PROPN iajs-2814 44 5 – – PUNCT iajs-2814 44 6 field field NOUN iajs-2814 44 7 , , PUNCT iajs-2814 44 8 then then ADV iajs-2814 44 9 β„³ β„³ PROPN iajs-2814 44 10 is be AUX iajs-2814 44 11 a a DET iajs-2814 44 12 Οƒ Οƒ PROPN iajs-2814 44 13 – – PUNCT iajs-2814 44 14 ring ring NOUN iajs-2814 44 15 . . PUNCT iajs-2814 45 1 proposition proposition NOUN iajs-2814 45 2 2.8 2.8 NUM iajs-2814 46 1 [ [ X iajs-2814 46 2 9 9 NUM iajs-2814 46 3 ] ] PUNCT iajs-2814 46 4 . . PUNCT iajs-2814 47 1 every every DET iajs-2814 47 2 Οƒ Οƒ PROPN iajs-2814 47 3 – – PUNCT iajs-2814 47 4 field field NOUN iajs-2814 47 5 is be AUX iajs-2814 47 6 π’«βˆ— π’«βˆ— NOUN iajs-2814 47 7 – – PUNCT iajs-2814 47 8 field field NOUN iajs-2814 47 9 . . PUNCT iajs-2814 48 1 ihjpas ihjpas PROPN iajs-2814 48 2 . . PUNCT iajs-2814 49 1 53 53 NUM iajs-2814 49 2 ( ( PUNCT iajs-2814 49 3 3)2022 3)2022 NOUN iajs-2814 49 4 157 157 NUM iajs-2814 49 5 proposition proposition NOUN iajs-2814 49 6 2.9 2.9 NUM iajs-2814 50 1 [ [ X iajs-2814 50 2 9 9 NUM iajs-2814 50 3 ] ] PUNCT iajs-2814 50 4 . . PUNCT iajs-2814 51 1 every every DET iajs-2814 51 2 Οƒ Οƒ PROPN iajs-2814 51 3 – – PUNCT iajs-2814 51 4 ring ring NOUN iajs-2814 51 5 is be AUX iajs-2814 51 6 π’«βˆ— π’«βˆ— NOUN iajs-2814 51 7 – – PUNCT iajs-2814 51 8 field field NOUN iajs-2814 51 9 . . PUNCT iajs-2814 52 1 2 2 X iajs-2814 52 2 . . X iajs-2814 52 3 the the DET iajs-2814 52 4 main main ADJ iajs-2814 52 5 results result NOUN iajs-2814 52 6 in in ADP iajs-2814 52 7 this this DET iajs-2814 52 8 section section NOUN iajs-2814 52 9 , , PUNCT iajs-2814 52 10 the the DET iajs-2814 52 11 basic basic ADJ iajs-2814 52 12 definitions definition NOUN iajs-2814 52 13 and and CCONJ iajs-2814 52 14 facts fact NOUN iajs-2814 52 15 related relate VERB iajs-2814 52 16 to to ADP iajs-2814 52 17 this this DET iajs-2814 52 18 work work NOUN iajs-2814 52 19 are be AUX iajs-2814 52 20 recalled recall VERB iajs-2814 52 21 , , PUNCT iajs-2814 52 22 starting start VERB iajs-2814 52 23 with with ADP iajs-2814 52 24 the the DET iajs-2814 52 25 following follow VERB iajs-2814 52 26 definition definition NOUN iajs-2814 52 27 : : PUNCT iajs-2814 52 28 definition definition NOUN iajs-2814 52 29 3.1 3.1 NUM iajs-2814 52 30 suppose suppose VERB iajs-2814 52 31 β„³ β„³ PROPN iajs-2814 52 32 is be AUX iajs-2814 52 33 a a DET iajs-2814 52 34 π’«βˆ— π’«βˆ— NOUN iajs-2814 52 35 – – PUNCT iajs-2814 52 36 field field NOUN iajs-2814 52 37 of of ADP iajs-2814 52 38 𝒰 𝒰 PROPN iajs-2814 52 39 and and CCONJ iajs-2814 52 40 Ο† Ο† NOUN iajs-2814 52 41 β‰  β‰  PROPN iajs-2814 52 42 ℬ ℬ PROPN iajs-2814 52 43 βŠ† βŠ† NUM iajs-2814 52 44 𝒰 𝒰 PROPN iajs-2814 52 45 , , PUNCT iajs-2814 52 46 then then ADV iajs-2814 52 47 a a DET iajs-2814 52 48 restriction restriction NOUN iajs-2814 52 49 of of ADP iajs-2814 52 50 β„³ β„³ PROPN iajs-2814 52 51 over over ADP iajs-2814 52 52 ℬ ℬ NOUN iajs-2814 52 53 is be AUX iajs-2814 52 54 defined define VERB iajs-2814 52 55 as as ADP iajs-2814 52 56 : : PUNCT iajs-2814 52 57 β„³|ℬ β„³|ℬ NOUN iajs-2814 52 58 = = PUNCT iajs-2814 52 59 { { PUNCT iajs-2814 52 60 n n CCONJ iajs-2814 52 61 : : PUNCT iajs-2814 52 62 n n PROPN iajs-2814 52 63 = = NUM iajs-2814 52 64 mβ‹‚ mβ‹‚ NOUN iajs-2814 52 65 ℬ ℬ NOUN iajs-2814 52 66 , , PUNCT iajs-2814 52 67 for for ADP iajs-2814 52 68 some some DET iajs-2814 52 69 m m NOUN iajs-2814 52 70 Ο΅ Ο΅ X iajs-2814 52 71 β„³ β„³ NOUN iajs-2814 52 72 } } PUNCT iajs-2814 52 73 . . PUNCT iajs-2814 53 1 proposition proposition NOUN iajs-2814 53 2 3.2 3.2 NUM iajs-2814 53 3 suppose suppose VERB iajs-2814 53 4 β„³ β„³ PROPN iajs-2814 53 5 is be AUX iajs-2814 53 6 π’«βˆ— π’«βˆ— NOUN iajs-2814 53 7 – – PUNCT iajs-2814 53 8 field field NOUN iajs-2814 53 9 of of ADP iajs-2814 53 10 𝒰 𝒰 PROPN iajs-2814 53 11 and and CCONJ iajs-2814 53 12 Ο† Ο† NOUN iajs-2814 53 13 β‰  β‰  PROPN iajs-2814 53 14 ℬ ℬ PROPN iajs-2814 53 15 βŠ† βŠ† NUM iajs-2814 53 16 𝒰 𝒰 NOUN iajs-2814 53 17 , , PUNCT iajs-2814 53 18 then then ADV iajs-2814 53 19 β„³|ℬ β„³|ℬ NOUN iajs-2814 53 20 is be AUX iajs-2814 53 21 π’«βˆ— π’«βˆ— NOUN iajs-2814 53 22 – – PUNCT iajs-2814 53 23 field field NOUN iajs-2814 53 24 on on ADP iajs-2814 53 25 ℬ. ℬ. NOUN iajs-2814 53 26 proof proof NOUN iajs-2814 53 27 . . PUNCT iajs-2814 54 1 since since SCONJ iajs-2814 54 2 φϡ φϡ PROPN iajs-2814 54 3 β„³ β„³ PROPN iajs-2814 54 4 and and CCONJ iajs-2814 54 5 Ο† Ο† NOUN iajs-2814 54 6 = = PUNCT iajs-2814 54 7 φ⋂ℬ φ⋂ℬ NOUN iajs-2814 54 8 , , PUNCT iajs-2814 54 9 then then ADV iajs-2814 54 10 φϡ φϡ INTJ iajs-2814 54 11 β„³|ℬ. β„³|ℬ. VERB iajs-2814 54 12 let let VERB iajs-2814 54 13 n1 n1 NOUN iajs-2814 54 14 , , PUNCT iajs-2814 54 15 n2Ο΅ n2Ο΅ ADV iajs-2814 54 16 β„³|ℬ β„³|ℬ NOUN iajs-2814 54 17 , , PUNCT iajs-2814 54 18 then then ADV iajs-2814 54 19 there there PRON iajs-2814 54 20 is be VERB iajs-2814 54 21 m1 m1 NOUN iajs-2814 54 22 , , PUNCT iajs-2814 54 23 m2Ο΅ m2Ο΅ NOUN iajs-2814 54 24 β„³ β„³ PROPN iajs-2814 54 25 such such ADJ iajs-2814 55 1 that that SCONJ iajs-2814 55 2 ni ni NOUN iajs-2814 55 3 = = NOUN iajs-2814 55 4 mi⋂ℬ mi⋂ℬ NOUN iajs-2814 55 5 where where SCONJ iajs-2814 55 6 i=1,2 i=1,2 ADJ iajs-2814 55 7 which which PRON iajs-2814 55 8 implies imply VERB iajs-2814 55 9 that that SCONJ iajs-2814 55 10 n1β‹‚n2= n1β‹‚n2= PROPN iajs-2814 55 11 ( ( PUNCT iajs-2814 55 12 m1⋂ℬ m1⋂ℬ PROPN iajs-2814 55 13 ) ) PUNCT iajs-2814 55 14 β‹‚(m2⋂ℬ β‹‚(m2⋂ℬ NOUN iajs-2814 55 15 ) ) PUNCT iajs-2814 55 16 = = PRON iajs-2814 55 17 ( ( PUNCT iajs-2814 55 18 m1β‹‚m2)⋂ℬ. m1β‹‚m2)⋂ℬ. NOUN iajs-2814 55 19 since since SCONJ iajs-2814 55 20 β„³ β„³ PROPN iajs-2814 55 21 is be AUX iajs-2814 55 22 a a DET iajs-2814 55 23 π’«βˆ— π’«βˆ— NOUN iajs-2814 55 24 – – PUNCT iajs-2814 55 25 field field NOUN iajs-2814 55 26 of of ADP iajs-2814 55 27 𝒰 𝒰 PROPN iajs-2814 55 28 , , PUNCT iajs-2814 55 29 then then ADV iajs-2814 55 30 , , PUNCT iajs-2814 55 31 m1β‹‚m2Ο΅ m1β‹‚m2Ο΅ ADJ iajs-2814 55 32 β„³. β„³. NOUN iajs-2814 55 33 thus thus ADV iajs-2814 55 34 n1β‹‚n2Ο΅ n1β‹‚n2Ο΅ NOUN iajs-2814 55 35 β„³|ℬ β„³|ℬ NOUN