OH ‟ O^U
XQHGULUHULVA H^ULUQEA
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
UHC@A KF OKGUAGU
8'8
SHULK
8'3
^SK^KSULKG
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LGELOA_
8'5
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8'1
@KIHSLUMB
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ 8'8
OMH^UAS " 8
SHULK
Shtlk4 H rhtlk ls h okbphrlskg kf tma slzas kf twk kr bkra quhgtltlas kf tma shba jlge -lg shba uglts, cy elvlslkg'
Gkta4 8' Qsuh``y h rhtlk ls axprassae lg tma `kwast tag fkr a'i' 864652
86
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86x8
65 86x5
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2845'
3' Krear kf tma tarbs lg tma rhtlk ls lbpkrthgt a'i' 341 143' ?' Shtlk axlsts catwaag quhgtltlas kf tma shba jlge a'i' rhtlk catwaag hia kf kga oml`e hge walimt kf hgktmar oml`e ohggkt ca fkuge' 5' Xuhgtltlas tk ca okbphrae bust ca lg tma shba uglts' A'i' Shtlk catwaag 81 blgutas hge 11 saokges 81411' Cut 81x6> 81x6> saokge saokgess 8=> 2 28=>488 11 saokges 88 1' Uk okbphra rhtlks okgvart tmab lgtk aqulvh`agt `lja frhotlkgs'
@L_U KF FKSBQ@HA 8' Shtlk kf twk gubcars h hge c ls h4c h ls oh``ae hgtaoaeagt -flrst tarb, hge c ls oh``ae okgsaquagt -saokge tarb,' a'i' lg tma rhtlk ?45 ? ls oh``ae hgtaoaeagt hge 5 ls oh``ae okgsaquagt' 3' Lgvarsa rhtlk4 Kga rhtlk ls tma lgvarsa kf hgktmar lf tmalr prkeuot ls 8' a'i' -l, Lgvarsa rhtlk kf h 4 c ls c 4 h' -ll, Lgvarsa rhtlk kf 34? ls ?43' ?' Shtlk kf irahtar lgaquh`lty4 H rhtlk h 4 c ls shle tk ca kf irahtar lgaquh`lty lf h c a'i' 145' 5' Shtlk kf `ass lgaquh`lty4 H rhtlk h 4 c ls shle tk ca kf `ass lgaquh`lty lga quh`lty lf h c a'i' 541' 1' Shtlk kf aquh`lty4 H rhtlk h4 c ls shle tk ca rhtlk kf aquh`lty lf h2c a'i' 545' 6' Okbpkuge rhtlk4 Uma okbpkuge rhtlk kf twk rhtlks h4c hge o4e ls ho4ce' A'i' tma okbpkuge rhtlk kf 34? hge 641 ls 3x64 ? x 1 2 541' 9' Eup`lohta rhtlk 4 H rhtlk okbpkugeae kf Lt sa`f ls oh``ae eup`lohta rhtlk' a'i' -l, Eup`lohta rhtlk kf h4c ls h 3 4 c3 ' -ll, Eup`lohta rhtlk kf ?45 ls <486' =' Urlp`lohta rhtlk4 Urlp`lohta rhtlk kf h4c ls
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ h? 4 c? ' a'i' Urlp`lohta rhtlk kf ?45 ls 39465'
<' _uc eup`lohta rhtlk4 _uc eup`lohta rhtlk kf h4 c ls 8>' _uc trlp`lohta rhtlk4 _uc trlp`lohta rhtlk kf h4 c ls
h 4 c a'i' suc eup`lohta rhtlk kf 5431 ls 341' ?
h 4 ? c a'i' suc trlp`lohta rhtlk kf 394= ls ?43' ?4 3'
Okbbagsurhc`a quhgtltlas4 Lf tma rhtlk kf twk slbl`hr quhgtltlas ohg ca axprassae hs h rhtlk kf 88' Okbbagsurhc`a twk lgtaiars tma quhgtltlas hra shle tk ca okbbagsurhc`a quhgtltlas' -l, a'i' rhtlk kf 8'3=48'1 2
8'3= 83= 8> 65 wmlom ls h rhtlk kf lgtaiars 65 hge 91' 2 x 2 8'1 8>> 81 91
843= 8'1 hra okbbagsurhc`a quhgtltlas' 83' Lgokbbagsurhc`a quhgtltlas4 Lf tma rhtlk kf twk slbl`hr quhgtltlas ohggkt ca axhot`y axprassae hs h rhtlk kf twk lgtaiars tma Xuhgtltlas hra shle tk ca lgokbbagsurhc`a
? 4 8 ohggkt ca axhot`y axprassae hs h rhtlk kf twk lgtaiars' _k
a'i' rhtlk
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okbbagsurhc`a quhgtltlas' 8?' Okgtlguae rhtlk4 Okgtlguae rhtlk ls tma ra`htlkg -kr okbphsslkg, catwaag tma bhigltueas kf tmraa kr bkra quhgtltlas kf tma shba jlge a'i' Uma okgtlguae rhtlk kf tmraa slbl`hr quhgtltlas h c o ls h4c4o'
85' h 4 c bh 4 bc b > ' 81' h 4 c 86' Lf
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h c o e