iajs-2814 55 36 let let VERB iajs-2814 55 37 n1 n1 NOUN iajs-2814 55 38 , , PUNCT iajs-2814 55 39 n2 n2 NOUN iajs-2814 55 40 , , PUNCT iajs-2814 55 41 … … PUNCT iajs-2814 56 1 Ο΅ Ο΅ SYM iajs-2814 56 2 β„³|ℬ β„³|ℬ NOUN iajs-2814 56 3 , , PUNCT iajs-2814 56 4 then then ADV iajs-2814 56 5 there there PRON iajs-2814 56 6 is be VERB iajs-2814 56 7 m1 m1 NOUN iajs-2814 56 8 , , PUNCT iajs-2814 56 9 m2 m2 PROPN iajs-2814 56 10 , , PUNCT iajs-2814 56 11 … … PUNCT iajs-2814 56 12 Ο΅ Ο΅ X iajs-2814 56 13 β„³ β„³ PROPN iajs-2814 56 14 such such ADJ iajs-2814 56 15 that that SCONJ iajs-2814 56 16 ni ni NOUN iajs-2814 56 17 = = NOUN iajs-2814 56 18 mi⋂ℬ mi⋂ℬ NOUN iajs-2814 56 19 where where SCONJ iajs-2814 56 20 i=1 i=1 PROPN iajs-2814 56 21 , , PUNCT iajs-2814 56 22 2 2 NUM iajs-2814 56 23 … … NUM iajs-2814 56 24 which which PRON iajs-2814 56 25 implies imply VERB iajs-2814 56 26 that that SCONJ iajs-2814 56 27 ⋃ ⋃ PROPN iajs-2814 56 28 , , PUNCT iajs-2814 56 29 ni ni NOUN iajs-2814 56 30 ∞ ∞ PROPN iajs-2814 56 31 i=1 i=1 PROPN iajs-2814 57 1 = = X iajs-2814 57 2 ⋃ ⋃ PROPN iajs-2814 57 3 ( ( PUNCT iajs-2814 57 4 mi mi NOUN iajs-2814 57 5 ∞ ∞ PROPN iajs-2814 57 6 i=1 i=1 PROPN iajs-2814 57 7 β‹‚ β‹‚ PROPN iajs-2814 57 8 ℬ)= ℬ)= PROPN iajs-2814 57 9 ( ( PUNCT iajs-2814 57 10 ⋃ ⋃ PROPN iajs-2814 57 11 mi mi NOUN iajs-2814 57 12 ∞ ∞ PROPN iajs-2814 57 13 i=1 i=1 PROPN iajs-2814 57 14 ) ) PUNCT iajs-2814 58 1 β‹‚ β‹‚ PROPN iajs-2814 58 2 ℬ ℬ X iajs-2814 58 3 .. .. PUNCT iajs-2814 58 4 since since SCONJ iajs-2814 58 5 β„³ β„³ PROPN iajs-2814 58 6 is be AUX iajs-2814 58 7 a a DET iajs-2814 58 8 π’«βˆ— π’«βˆ— NOUN iajs-2814 58 9 – – PUNCT iajs-2814 58 10 field field NOUN iajs-2814 58 11 of of ADP iajs-2814 58 12 a a DET iajs-2814 58 13 set set ADJ iajs-2814 58 14 𝒰 𝒰 NOUN iajs-2814 58 15 , , PUNCT iajs-2814 58 16 then then ADV iajs-2814 58 17 ⋃ ⋃ PUNCT iajs-2814 58 18 mi mi PROPN iajs-2814 58 19 ∞ ∞ PROPN iajs-2814 58 20 i=1 i=1 PROPN iajs-2814 58 21 Ο΅ Ο΅ X iajs-2814 58 22 β„³ β„³ PROPN iajs-2814 58 23 and and CCONJ iajs-2814 58 24 hence hence ADV iajs-2814 58 25 ⋃ ⋃ VERB iajs-2814 58 26 ni ni PROPN iajs-2814 58 27 ∞ ∞ PROPN iajs-2814 58 28 i=1 i=1 PROPN iajs-2814 59 1 Ο΅ Ο΅ X iajs-2814 59 2 β„³|ℬ. β„³|ℬ. PROPN iajs-2814 59 3 thus thus ADV iajs-2814 59 4 , , PUNCT iajs-2814 59 5 β„³|ℬis β„³|ℬis ADJ iajs-2814 59 6 a a DET iajs-2814 59 7 π’«βˆ— π’«βˆ— NOUN iajs-2814 59 8 – – PUNCT iajs-2814 59 9 field field NOUN iajs-2814 59 10 on on ADP iajs-2814 59 11 ℬ. ℬ. NOUN iajs-2814 59 12 proposition proposition NOUN iajs-2814 59 13 3.3 3.3 NUM iajs-2814 59 14 if if SCONJ iajs-2814 59 15 β„³ β„³ PROPN iajs-2814 59 16 is be AUX iajs-2814 59 17 π’«βˆ— π’«βˆ— NOUN iajs-2814 59 18 – – PUNCT iajs-2814 59 19 field field NOUN iajs-2814 59 20 of of ADP iajs-2814 59 21 𝒰 𝒰 PROPN iajs-2814 59 22 and and CCONJ iajs-2814 59 23 c c NOUN iajs-2814 59 24 βŠ† βŠ† NUM iajs-2814 59 25 ℬ ℬ PROPN iajs-2814 59 26 βŠ† βŠ† NUM iajs-2814 59 27 𝒰 𝒰 NOUN iajs-2814 59 28 such such ADJ iajs-2814 59 29 that that SCONJ iajs-2814 59 30 cΟ΅β„³ cΟ΅β„³ PROPN iajs-2814 59 31 , , PUNCT iajs-2814 59 32 then then ADV iajs-2814 59 33 cΟ΅β„³|ℬ. cΟ΅β„³|ℬ. NOUN iajs-2814 59 34 proof proof NOUN iajs-2814 59 35 . . PUNCT iajs-2814 60 1 clearly clearly ADV iajs-2814 60 2 . . PUNCT iajs-2814 61 1 the the DET iajs-2814 61 2 following follow VERB iajs-2814 61 3 examples example NOUN iajs-2814 61 4 explain explain VERB iajs-2814 61 5 that that SCONJ iajs-2814 61 6 if if SCONJ iajs-2814 61 7 β„³ β„³ PROPN iajs-2814 61 8 is be AUX iajs-2814 61 9 a a DET iajs-2814 61 10 π’«βˆ— π’«βˆ— NOUN iajs-2814 61 11 – – PUNCT iajs-2814 61 12 field field NOUN iajs-2814 61 13 of of ADP iajs-2814 61 14 a a DET iajs-2814 61 15 set set ADJ iajs-2814 61 16 𝒰 𝒰 NOUN iajs-2814 61 17 , , PUNCT iajs-2814 61 18 then then ADV iajs-2814 61 19 it it PRON iajs-2814 61 20 is be AUX iajs-2814 61 21 not not PART iajs-2814 61 22 necessarily necessarily ADV iajs-2814 61 23 that that PRON iajs-2814 61 24 : : PUNCT iajs-2814 61 25 1β„³|ℬ 1β„³|ℬ NUM iajs-2814 61 26 βŠ† βŠ† NUM iajs-2814 61 27 β„³. β„³. PROPN iajs-2814 61 28 2β„³ 2β„³ PROPN iajs-2814 61 29 βŠ† βŠ† NUM iajs-2814 61 30 β„³|ℬ β„³|ℬ PROPN iajs-2814 61 31 example example NOUN iajs-2814 61 32 3.4 3.4 NUM iajs-2814 61 33 let let VERB iajs-2814 61 34 𝒰 𝒰 PROPN iajs-2814 61 35 = = NOUN iajs-2814 61 36 { { PUNCT iajs-2814 61 37 1,2,3,4}and 1,2,3,4}and NUM iajs-2814 61 38 β„³ β„³ NOUN iajs-2814 61 39 = = NOUN iajs-2814 61 40 { { PUNCT iajs-2814 61 41 Ο†,{1,3},{1,2,3},{1,3,4},𝒰 Ο†,{1,3},{1,2,3},{1,3,4},𝒰 NOUN iajs-2814 61 42 } } PUNCT iajs-2814 61 43 . . PUNCT iajs-2814 62 1 then then ADV iajs-2814 62 2 , , PUNCT iajs-2814 62 3 β„³ β„³ PROPN iajs-2814 62 4 is be AUX iajs-2814 62 5 a a DET iajs-2814 62 6 π’«βˆ— π’«βˆ— NOUN iajs-2814 62 7 – – PUNCT iajs-2814 62 8 field field NOUN iajs-2814 62 9 of of ADP iajs-2814 62 10 𝒰. 𝒰. PROPN iajs-2814 62 11 if if SCONJ iajs-2814 62 12 ℬ ℬ NOUN iajs-2814 62 13 = = NOUN iajs-2814 62 14 { { PUNCT iajs-2814 62 15 2,3,4 2,3,4 NUM iajs-2814 62 16 } } PUNCT iajs-2814 62 17 , , PUNCT iajs-2814 62 18 then then ADV iajs-2814 62 19 β„³|ℬ β„³|ℬ NOUN iajs-2814 63 1 = = NOUN iajs-2814 63 2 { { PUNCT iajs-2814 63 3 Ο†,{3},{2,3},{3,4 Ο†,{3},{2,3},{3,4 ADJ iajs-2814 63 4 } } PUNCT iajs-2814 63 5 , , PUNCT iajs-2814 63 6 ℬ ℬ NOUN iajs-2814 63 7 } } PUNCT iajs-2814 63 8 . . PUNCT iajs-2814 64 1 it it PRON iajs-2814 64 2 is be AUX iajs-2814 64 3 clear clear ADJ iajs-2814 64 4 that that SCONJ iajs-2814 64 5 β„³|ℬ β„³|ℬ NOUN iajs-2814 64 6 ⊈ ⊈ NUM iajs-2814 64 7 β„³ β„³ NOUN iajs-2814 64 8 , , PUNCT iajs-2814 64 9 since since SCONJ iajs-2814 64 10 { { PUNCT iajs-2814 64 11 3}∈ 3}∈ NUM iajs-2814 64 12 β„³|ℬ β„³|ℬ NOUN iajs-2814 64 13 but but CCONJ iajs-2814 64 14 { { PUNCT iajs-2814 64 15 3}βˆ‰ 3}βˆ‰ NUM iajs-2814 64 16 β„³. β„³. PROPN iajs-2814 64 17 example example NOUN iajs-2814 64 18 3.5 3.5 NUM iajs-2814 64 19 let let VERB iajs-2814 64 20 𝒰 𝒰 PROPN iajs-2814 64 21 = = NOUN iajs-2814 64 22 { { PUNCT iajs-2814 64 23 1,2,3,4}and 1,2,3,4}and NUM iajs-2814 64 24 β„³ β„³ NOUN iajs-2814 64 25 = = NOUN iajs-2814 64 26 { { PUNCT iajs-2814 64 27 Ο†,{1,2},{1,2,3},{1,2,4},𝒰 Ο†,{1,2},{1,2,3},{1,2,4},𝒰 PRON iajs-2814 64 28 } } PUNCT iajs-2814 64 29 . . PUNCT iajs-2814 65 1 then then ADV iajs-2814 65 2 , , PUNCT iajs-2814 65 3 β„³ β„³ PROPN iajs-2814 65 4 is be AUX iajs-2814 65 5 a a DET iajs-2814 65 6 π’«βˆ— π’«βˆ— NOUN iajs-2814 65 7 – – PUNCT iajs-2814 65 8 field field NOUN iajs-2814 65 9 of of ADP iajs-2814 65 10 𝒰. 𝒰. PROPN iajs-2814 65 11 if if SCONJ iajs-2814 65 12 ℬ ℬ NOUN iajs-2814 65 13 = = NOUN iajs-2814 65 14 { { PUNCT iajs-2814 65 15 2,3,4 2,3,4 NUM iajs-2814 65 16 } } PUNCT iajs-2814 65 17 , , PUNCT iajs-2814 65 18 then then ADV iajs-2814 65 19 β„³|ℬ β„³|ℬ NOUN iajs-2814 65 20 = = NOUN iajs-2814 65 21 { { PUNCT iajs-2814 65 22 Ο†,{2},{2,3},{2,4 Ο†,{2},{2,3},{2,4 NOUN iajs-2814 65 23 } } PUNCT iajs-2814 65 24 , , PUNCT iajs-2814 65 25 ℬ ℬ NOUN iajs-2814 65 26 } } PUNCT iajs-2814 65 27 . . PUNCT iajs-2814 66 1 it it PRON iajs-2814 66 2 is be AUX iajs-2814 66 3 clear clear ADJ iajs-2814 66 4 that that SCONJ iajs-2814 66 5 β„³ β„³ PROPN iajs-2814 66 6 ⊈ ⊈ PROPN iajs-2814 66 7 β„³|ℬ β„³|ℬ NOUN iajs-2814 66 8 , , PUNCT iajs-2814 66 9 since since SCONJ iajs-2814 66 10 { { PUNCT iajs-2814 66 11 1 1 NUM iajs-2814 66 12 , , PUNCT iajs-2814 66 13 2}∈ 2}∈ PROPN iajs-2814 66 14 β„³ β„³ PROPN iajs-2814 66 15 but but CCONJ iajs-2814 66 16 { { PUNCT iajs-2814 66 17 1,2}βˆ‰ 1,2}βˆ‰ NUM iajs-2814 66 18 β„³|ℬ. β„³|ℬ. NOUN iajs-2814 66 19 ihjpas ihjpas PROPN iajs-2814 66 20 . . PUNCT iajs-2814 67 1 53 53 NUM iajs-2814 67 2 ( ( PUNCT iajs-2814 67 3 3)2022 3)2022 NOUN iajs-2814 67 4 158 158 NUM iajs-2814 67 5 proposition proposition NOUN iajs-2814 67 6 3.6 3.6 NUM iajs-2814 67 7 if if SCONJ iajs-2814 67 8 β„³ β„³ PROPN iajs-2814 67 9 is be AUX iajs-2814 67 10 π’«βˆ— π’«βˆ— NOUN iajs-2814 67 11 – – PUNCT iajs-2814 67 12 field field NOUN iajs-2814 67 13 on on ADP iajs-2814 67 14 𝒰 𝒰 PROPN iajs-2814 67 15 and and CCONJ iajs-2814 67 16 Ο† Ο† NOUN iajs-2814 67 17 β‰  β‰  PROPN iajs-2814 67 18 ℬ ℬ NOUN iajs-2814 67 19 βŠ† βŠ† NUM iajs-2814 67 20 𝒰 𝒰 NOUN iajs-2814 67 21 such such ADJ iajs-2814 67 22 that that PRON iajs-2814 67 23 ℬϡ ℬϡ ADP iajs-2814 67 24 β„³. β„³. PROPN iajs-2814 67 25 then then ADV iajs-2814 67 26 β„³|ℬ= β„³|ℬ= PROPN iajs-2814 67 27 { { PUNCT iajs-2814 67 28 c c NOUN iajs-2814 67 29 βŠ† βŠ† NUM iajs-2814 67 30 ℬ ℬ NOUN iajs-2814 67 31 : : PUNCT iajs-2814 67 32 cΟ΅ cΟ΅ NOUN iajs-2814 67 33 β„³ β„³ PROPN iajs-2814 67 34 } } PUNCT iajs-2814 67 35 . . PUNCT iajs-2814 68 1 proof proof NOUN iajs-2814 68 2 . . PUNCT iajs-2814 69 1 assume assume VERB iajs-2814 69 2 that that SCONJ iajs-2814 69 3 nΟ΅ nΟ΅ PRON iajs-2814 69 4 β„³|ℬ β„³|ℬ NOUN iajs-2814 69 5 , , PUNCT iajs-2814 69 6 then then ADV iajs-2814 69 7 n n CCONJ iajs-2814 69 8 = = NOUN iajs-2814 69 9 mβ‹‚ mβ‹‚ NOUN iajs-2814 69 10 ℬ ℬ NOUN iajs-2814 69 11 , , PUNCT iajs-2814 69 12 for for ADP iajs-2814 69 13 some some DET iajs-2814 69 14 mΟ΅ mΟ΅ NOUN iajs-2814 69 15 β„³ β„³ NOUN iajs-2814 69 16 and and CCONJ iajs-2814 69 17 thus thus ADV iajs-2814 69 18 nΟ΅ nΟ΅ PRON iajs-2814 69 19 β„³. β„³. PROPN iajs-2814 69 20 hence hence ADV iajs-2814 69 21 , , PUNCT iajs-2814 69 22 nΟ΅{c nΟ΅{c NOUN iajs-2814 69 23 βŠ† βŠ† NUM iajs-2814 69 24 ℬ ℬ NOUN iajs-2814 69 25 : : PUNCT iajs-2814 69 26 cΟ΅ cΟ΅ NOUN iajs-2814 69 27 β„³ β„³ PROPN iajs-2814 69 28 } } PUNCT iajs-2814 69 29 . . PUNCT iajs-2814 70 1 therefore therefore ADV iajs-2814 70 2 , , PUNCT iajs-2814 70 3 β„³|ℬ β„³|ℬ NOUN iajs-2814 70 4 βŠ† βŠ† NUM iajs-2814 70 5 { { PUNCT iajs-2814 70 6 c c NOUN iajs-2814 70 7 βŠ† βŠ† NUM iajs-2814 70 8 ℬ ℬ NOUN iajs-2814 70 9 : : PUNCT iajs-2814 70 10 cΟ΅ cΟ΅ NOUN iajs-2814 70 11 β„³ β„³ PROPN iajs-2814 70 12 } } PUNCT iajs-2814 70 13 . . PUNCT iajs-2814 71 1 let let VERB iajs-2814 71 2 dΟ΅ dΟ΅ VERB iajs-2814 71 3 { { PUNCT iajs-2814 71 4 c c NOUN iajs-2814 71 5 βŠ† βŠ† NUM iajs-2814 71 6 ℬ ℬ NOUN iajs-2814 71 7 : : PUNCT iajs-2814 71 8 cΟ΅ cΟ΅ NOUN iajs-2814 71 9 β„³ β„³ PROPN iajs-2814 71 10 } } PUNCT iajs-2814 71 11 . . PUNCT iajs-2814 72 1 then then ADV iajs-2814 72 2 d d X iajs-2814 72 3 βŠ† βŠ† NUM iajs-2814 72 4 ℬ ℬ NOUN iajs-2814 72 5 and and CCONJ iajs-2814 72 6 d d X iajs-2814 72 7 Ο΅ Ο΅ X iajs-2814 72 8 β„³ β„³ PROPN iajs-2814 72 9 , , PUNCT iajs-2814 72 10 hence hence ADV iajs-2814 72 11 d d NOUN iajs-2814 72 12 = = PUNCT iajs-2814 72 13 dβ‹‚ dβ‹‚ NOUN iajs-2814 72 14 ℬ ℬ NOUN iajs-2814 72 15 , , PUNCT iajs-2814 72 16 but but CCONJ iajs-2814 72 17 dΟ΅ dΟ΅ PROPN iajs-2814 72 18 β„³ β„³ PROPN iajs-2814 72 19 , , PUNCT iajs-2814 72 20 then then ADV iajs-2814 72 21 dΟ΅ dΟ΅ PROPN iajs-2814 72 22 β„³|ℬ. β„³|ℬ. VERB iajs-2814 72 23 so so ADV iajs-2814 72 24 , , PUNCT iajs-2814 72 25 we we PRON iajs-2814 72 26 get get VERB iajs-2814 72 27 { { PUNCT iajs-2814 72 28 c c NOUN iajs-2814 72 29 βŠ† βŠ† NUM iajs-2814 72 30 ℬ ℬ NOUN iajs-2814 72 31 : : PUNCT iajs-2814 72 32 c c NOUN iajs-2814 72 33 Ο΅ Ο΅ X iajs-2814 72 34 β„³}βŠ† β„³}βŠ† PROPN iajs-2814 72 35 β„³|ℬ. β„³|ℬ. PROPN iajs-2814 72 36 consequentially consequentially ADV iajs-2814 72 37 , , PUNCT iajs-2814 72 38 β„³|ℬ={c β„³|ℬ={c VERB iajs-2814 72 39 βŠ† βŠ† NUM iajs-2814 72 40 ℬ ℬ NOUN iajs-2814 72 41 : : PUNCT iajs-2814 72 42 cΟ΅ cΟ΅ NOUN iajs-2814 72 43 β„³ β„³ PROPN iajs-2814 72 44 } } PUNCT iajs-2814 72 45 . . PUNCT iajs-2814 73 1 corollary corollary ADJ iajs-2814 73 2 3.7 3.7 NUM iajs-2814 73 3 if if SCONJ iajs-2814 73 4 β„³ β„³ PROPN iajs-2814 73 5 is be AUX iajs-2814 73 6 π’«βˆ— π’«βˆ— NOUN iajs-2814 73 7 – – PUNCT iajs-2814 73 8 field field NOUN iajs-2814 73 9 on on ADP iajs-2814 73 10 𝒰 𝒰 PROPN iajs-2814 73 11 and and CCONJ iajs-2814 73 12 Ο† Ο† NOUN iajs-2814 73 13 β‰  β‰  PROPN iajs-2814 73 14 ℬ ℬ NOUN iajs-2814 73 15 βŠ† βŠ† NUM iajs-2814 73 16 𝒰 𝒰 NOUN iajs-2814 73 17 such such ADJ iajs-2814 73 18 that that PRON iajs-2814 73 19 ℬϡ ℬϡ ADP iajs-2814 73 20 β„³. β„³. PROPN iajs-2814 73 21 then then ADV iajs-2814 73 22 , , PUNCT iajs-2814 73 23 β„³|ℬ β„³|ℬ NOUN iajs-2814 73 24 βŠ† βŠ† NUM iajs-2814 73 25 β„³. β„³. PROPN iajs-2814 73 26 proof proof NOUN iajs-2814 73 27 . . PUNCT iajs-2814 74 1 the the DET iajs-2814 74 2 proof proof NOUN iajs-2814 74 3 follows follow VERB iajs-2814 74 4 proposition proposition NOUN iajs-2814 74 5 3.6 3.6 NUM iajs-2814 74 6 . . PUNCT iajs-2814 75 1 definition definition NOUN iajs-2814 75 2 3.8 3.8 NUM iajs-2814 75 3 if if SCONJ iajs-2814 75 4 𝒰 𝒰 PROPN iajs-2814 75 5 is be AUX iajs-2814 75 6 a a DET iajs-2814 75 7 universal universal ADJ iajs-2814 75 8 set set NOUN iajs-2814 75 9 and and CCONJ iajs-2814 75 10 ℐ ℐ PRON iajs-2814 75 11 βŠ† βŠ† NUM iajs-2814 75 12 p(𝒰 p(𝒰 X iajs-2814 75 13 ) ) PUNCT iajs-2814 75 14 and and CCONJ iajs-2814 75 15 Ο† Ο† NOUN iajs-2814 75 16 β‰  β‰  PROPN iajs-2814 75 17 ℬ ℬ PROPN iajs-2814 75 18 βŠ† βŠ† NUM iajs-2814 75 19 𝒰 𝒰 PROPN iajs-2814 75 20 , , PUNCT iajs-2814 75 21 then then ADV iajs-2814 75 22 a a DET iajs-2814 75 23 restriction restriction NOUN iajs-2814 75 24 of of ADP iajs-2814 75 25 ℐ ℐ PRON iajs-2814 75 26 on on ADP iajs-2814 75 27 ℬ ℬ PROPN iajs-2814 75 28 is be AUX iajs-2814 75 29 defined define VERB iajs-2814 75 30 as as ADP iajs-2814 75 31 : : PUNCT iajs-2814 75 32 ℐ|ℬ ℐ|ℬ X iajs-2814 75 33 = = X iajs-2814 75 34 { { PUNCT iajs-2814 75 35 n n CCONJ iajs-2814 75 36 : : PUNCT iajs-2814 75 37 n n CCONJ iajs-2814 75 38 = = SYM iajs-2814 75 39 mβ‹‚ mβ‹‚ NOUN iajs-2814 75 40 ℬ ℬ NOUN iajs-2814 75 41 , , PUNCT iajs-2814 75 42 for for ADP iajs-2814 75 43 some some DET iajs-2814 75 44 mΟ΅ mΟ΅ NOUN iajs-2814 75 45 ℐ ℐ NOUN iajs-2814 75 46 } } PUNCT iajs-2814 75 47 . . PUNCT iajs-2814 76 1 proposition proposition NOUN iajs-2814 76 2 3.9 3.9 NUM iajs-2814 76 3 if if SCONJ iajs-2814 76 4 ℐ ℐ PROPN iajs-2814 76 5 βŠ† βŠ† NUM iajs-2814 76 6 p(𝒰 p(𝒰 X iajs-2814 76 7 ) ) PUNCT iajs-2814 76 8 and and CCONJ iajs-2814 76 9 Ο† Ο† NOUN iajs-2814 76 10 β‰  β‰  PROPN iajs-2814 76 11 ℬ ℬ NOUN iajs-2814 76 12 βŠ† βŠ† NUM iajs-2814 76 13 𝒰. 𝒰. NOUN iajs-2814 76 14 assume assume VERB iajs-2814 76 15 β„³ β„³ PROPN iajs-2814 76 16 is be AUX iajs-2814 76 17 a a DET iajs-2814 76 18 π’«βˆ— π’«βˆ— NOUN iajs-2814 76 19 – – PUNCT iajs-2814 76 20 field field NOUN iajs-2814 76 21 of of ADP iajs-2814 76 22 𝒰 𝒰 PROPN iajs-2814 76 23 that that PRON iajs-2814 76 24 contains contain VERB iajs-2814 76 25 ℐ ℐ PRON iajs-2814 76 26 and and CCONJ iajs-2814 76 27 ℬϡ ℬϡ PROPN iajs-2814 76 28 β„³ β„³ PROPN iajs-2814 76 29 , , PUNCT iajs-2814 76 30 then then ADV iajs-2814 76 31 π’«βˆ—(ℐ)|ℬ π’«βˆ—(ℐ)|ℬ PROPN iajs-2814 76 32 is be AUX iajs-2814 76 33 a a DET iajs-2814 76 34 π’«βˆ— π’«βˆ— NOUN iajs-2814 76 35 – – PUNCT iajs-2814 76 36 field field NOUN iajs-2814 76 37 of of ADP iajs-2814 76 38 ℬ. ℬ. NOUN iajs-2814 76 39 proof proof NOUN iajs-2814 76 40 . . PUNCT iajs-2814 77 1 the the DET iajs-2814 77 2 proof proof NOUN iajs-2814 77 3 is be AUX iajs-2814 77 4 done do VERB iajs-2814 77 5 by by ADP iajs-2814 77 6 proposition proposition NOUN iajs-2814 77 7 2.6 2.6 NUM iajs-2814 77 8 and and CCONJ iajs-2814 77 9 3.2 3.2 NUM iajs-2814 77 10 theorem theorem VERB iajs-2814 77 11 3.10 3.10 NUM iajs-2814 77 12 assume assume VERB iajs-2814 77 13 ℐ ℐ PROPN iajs-2814 77 14 βŠ† βŠ† NUM iajs-2814 77 15 p(𝒰 p(𝒰 X iajs-2814 77 16 ) ) PUNCT iajs-2814 77 17 and and CCONJ iajs-2814 77 18 Ο† Ο† NOUN iajs-2814 77 19 β‰  β‰  PROPN iajs-2814 77 20 ℬ ℬ PROPN iajs-2814 77 21 βŠ† βŠ† NUM iajs-2814 77 22 𝒰 𝒰 NOUN iajs-2814 77 23 , , PUNCT iajs-2814 77 24 then then ADV iajs-2814 77 25 π’«βˆ— π’«βˆ— NOUN iajs-2814 77 26 ( ( PUNCT iajs-2814 77 27 ℐ|ℬ ℐ|ℬ ADJ iajs-2814 77 28 ) ) PUNCT iajs-2814 77 29 is be AUX iajs-2814 77 30 the the DET iajs-2814 77 31 smallest small ADJ iajs-2814 77 32 π’«βˆ— π’«βˆ— NOUN iajs-2814 77 33 – – PUNCT iajs-2814 77 34 field field NOUN iajs-2814 77 35 on on ADP iajs-2814 77 36 ℬ ℬ NOUN iajs-2814 77 37 that that PRON iajs-2814 77 38 contain contain VERB iajs-2814 77 39 ℐ|ℬ ℐ|ℬ ADJ iajs-2814 77 40 , , PUNCT iajs-2814 77 41 where where SCONJ iajs-2814 77 42 π’«βˆ— π’«βˆ— NOUN iajs-2814 77 43 ( ( PUNCT iajs-2814 77 44 ℐ|ℬ ℐ|ℬ ADJ iajs-2814 77 45 ) ) PUNCT iajs-2814 77 46 = = SYM iajs-2814 77 47 β‹‚{β„³i|ℬ β‹‚{β„³i|ℬ NOUN iajs-2814 77 48 : : PUNCT iajs-2814 77 49 β„³i|ℬ β„³i|ℬ NOUN iajs-2814 77 50 is be AUX iajs-2814 77 51 a a DET iajs-2814 77 52 π’«βˆ— π’«βˆ— NOUN iajs-2814 77 53 – – PUNCT iajs-2814 77 54 field field NOUN iajs-2814 77 55 of of ADP iajs-2814 77 56 ℬ ℬ NOUN iajs-2814 77 57 and and CCONJ iajs-2814 77 58 β„³i|ℬ β„³i|ℬ NOUN iajs-2814 77 59 βŠ‡ βŠ‡ X iajs-2814 77 60 ℐ|ℬ ℐ|ℬ ADJ iajs-2814 77 61 , , PUNCT iajs-2814 77 62 βˆ€i βˆ€i NOUN iajs-2814 77 63 ∈ ∈ NOUN iajs-2814 77 64 ΞΉ ΞΉ X iajs-2814 77 65 } } PUNCT iajs-2814 77 66 . . PUNCT iajs-2814 78 1 proof proof NOUN iajs-2814 78 2 . . PUNCT iajs-2814 79 1 in in ADP iajs-2814 79 2 the the DET iajs-2814 79 3 same same ADJ iajs-2814 79 4 way way NOUN iajs-2814 79 5 as as ADP iajs-2814 79 6 in in ADP iajs-2814 79 7 proposition proposition NOUN iajs-2814 79 8 2.4 2.4 NUM iajs-2814 79 9 , , PUNCT iajs-2814 79 10 we we PRON iajs-2814 79 11 can can AUX iajs-2814 79 12 prove prove VERB iajs-2814 79 13 that that PRON iajs-2814 79 14 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 79 15 ) ) PUNCT iajs-2814 79 16 is be AUX iajs-2814 79 17 a a DET iajs-2814 79 18 π’«βˆ— π’«βˆ— NOUN iajs-2814 79 19 – – PUNCT iajs-2814 79 20 field field NOUN iajs-2814 79 21 on on ADP iajs-2814 79 22 ℬ. ℬ. PROPN iajs-2814 79 23 to to PART iajs-2814 79 24 prove prove VERB iajs-2814 79 25 that that SCONJ iajs-2814 79 26 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 79 27 ) ) PUNCT iajs-2814 79 28 βŠ‡ βŠ‡ PROPN iajs-2814 79 29 ℐ|ℬ ℐ|ℬ X iajs-2814 79 30 , , PUNCT iajs-2814 79 31 assume assume VERB iajs-2814 79 32 that that SCONJ iajs-2814 79 33 β„³i|ℬ β„³i|ℬ NOUN iajs-2814 79 34 is be AUX iajs-2814 79 35 a a DET iajs-2814 79 36 π’«βˆ— π’«βˆ— NOUN iajs-2814 79 37 – – PUNCT iajs-2814 79 38 field field NOUN iajs-2814 79 39 on on ADP iajs-2814 79 40 ℬ ℬ PROPN iajs-2814 79 41 and and CCONJ iajs-2814 79 42 β„³i|ℬ β„³i|ℬ NOUN iajs-2814 79 43 βŠ‡ βŠ‡ X iajs-2814 79 44 ℐ|ℬ ℐ|ℬ ADJ iajs-2814 79 45 , , PUNCT iajs-2814 79 46 βˆ€i βˆ€i NOUN iajs-2814 79 47 ∈ ∈ PROPN iajs-2814 79 48 ΞΉ ΞΉ NOUN iajs-2814 79 49 , , PUNCT iajs-2814 79 50 then then ADV iajs-2814 79 51 ℐ|ℬ ℐ|ℬ PROPN iajs-2814 80 1 βŠ† βŠ† NUM iajs-2814 80 2 β‹‚ β‹‚ PROPN iajs-2814 80 3 β„³i|ℬ β„³i|ℬ NOUN iajs-2814 80 4 i∈ι i∈ι NOUN iajs-2814 80 5 ; ; PUNCT iajs-2814 80 6 hence hence ADV iajs-2814 80 7 ℐ|ℬ ℐ|ℬ X iajs-2814 80 8 βŠ† βŠ† NUM iajs-2814 80 9 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ PROPN iajs-2814 80 10 ) ) PUNCT iajs-2814 80 11 . . PUNCT iajs-2814 81 1 now now ADV iajs-2814 81 2 , , PUNCT iajs-2814 81 3 let let VERB iajs-2814 81 4 β„³βˆ—|ℬ β„³βˆ—|ℬ PRON iajs-2814 81 5 be be AUX iajs-2814 81 6 a a DET iajs-2814 81 7 π’«βˆ— π’«βˆ— NOUN iajs-2814 81 8 – – PUNCT iajs-2814 81 9 field field NOUN iajs-2814 81 10 on on ADP iajs-2814 81 11 ℬ ℬ DET iajs-2814 81 12 such such ADJ iajs-2814 81 13 that that SCONJ iajs-2814 81 14 β„³βˆ—|ℬ β„³βˆ—|ℬ PROPN iajs-2814 81 15 βŠ‡ βŠ‡ PROPN iajs-2814 81 16 ℐ|ℬ. ℐ|ℬ. PROPN iajs-2814 81 17 then then ADV iajs-2814 81 18 , , PUNCT iajs-2814 81 19 β„³βˆ—|ℬ β„³βˆ—|ℬ PROPN iajs-2814 81 20 βŠ‡ βŠ‡ PROPN iajs-2814 81 21 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ PROPN iajs-2814 81 22 ) ) PUNCT iajs-2814 81 23 . . PUNCT iajs-2814 82 1 therefore therefore ADV iajs-2814 82 2 , , PUNCT iajs-2814 82 3 𝒫(ℐ|ℬ 𝒫(ℐ|ℬ NOUN iajs-2814 82 4 ) ) PUNCT iajs-2814 82 5 is be AUX iajs-2814 82 6 the the DET iajs-2814 82 7 smallest small ADJ iajs-2814 82 8 π’«βˆ— π’«βˆ— NOUN iajs-2814 82 9 – – PUNCT iajs-2814 82 10 field field NOUN iajs-2814 82 11 on on ADP iajs-2814 82 12 ℬ ℬ NOUN iajs-2814 82 13 containing contain VERB iajs-2814 82 14 ℐ|ℬ. ℐ|ℬ. NOUN iajs-2814 82 15 theorem theorem NOUN iajs-2814 82 16 3.11 3.11 NUM iajs-2814 82 17 if if SCONJ iajs-2814 82 18 ℐ ℐ PRON iajs-2814 82 19 βŠ† βŠ† NUM iajs-2814 82 20 p(𝒰 p(𝒰 X iajs-2814 82 21 ) ) PUNCT iajs-2814 82 22 and and CCONJ iajs-2814 82 23 Ο† Ο† NOUN iajs-2814 82 24 β‰  β‰  PROPN iajs-2814 82 25 ℬ ℬ PROPN iajs-2814 82 26 βŠ† βŠ† NUM iajs-2814 82 27 𝒰 𝒰 NOUN iajs-2814 82 28 , , PUNCT iajs-2814 82 29 define define VERB iajs-2814 82 30 a a DET iajs-2814 82 31 class class NOUN iajs-2814 82 32 β„³ β„³ NOUN iajs-2814 82 33 by by ADP iajs-2814 82 34 : : PUNCT iajs-2814 82 35 β„³ β„³ PROPN iajs-2814 82 36 = = PROPN iajs-2814 82 37 { { PUNCT iajs-2814 82 38 m m PROPN iajs-2814 82 39 βŠ† βŠ† NUM iajs-2814 82 40 𝒰 𝒰 NOUN iajs-2814 82 41 : : PUNCT iajs-2814 82 42 mβ‹‚ mβ‹‚ ADP iajs-2814 82 43 ℬ ℬ PROPN iajs-2814 82 44 Ο΅ Ο΅ X iajs-2814 82 45 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 82 46 ) ) PUNCT iajs-2814 82 47 } } PUNCT iajs-2814 82 48 . . PUNCT iajs-2814 83 1 then then ADV iajs-2814 83 2 β„³ β„³ PROPN iajs-2814 83 3 is be AUX iajs-2814 83 4 a a DET iajs-2814 83 5 π’«βˆ— π’«βˆ— NOUN iajs-2814 83 6 – – PUNCT iajs-2814 83 7 field field NOUN iajs-2814 83 8 on on ADP iajs-2814 83 9 a a DET iajs-2814 83 10 set set NOUN iajs-2814 83 11 𝒰. 𝒰. PROPN iajs-2814 83 12 ihjpas ihjpas PROPN iajs-2814 83 13 . . PUNCT iajs-2814 84 1 53 53 NUM iajs-2814 84 2 ( ( PUNCT iajs-2814 84 3 3)2022 3)2022 NOUN iajs-2814 84 4 159 159 NUM iajs-2814 84 5 proof proof NOUN iajs-2814 84 6 . . PUNCT iajs-2814 85 1 by by ADP iajs-2814 85 2 theorem theorem NOUN iajs-2814 85 3 3.10 3.10 NUM iajs-2814 85 4 , , PUNCT iajs-2814 85 5 we we PRON iajs-2814 85 6 have have VERB iajs-2814 85 7 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 85 8 ) ) PUNCT iajs-2814 85 9 as as ADP iajs-2814 85 10 a a DET iajs-2814 85 11 π’«βˆ— π’«βˆ— NOUN iajs-2814 85 12 – – PUNCT iajs-2814 85 13 field field NOUN iajs-2814 85 14 on on ADP iajs-2814 85 15 ℬ ℬ NOUN iajs-2814 85 16 , , PUNCT iajs-2814 85 17 so so ADV iajs-2814 85 18 Ο† Ο† PROPN iajs-2814 85 19 Ο΅ Ο΅ X iajs-2814 85 20 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ PROPN iajs-2814 85 21 ) ) PUNCT iajs-2814 85 22 . . PUNCT iajs-2814 86 1 since since SCONJ iajs-2814 86 2 Ο†= Ο†= NOUN iajs-2814 86 3 Ο† Ο† PROPN iajs-2814 86 4 β‹‚ β‹‚ PROPN iajs-2814 86 5 ℬ ℬ PROPN iajs-2814 86 6 , , PUNCT iajs-2814 86 7 then then ADV iajs-2814 86 8 we we PRON iajs-2814 86 9 get get VERB iajs-2814 86 10 φϡℳ. φϡℳ. PROPN iajs-2814 86 11 assume assume VERB iajs-2814 86 12 that that SCONJ iajs-2814 86 13 m1 m1 PROPN iajs-2814 86 14 , , PUNCT iajs-2814 86 15 m2 m2 PROPN iajs-2814 86 16 Ο΅ Ο΅ PROPN iajs-2814 86 17 β„³. β„³. PROPN iajs-2814 86 18 then then ADV iajs-2814 86 19 ( ( PUNCT iajs-2814 86 20 mi⋂ℬ)Ο΅ mi⋂ℬ)Ο΅ PROPN iajs-2814 86 21 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 86 22 ) ) PUNCT iajs-2814 86 23 , , PUNCT iajs-2814 86 24 for for ADP iajs-2814 86 25 each each DET iajs-2814 86 26 i=1,2 i=1,2 NOUN iajs-2814 86 27 . . PUNCT iajs-2814 87 1 now now ADV iajs-2814 87 2 , , PUNCT iajs-2814 87 3 ( ( PUNCT iajs-2814 87 4 m1β‹‚m2)⋂ℬ m1β‹‚m2)⋂ℬ NOUN iajs-2814 87 5 = = SYM iajs-2814 87 6 ( ( PUNCT iajs-2814 87 7 m1⋂ℬ m1⋂ℬ NOUN iajs-2814 87 8 ) ) PUNCT iajs-2814 87 9 β‹‚(m2⋂ℬ β‹‚(m2⋂ℬ PROPN iajs-2814 87 10 ) ) PUNCT iajs-2814 87 11 . . PUNCT iajs-2814 88 1 since since SCONJ iajs-2814 88 2 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ PROPN iajs-2814 88 3 ) ) PUNCT iajs-2814 88 4 } } PUNCT iajs-2814 88 5 is be AUX iajs-2814 88 6 a a DET iajs-2814 88 7 π’«βˆ— π’«βˆ— NOUN iajs-2814 88 8 – – PUNCT iajs-2814 88 9 field field NOUN iajs-2814 88 10 on on ADP iajs-2814 88 11 ℬ ℬ NOUN iajs-2814 88 12 , , PUNCT iajs-2814 88 13 then then ADV iajs-2814 88 14 ( ( PUNCT iajs-2814 88 15 m1⋂ℬ m1⋂ℬ NOUN iajs-2814 88 16 ) ) PUNCT iajs-2814 88 17 β‹‚(m2⋂ℬ β‹‚(m2⋂ℬ NOUN iajs-2814 88 18 ) ) PUNCT iajs-2814 88 19 Ο΅π’«βˆ—(ℐ|ℬ Ο΅π’«βˆ—(ℐ|ℬ NUM iajs-2814 88 20 ) ) PUNCT iajs-2814 88 21 and and CCONJ iajs-2814 88 22 hence hence ADV iajs-2814 88 23 ( ( PUNCT iajs-2814 88 24 m1β‹‚m2)β‹‚β„¬Ο΅π’«βˆ—(ℐ|ℬ m1β‹‚m2)β‹‚β„¬Ο΅π’«βˆ—(ℐ|ℬ NOUN iajs-2814 88 25 ) ) PUNCT iajs-2814 88 26 , , PUNCT iajs-2814 88 27 thus thus ADV iajs-2814 88 28 m1β‹‚m2Ο΅ m1β‹‚m2Ο΅ VERB iajs-2814 88 29 β„³. β„³. PROPN iajs-2814 88 30 let let VERB iajs-2814 88 31 m1 m1 PROPN iajs-2814 88 32 , , PUNCT iajs-2814 88 33 m2 m2 PROPN iajs-2814 88 34 , , PUNCT iajs-2814 88 35 … … PUNCT iajs-2814 88 36 Ο΅ Ο΅ X iajs-2814 88 37 β„³. β„³. PROPN iajs-2814 88 38 then then ADV iajs-2814 88 39 ( ( PUNCT iajs-2814 88 40 mi⋂ℬ)Ο΅ mi⋂ℬ)Ο΅ PROPN iajs-2814 88 41 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 88 42 ) ) PUNCT iajs-2814 88 43 , , PUNCT iajs-2814 88 44 for for ADP iajs-2814 88 45 i=1,2 i=1,2 ADJ iajs-2814 88 46 , , PUNCT iajs-2814 88 47 … … PUNCT iajs-2814 88 48 since since SCONJ iajs-2814 88 49 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 88 50 ) ) PUNCT iajs-2814 88 51 } } PUNCT iajs-2814 88 52 is be AUX iajs-2814 88 53 π’«βˆ— π’«βˆ— NOUN iajs-2814 88 54 – – PUNCT iajs-2814 88 55 field field NOUN iajs-2814 88 56 on on ADP iajs-2814 88 57 ℬ ℬ NOUN iajs-2814 88 58 , , PUNCT iajs-2814 88 59 then then ADV iajs-2814 88 60 ⋃ ⋃ PROPN iajs-2814 88 61 ( ( PUNCT iajs-2814 88 62 mi mi NOUN iajs-2814 88 63 ∞ ∞ PROPN iajs-2814 88 64 i=1 i=1 PROPN iajs-2814 88 65 β‹‚ β‹‚ PROPN iajs-2814 88 66 ℬ)Ο΅π’«βˆ—(ℐ|ℬ ℬ)Ο΅π’«βˆ—(ℐ|ℬ NOUN iajs-2814 88 67 ) ) PUNCT iajs-2814 88 68 . . PUNCT iajs-2814 89 1 now now ADV iajs-2814 89 2 , , PUNCT iajs-2814 89 3 ( ( PUNCT iajs-2814 89 4 ⋃ ⋃ NOUN iajs-2814 89 5 mi mi NOUN iajs-2814 89 6 ∞ ∞ PROPN iajs-2814 89 7 i=1 i=1 PROPN iajs-2814 89 8 ) ) PUNCT iajs-2814 90 1 ⋂ℬ ⋂ℬ NOUN iajs-2814 90 2 = = SYM iajs-2814 91 1 ⋃ ⋃ PROPN iajs-2814 91 2 ( ( PUNCT iajs-2814 91 3 mi mi NOUN iajs-2814 91 4 ∞ ∞ PROPN iajs-2814 91 5 i=1 i=1 PROPN iajs-2814 91 6 β‹‚ β‹‚ PROPN iajs-2814 91 7 ℬ)Ο΅π’«βˆ—(ℐ|ℬ ℬ)Ο΅π’«βˆ—(ℐ|ℬ X iajs-2814 91 8 ) ) PUNCT iajs-2814 91 9 , , PUNCT iajs-2814 91 10 thus thus ADV iajs-2814 91 11 ⋃ ⋃ PUNCT iajs-2814 91 12 mi mi X iajs-2814 91 13 ∞ ∞ PROPN iajs-2814 91 14 i=1 i=1 PROPN iajs-2814 91 15 Ο΅ Ο΅ ADP iajs-2814 91 16 β„³. β„³. PROPN iajs-2814 91 17 therefore therefore ADV iajs-2814 91 18 , , PUNCT iajs-2814 91 19 β„³ β„³ PROPN iajs-2814 91 20 is be AUX iajs-2814 91 21 π’«βˆ— π’«βˆ— NOUN iajs-2814 91 22 – – PUNCT iajs-2814 91 23 field field NOUN iajs-2814 91 24 on on ADP iajs-2814 91 25 a a DET iajs-2814 91 26 universal universal ADJ iajs-2814 91 27 set set VERB iajs-2814 91 28 𝒰. 𝒰. NOUN iajs-2814 91 29 theorem theorem VERB iajs-2814 91 30 3.12 3.12 NUM iajs-2814 91 31 if if SCONJ iajs-2814 91 32 𝒰 𝒰 PROPN iajs-2814 91 33 is be AUX iajs-2814 91 34 a a DET iajs-2814 91 35 universal universal ADJ iajs-2814 91 36 set set NOUN iajs-2814 91 37 and and CCONJ iajs-2814 91 38 ℐ ℐ PRON iajs-2814 91 39 βŠ† βŠ† NUM iajs-2814 91 40 p(𝒰 p(𝒰 NOUN iajs-2814 91 41 ) ) PUNCT iajs-2814 91 42 such such ADJ iajs-2814 91 43 that that SCONJ iajs-2814 91 44 Ο† Ο† PROPN iajs-2814 91 45 β‰  β‰  PROPN iajs-2814 91 46 ℬ ℬ NOUN iajs-2814 91 47 βŠ† βŠ† NUM iajs-2814 91 48 𝒰 𝒰 PROPN iajs-2814 91 49 , , PUNCT iajs-2814 91 50 then then ADV iajs-2814 91 51 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 91 52 ) ) PUNCT iajs-2814 91 53 = = SYM iajs-2814 91 54 π’«βˆ—(ℐ)|ℬ. π’«βˆ—(ℐ)|ℬ. ADJ iajs-2814 91 55 proof proof NOUN iajs-2814 91 56 . . PUNCT iajs-2814 92 1 by by ADP iajs-2814 92 2 proposition proposition NOUN iajs-2814 92 3 2.6 2.6 NUM iajs-2814 92 4 , , PUNCT iajs-2814 92 5 we we PRON iajs-2814 92 6 have have VERB iajs-2814 92 7 π’«βˆ—(ℐ π’«βˆ—(ℐ NUM iajs-2814 92 8 ) ) PUNCT iajs-2814 93 1 is be AUX iajs-2814 93 2 π’«βˆ— π’«βˆ— NOUN iajs-2814 93 3 – – PUNCT iajs-2814 93 4 field field NOUN iajs-2814 93 5 on on ADP iajs-2814 93 6 𝒰. 𝒰. PROPN iajs-2814 93 7 so so ADV iajs-2814 93 8 , , PUNCT iajs-2814 93 9 we we PRON iajs-2814 93 10 get get AUX iajs-2814 93 11 π’«βˆ—(ℐ)|ℬ π’«βˆ—(ℐ)|ℬ VERB iajs-2814 93 12 is be AUX iajs-2814 93 13 π‘Ž π‘Ž DET iajs-2814 93 14 π’«βˆ— π’«βˆ— NOUN iajs-2814 93 15 – – PUNCT iajs-2814 93 16 field field NOUN iajs-2814 93 17 on on ADP iajs-2814 93 18 ℬ ℬ NOUN iajs-2814 93 19 by by ADP iajs-2814 93 20 proposition proposition NOUN iajs-2814 93 21 3.2 3.2 NUM iajs-2814 93 22 . . PUNCT iajs-2814 94 1 assumeπ‘‘β„Žπ‘Žπ‘‘ assumeπ‘‘β„Žπ‘Žπ‘‘ PROPN iajs-2814 94 2 nϡℐ|ℬ. nϡℐ|ℬ. PROPN iajs-2814 94 3 then then ADV iajs-2814 94 4 n n NOUN iajs-2814 94 5 = = SYM iajs-2814 94 6 mβ‹‚ mβ‹‚ NOUN iajs-2814 94 7 ℬ ℬ NOUN iajs-2814 94 8 for for ADP iajs-2814 94 9 some some DET iajs-2814 94 10 mΟ΅ mΟ΅ NOUN iajs-2814 94 11 ℐ. ℐ. PROPN iajs-2814 94 12 but but CCONJ iajs-2814 94 13 ℐ ℐ ADV iajs-2814 94 14 βŠ† βŠ† PROPN iajs-2814 94 15 π’«βˆ—(ℐ π’«βˆ—(ℐ NUM iajs-2814 94 16 ) ) PUNCT iajs-2814 94 17 , , PUNCT iajs-2814 94 18 so so SCONJ iajs-2814 94 19 we we PRON iajs-2814 94 20 have have VERB iajs-2814 94 21 mΟ΅ mΟ΅ ADP iajs-2814 94 22 π’«βˆ—(ℐ π’«βˆ—(ℐ NUM iajs-2814 94 23 ) ) PUNCT iajs-2814 94 24 and and CCONJ iajs-2814 94 25 thus thus ADV iajs-2814 94 26 nΟ΅ nΟ΅ PRON iajs-2814 94 27 π’«βˆ—(ℐ)|ℬ. π’«βˆ—(ℐ)|ℬ. NOUN iajs-2814 94 28 hence hence ADV iajs-2814 94 29 ℐ|ℬ ℐ|ℬ NUM iajs-2814 94 30 βŠ† βŠ† NUM iajs-2814 94 31 π’«βˆ—(ℐ)|ℬ. π’«βˆ—(ℐ)|ℬ. PROPN iajs-2814 94 32 therefore therefore ADV iajs-2814 94 33 , , PUNCT iajs-2814 94 34 π’«βˆ—(ℐ)|ℬ π’«βˆ—(ℐ)|ℬ VERB iajs-2814 94 35 is be AUX iajs-2814 94 36 a a DET iajs-2814 94 37 π’«βˆ— π’«βˆ— NOUN iajs-2814 94 38 – – PUNCT iajs-2814 94 39 field field NOUN iajs-2814 94 40 on on ADP iajs-2814 94 41 ℬ ℬ NOUN iajs-2814 94 42 that that SCONJ iajs-2814 94 43 containing contain VERB iajs-2814 94 44 ℐ|ℬ. ℐ|ℬ. INTJ iajs-2814 94 45 by by ADP iajs-2814 94 46 theorem theorem NOUN iajs-2814 94 47 3.10 3.10 NUM iajs-2814 94 48 , , PUNCT iajs-2814 94 49 we we PRON iajs-2814 94 50 have have VERB iajs-2814 94 51 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 94 52 ) ) PUNCT iajs-2814 94 53 is be AUX iajs-2814 94 54 the the DET iajs-2814 94 55 smallest small ADJ iajs-2814 94 56 π’«βˆ— π’«βˆ— NOUN iajs-2814 94 57 – – PUNCT iajs-2814 94 58 field field NOUN iajs-2814 94 59 on on ADP iajs-2814 94 60 ℬ ℬ PROPN iajs-2814 94 61 that that SCONJ iajs-2814 94 62 containing contain VERB iajs-2814 94 63 ℐ|ℬ ℐ|ℬ NOUN iajs-2814 94 64 , , PUNCT iajs-2814 94 65 which which PRON iajs-2814 94 66 implies imply VERB iajs-2814 94 67 that that SCONJ iajs-2814 94 68 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 94 69 ) ) PUNCT iajs-2814 94 70 βŠ† βŠ† NUM iajs-2814 94 71 π’«βˆ—(ℐ)|ℬ. π’«βˆ—(ℐ)|ℬ. PROPN iajs-2814 94 72 now now ADV iajs-2814 94 73 , , PUNCT iajs-2814 94 74 if if SCONJ iajs-2814 94 75 we we PRON iajs-2814 94 76 define define VERB iajs-2814 94 77 a a DET iajs-2814 94 78 class class NOUN iajs-2814 94 79 β„³ β„³ NOUN iajs-2814 94 80 by by ADP iajs-2814 94 81 β„³= β„³= PROPN iajs-2814 94 82 { { PUNCT iajs-2814 94 83 c c PROPN iajs-2814 94 84 βŠ† βŠ† NUM iajs-2814 94 85 𝒰 𝒰 NOUN iajs-2814 94 86 : : PUNCT iajs-2814 94 87 cβ‹‚ cβ‹‚ VERB iajs-2814 94 88 ℬ ℬ PROPN iajs-2814 94 89 Ο΅ Ο΅ X iajs-2814 94 90 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ PROPN iajs-2814 94 91 ) ) PUNCT iajs-2814 94 92 } } PUNCT iajs-2814 94 93 , , PUNCT iajs-2814 94 94 then then ADV iajs-2814 94 95 in in ADP iajs-2814 94 96 theorem theorem NOUN iajs-2814 94 97 3.11 3.11 NUM iajs-2814 94 98 , , PUNCT iajs-2814 94 99 we we PRON iajs-2814 94 100 have have VERB iajs-2814 94 101 β„³ β„³ PROPN iajs-2814 94 102 as as ADP iajs-2814 94 103 π‘Ž π‘Ž DET iajs-2814 94 104 π’«βˆ— π’«βˆ— NOUN iajs-2814 94 105 – – PUNCT iajs-2814 94 106 field field NOUN iajs-2814 94 107 on on ADP iajs-2814 94 108 𝒰. 𝒰. PROPN iajs-2814 94 109 let let VERB iajs-2814 94 110 cΟ΅ cΟ΅ VERB iajs-2814 94 111 ℐ ℐ PROPN iajs-2814 94 112 , , PUNCT iajs-2814 94 113 then then ADV iajs-2814 94 114 ( ( PUNCT iajs-2814 94 115 c c NOUN iajs-2814 94 116 ∩ ∩ ADJ iajs-2814 94 117 ℬ ℬ NOUN iajs-2814 94 118 ) ) PUNCT iajs-2814 94 119 ϡℐ|ℬ ϡℐ|ℬ NOUN iajs-2814 94 120 , , PUNCT iajs-2814 94 121 but but CCONJ iajs-2814 94 122 ℐ|ℬ ℐ|ℬ NUM iajs-2814 94 123 βŠ† βŠ† NUM iajs-2814 94 124 𝒫(ℐ|ℬ 𝒫(ℐ|ℬ NOUN iajs-2814 94 125 ) ) PUNCT iajs-2814 94 126 implies imply VERB iajs-2814 94 127 that that SCONJ iajs-2814 94 128 ( ( PUNCT iajs-2814 94 129 c c NOUN iajs-2814 94 130 ∩ ∩ NOUN iajs-2814 94 131 ℬ)Ο΅ ℬ)Ο΅ X iajs-2814 94 132 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 94 133 ) ) PUNCT iajs-2814 94 134 , , PUNCT iajs-2814 94 135 hence hence ADV iajs-2814 94 136 cΟ΅ cΟ΅ VERB iajs-2814 94 137 β„³ β„³ PROPN iajs-2814 94 138 and and CCONJ iajs-2814 94 139 ℐ ℐ PRON iajs-2814 94 140 βŠ† βŠ† NUM iajs-2814 94 141 β„³. β„³. PROPN iajs-2814 94 142 now now ADV iajs-2814 94 143 , , PUNCT iajs-2814 94 144 if if SCONJ iajs-2814 94 145 we we PRON iajs-2814 94 146 assume assume VERB iajs-2814 94 147 that that SCONJ iajs-2814 94 148 nΟ΅ nΟ΅ PRON iajs-2814 94 149 π’«βˆ—(ℐ)|ℬ π’«βˆ—(ℐ)|ℬ NUM iajs-2814 94 150 , , PUNCT iajs-2814 94 151 then then ADV iajs-2814 94 152 n= n= ADJ iajs-2814 94 153 m m VERB iajs-2814 94 154 ∩ ∩ ADJ iajs-2814 94 155 ℬ ℬ NOUN iajs-2814 94 156 , , PUNCT iajs-2814 94 157 for for ADP iajs-2814 94 158 some some DET iajs-2814 94 159 mΟ΅ mΟ΅ NOUN iajs-2814 94 160 𝒫(ℐ 𝒫(ℐ NOUN iajs-2814 94 161 ) ) PUNCT iajs-2814 94 162 . . PUNCT iajs-2814 95 1 but but CCONJ iajs-2814 95 2 π’«βˆ—(ℐ π’«βˆ—(ℐ NUM iajs-2814 95 3 ) ) PUNCT iajs-2814 95 4 βŠ† βŠ† NUM iajs-2814 95 5 β„³ β„³ PROPN iajs-2814 95 6 , , PUNCT iajs-2814 95 7 then then ADV iajs-2814 95 8 mΟ΅ mΟ΅ PROPN iajs-2814 95 9 β„³ β„³ PROPN iajs-2814 95 10 , , PUNCT iajs-2814 95 11 hence hence ADV iajs-2814 95 12 nΟ΅ nΟ΅ PRON iajs-2814 95 13 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 95 14 ) ) PUNCT iajs-2814 95 15 . . PUNCT iajs-2814 96 1 consequentially consequentially ADV iajs-2814 96 2 , , PUNCT iajs-2814 96 3 π’«βˆ—(ℐ)|ℬ π’«βˆ—(ℐ)|ℬ VERB iajs-2814 96 4 βŠ† βŠ† NUM iajs-2814 96 5 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ NOUN iajs-2814 96 6 ) ) PUNCT iajs-2814 96 7 . . PUNCT iajs-2814 97 1 this this PRON iajs-2814 97 2 completes complete VERB iajs-2814 97 3 the the DET iajs-2814 97 4 proof proof NOUN iajs-2814 97 5 . . PUNCT iajs-2814 98 1 3 3 X iajs-2814 98 2 . . X iajs-2814 98 3 conclusions conclusion NOUN iajs-2814 98 4 we we PRON iajs-2814 98 5 tried try VERB iajs-2814 98 6 to to PART iajs-2814 98 7 define define VERB iajs-2814 98 8 the the DET iajs-2814 98 9 concept concept NOUN iajs-2814 98 10 of of ADP iajs-2814 98 11 measure measure NOUN iajs-2814 98 12 relative relative ADJ iajs-2814 98 13 to to ADP iajs-2814 98 14 the the DET iajs-2814 98 15 π’«βˆ— π’«βˆ— NOUN iajs-2814 98 16 – – PUNCT iajs-2814 98 17 field field NOUN iajs-2814 98 18 β„³ β„³ NOUN iajs-2814 98 19 of of ADP iajs-2814 98 20 𝒰 𝒰 PROPN iajs-2814 98 21 and and CCONJ iajs-2814 98 22 also also ADV iajs-2814 98 23 define define VERB iajs-2814 98 24 the the DET iajs-2814 98 25 idea idea NOUN iajs-2814 98 26 of of ADP iajs-2814 98 27 the the DET iajs-2814 98 28 restriction restriction NOUN iajs-2814 98 29 of of ADP iajs-2814 98 30 measure measure NOUN iajs-2814 98 31 on on ADP iajs-2814 98 32 β„³|ℬ β„³|ℬ NOUN iajs-2814 98 33 of of ADP iajs-2814 98 34 a a DET iajs-2814 98 35 set set NOUN iajs-2814 98 36 ℬ. ℬ. NOUN iajs-2814 98 37 also also ADV iajs-2814 98 38 , , PUNCT iajs-2814 98 39 we we PRON iajs-2814 98 40 discuss discuss VERB iajs-2814 98 41 many many ADJ iajs-2814 98 42 properties property NOUN iajs-2814 98 43 of of ADP iajs-2814 98 44 these these DET iajs-2814 98 45 notions notion NOUN iajs-2814 98 46 . . PUNCT iajs-2814 99 1 in in ADP iajs-2814 99 2 this this DET iajs-2814 99 3 article article NOUN iajs-2814 99 4 , , PUNCT iajs-2814 99 5 the the DET iajs-2814 99 6 idea idea NOUN iajs-2814 99 7 of of ADP iajs-2814 99 8 π’«βˆ— π’«βˆ— NOUN iajs-2814 99 9 – – PUNCT iajs-2814 99 10 field field NOUN iajs-2814 99 11 is be AUX iajs-2814 99 12 given give VERB iajs-2814 99 13 to to PART iajs-2814 99 14 refer refer VERB iajs-2814 99 15 to to ADP iajs-2814 99 16 the the DET iajs-2814 99 17 generalization generalization NOUN iajs-2814 99 18 of of ADP iajs-2814 99 19 each each DET iajs-2814 99 20 Οƒ Οƒ PROPN iajs-2814 99 21 – – PUNCT iajs-2814 99 22 field field NOUN iajs-2814 99 23 and and CCONJ iajs-2814 99 24 Οƒ Οƒ PROPN iajs-2814 99 25 – – PUNCT iajs-2814 99 26 ring ring NOUN iajs-2814 99 27 . . PUNCT iajs-2814 100 1 furthermore furthermore ADV iajs-2814 100 2 , , PUNCT iajs-2814 100 3 some some DET iajs-2814 100 4 properties property NOUN iajs-2814 100 5 of of ADP iajs-2814 100 6 the the DET iajs-2814 100 7 purposed purposed ADJ iajs-2814 100 8 notion notion NOUN iajs-2814 100 9 are be AUX iajs-2814 100 10 proven prove VERB iajs-2814 100 11 as as SCONJ iajs-2814 100 12 explained explain VERB iajs-2814 100 13 below below ADV iajs-2814 100 14 : : PUNCT iajs-2814 100 15 1 1 X iajs-2814 100 16 . . X iajs-2814 100 17 let let VERB iajs-2814 100 18 β„³ β„³ PRON iajs-2814 100 19 be be AUX iajs-2814 100 20 a a DET iajs-2814 100 21 π’«βˆ— π’«βˆ— NOUN iajs-2814 100 22 – – PUNCT iajs-2814 100 23 field field NOUN iajs-2814 100 24 of of ADP iajs-2814 100 25 a a DET iajs-2814 100 26 set set VERB iajs-2814 100 27 𝒰 𝒰 NOUN iajs-2814 100 28 and and CCONJ iajs-2814 100 29 let let VERB iajs-2814 100 30 ℬ ℬ PRON iajs-2814 100 31 be be AUX iajs-2814 100 32 a a DET iajs-2814 100 33 nonempty nonempty ADJ iajs-2814 100 34 subset subset NOUN iajs-2814 100 35 of of ADP iajs-2814 100 36 𝒰. 𝒰. PROPN iajs-2814 100 37 then then ADV iajs-2814 100 38 , , PUNCT iajs-2814 100 39 β„³|ℬ β„³|ℬ NOUN iajs-2814 100 40 is be AUX iajs-2814 100 41 a a DET iajs-2814 100 42 π’«βˆ— π’«βˆ— NOUN iajs-2814 100 43 – – PUNCT iajs-2814 100 44 field field NOUN iajs-2814 100 45 of of ADP iajs-2814 100 46 a a DET iajs-2814 100 47 set set NOUN iajs-2814 100 48 ℬ. ℬ. NOUN iajs-2814 100 49 2 2 NUM iajs-2814 100 50 . . PUNCT iajs-2814 100 51 assume assume VERB iajs-2814 100 52 π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘Žπ‘‘ PROPN iajs-2814 100 53 β„³ β„³ PROPN iajs-2814 100 54 is be AUX iajs-2814 100 55 a a DET iajs-2814 100 56 π’«βˆ— π’«βˆ— NOUN iajs-2814 100 57 – – PUNCT iajs-2814 100 58 field field NOUN iajs-2814 100 59 on on ADP iajs-2814 100 60 𝒰 𝒰 PROPN iajs-2814 100 61 and and CCONJ iajs-2814 100 62 a a DET iajs-2814 100 63 βŠ† βŠ† NUM iajs-2814 100 64 ℬ ℬ NOUN iajs-2814 100 65 βŠ† βŠ† NUM iajs-2814 100 66 𝒰. 𝒰. NOUN iajs-2814 100 67 if if SCONJ iajs-2814 100 68 aΟ΅ aΟ΅ ADP iajs-2814 100 69 β„³ β„³ PROPN iajs-2814 100 70 , , PUNCT iajs-2814 100 71 then then ADV iajs-2814 100 72 aΟ΅ aΟ΅ ADP iajs-2814 100 73 β„³|ℬ. β„³|ℬ. NOUN iajs-2814 100 74 3 3 X iajs-2814 100 75 . . PUNCT iajs-2814 101 1 if if SCONJ iajs-2814 101 2 β„³ β„³ PROPN iajs-2814 101 3 is be AUX iajs-2814 101 4 a a DET iajs-2814 101 5 π’«βˆ— π’«βˆ— NOUN iajs-2814 101 6 – – PUNCT iajs-2814 101 7 field field NOUN iajs-2814 101 8 and and CCONJ iajs-2814 101 9 ℬ ℬ NOUN iajs-2814 101 10 be be VERB iajs-2814 101 11 a a DET iajs-2814 101 12 nonempty nonempty ADJ iajs-2814 101 13 subset subset NOUN iajs-2814 101 14 of of ADP iajs-2814 101 15 𝒰 𝒰 PROPN iajs-2814 101 16 such such ADJ iajs-2814 101 17 that that SCONJ iajs-2814 101 18 ℬϡ ℬϡ ADP iajs-2814 101 19 β„³. β„³. PROPN iajs-2814 101 20 then then ADV iajs-2814 101 21 β„³|ℬ= β„³|ℬ= PROPN iajs-2814 101 22 { { PUNCT iajs-2814 101 23 a a DET iajs-2814 101 24 βŠ† βŠ† NUM iajs-2814 101 25 ℬ ℬ NOUN iajs-2814 101 26 : : PUNCT iajs-2814 101 27 aΟ΅ aΟ΅ ADP iajs-2814 101 28 β„³ β„³ PROPN iajs-2814 101 29 } } PUNCT iajs-2814 101 30 . . PUNCT iajs-2814 102 1 4 4 X iajs-2814 102 2 . . X iajs-2814 102 3 suppose suppose VERB iajs-2814 102 4 that that SCONJ iajs-2814 102 5 β„³ β„³ PROPN iajs-2814 102 6 is be AUX iajs-2814 102 7 a a DET iajs-2814 102 8 π’«βˆ— π’«βˆ— NOUN iajs-2814 102 9 – – PUNCT iajs-2814 102 10 field field NOUN iajs-2814 102 11 and and CCONJ iajs-2814 102 12 ℬ ℬ PROPN iajs-2814 102 13 βŠ† βŠ† NUM iajs-2814 102 14 𝒰 𝒰 NOUN iajs-2814 102 15 such such ADJ iajs-2814 102 16 that that SCONJ iajs-2814 102 17 ℬϡ ℬϡ ADP iajs-2814 102 18 β„³. β„³. PROPN iajs-2814 102 19 then then ADV iajs-2814 102 20 β„³|ℬ β„³|ℬ NOUN iajs-2814 102 21 βŠ† βŠ† NUM iajs-2814 102 22 β„³. β„³. PROPN iajs-2814 102 23 5 5 NUM iajs-2814 102 24 . . PUNCT iajs-2814 103 1 if if SCONJ iajs-2814 103 2 ℐ ℐ PRON iajs-2814 103 3 βŠ† βŠ† NUM iajs-2814 103 4 p(𝒰 p(𝒰 X iajs-2814 103 5 ) ) PUNCT iajs-2814 103 6 and and CCONJ iajs-2814 103 7 Ο† Ο† NOUN iajs-2814 103 8 β‰  β‰  PROPN iajs-2814 103 9 ℬ ℬ NOUN iajs-2814 103 10 βŠ† βŠ† NUM iajs-2814 103 11 𝒰 𝒰 NOUN iajs-2814 103 12 and and CCONJ iajs-2814 103 13 π’«βˆ—(ℐ)|ℬ π’«βˆ—(ℐ)|ℬ NUM iajs-2814 103 14 is be AUX iajs-2814 103 15 a a DET iajs-2814 103 16 π’«βˆ— π’«βˆ— NOUN iajs-2814 103 17 – – PUNCT iajs-2814 103 18 field field NOUN iajs-2814 103 19 on on ADP iajs-2814 103 20 ℬ. ℬ. PROPN iajs-2814 103 21 then then ADV iajs-2814 103 22 , , PUNCT iajs-2814 103 23 π’«βˆ—(ℐ|ℬ π’«βˆ—(ℐ|ℬ PROPN iajs-2814 103 24 ) ) PUNCT iajs-2814 103 25 = = SYM iajs-2814 103 26 π’«βˆ—(ℐ)|ℬ. π’«βˆ—(ℐ)|ℬ. PROPN iajs-2814 103 27 ihjpas ihjpas PROPN iajs-2814 103 28 . . PUNCT iajs-2814 104 1 53 53 NUM iajs-2814 104 2 ( ( PUNCT iajs-2814 104 3 3)2022 3)2022 NOUN iajs-2814 104 4 160 160 NUM iajs-2814 104 5 references reference NOUN iajs-2814 104 6 1 1 NUM iajs-2814 104 7 . . PUNCT iajs-2814 105 1 wang wang PROPN iajs-2814 105 2 , , PUNCT iajs-2814 105 3 z. z. PROPN iajs-2814 105 4 ; ; PUNCT iajs-2814 105 5 blir blir NOUN iajs-2814 105 6 , , PUNCT iajs-2814 105 7 g.j g.j PROPN iajs-2814 105 8 . . PROPN iajs-2814 105 9 fuzzy fuzzy ADJ iajs-2814 105 10 measure measure NOUN iajs-2814 105 11 theory theory NOUN iajs-2814 105 12 . . PUNCT iajs-2814 106 1 springer springer NOUN iajs-2814 106 2 science science PROPN iajs-2814 106 3 and and CCONJ iajs-2814 106 4 business business NOUN iajs-2814 106 5 media medium NOUN iajs-2814 106 6 , , PUNCT iajs-2814 106 7 llc llc PROPN iajs-2814 106 8 , , PUNCT iajs-2814 106 9 new new PROPN iajs-2814 106 10 york york PROPN iajs-2814 106 11 , , PUNCT iajs-2814 106 12 1992 1992 NUM iajs-2814 106 13 ; ; PUNCT iajs-2814 106 14 isbn isbn ADJ iajs-2814 106 15 978 978 NUM iajs-2814 106 16 - - SYM iajs-2814 106 17 1 1 NUM iajs-2814 106 18 - - PUNCT iajs-2814 106 19 4419 4419 NUM iajs-2814 106 20 - - PUNCT iajs-2814 106 21 3225 3225 NUM iajs-2814 106 22 - - SYM iajs-2814 106 23 9 9 NUM iajs-2814 106 24 . . NOUN iajs-2814 106 25 2 2 NUM iajs-2814 106 26 . . X iajs-2814 107 1 ahmed ahmed PROPN iajs-2814 107 2 , , PUNCT iajs-2814 107 3 i.s i.s PROPN iajs-2814 107 4 . . 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