h
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c
b b
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a f
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bh*go*pa bc*ge*pf
Uypa ‟84 Uk flge hgtaoaeagt+okgsaquagt'
Bu`tlp`a Omkloa Xuastlkgs 8' Uma rhtlk kf twk quhgtltlas ls ? 4 5' Lf tma hgtaoaeagt ls 81 tma okgsaquagt ls h' 86 c' 6> o' 33 e' 3> ? ?x1 81 _k`utlkg4 Shtlk kf twk quhgtltlas 2 ?452 2 ' Hs hgtaoaeagt ls 81 okgsaquagt ls 3>' 2 5 5x1 3> 3' Uma rhtlk kf twk quhgtltlas ls 341' Lf tma hgtaoaeagt ls 8>, tma okgsaquagt ls …' h' 5 c' 31
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
OMH^UAS " 8
o' 3> e' 1> ?' Uma rhtlk kf twk quhgtltlas ls 945' Lf tma okgsaquagt ls 3=, tma hgtaoaeagt ls …'' h' 5< c' 3= o' 85 e' 35'1
Uypa ‟34 Uk parfkrb kparhtlkg kg rhtlk' 8' Uma lgvarsa rhtlk kf 88 4 81 ls 81488
h' 81 4 88 c' 88 88 4 81 81 o' 838 4 331
e' gkga kf tmasa 3' Uma rhtlk kf tma quhgtltlas ls 1 4 9' Lf tma okgsaquagt kf lts lgvarsa rhtlk ls 1 tma hgtaoaeagt ls h' 1 c' 1 o' 9 e' gkga kf tmasa
Uypa ‟?4 Uk flge okbpkuge rhtlk kf ilvag rhtlks' 8' Uma rhtlk okbpkugeae kf 3 4 ? < 4 5 1 4 6 hge = 4 8> ls h' 8 4 8 c' 841 o' ?4= e' gkga kf tmasa 3 < 1 = 93> _k`utlkg4 Okbpkuge rhtlk kf 34? <45 146 hge hge =48> ls x x x 2 2848' ? 5 6 8> 93> 3' Uma okbpkuge rhtlk kf ?45 8>438 hge 85489 ls …' h' ?1488 c' 8>488 o' 1488 e' 14?? ?' Uma okbpkuge rhtlk kf 549 ?41 146 hge 8541 ls h' 549 h' 149 c' 145 o' 541
Uypa ‟54 Uk flge eup`lohta trlp`lohta suc eup`lohta suc trlp`lohta rhtlk kf ilvag rhtlks4 8' Uma rhtlk okbpkugeae kf eup`lohta rhtlk kf 5 4 1 trlp`lohta rhtlk kf 8 4 ? suc eup`lohta rhtlk kf =8 4 316 hge suc trlp`lohta rhtlk kf 831 4 183 ls h' 5 4 183 c' ? 4 ?3
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ o' 8 4 83 e' gkga kf tmasa 3
5 86 _k`utlkg4 Eup`lohta rhtlk kf 541 2 ' Urlp`lohta rhtlk kf 84? 2 1 31 =8
_uc Eup`lohta rhtlk kf =84316 2 Saqulrae okbpkuge rhtlk 2
86 31
316
x
8
< 86
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8 8 ? 39 '
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831 183
1 =
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3' Uma eup`lohta rhtlk kf ? 4 5 ls h' ?43 c' 5 4 ? o' < 4 86 e' gkga kf tmasa ?' Uma suc eup`lohta rhtlk kf 31 4 ?6 ls h' 6 4 1 c' ?6 4 31 o' 1> 4 93 e' 1 4 6 5' Uma trlp`lohta rhtlk kf 3 4 ? ls h' = 4 39 c' 6 4 < o' ? 4 3 e' gkga kf tmasa 1' Uma suc trlp`lohta rhtlk kf = 4 39 ls h' 39 4 = c' 35 4 =8 o' 3 4 ? e' gkga kf tmasa 6' Uma rhtlk okbpkugeae kf 5 4 < hge tma eup`lohta rhtlk kf ? 4 5 ls h' 845 c' 84? o' ?48 e' gkga kf tmasa 9' Uma rhtlk okbpkugeae kf 5 4 < tma eup`lohta rhtlk kf ? 4 5 tma trlp`lohta rhtlk kf 3 4 ? hge < 4 9 ls h' 349 c' 943 o' 3438 e' gkga kf tmasa =' Uma rhtlk okbpkugeae kf 541 hge suc eup`lohta rhtlk kf h4q ls =481' Umag vh`ua kf h ls …' -Eao', h' 3 c' ? o' 5 e' <
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ Uypa ‟14 Mhrear subs kg eup`lohta suc eup`lohta rhtlks4 8' Lf p 4 q ls tma suc eup`lohta rhtlk kf p ‟ x 3 4 q ‟ x 3 tmag x 3 ls h' [[ [[p p [[[[
p*q c' [[ [[q q [[[[
p*q o' [[p [[pq q [[[[
p*q e' gkga kf tmasa
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pq- p q ,
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3
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pq p q
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Uypa ‟64 Frkb ilvag vh`ua kf rhtlk avh`uhta 8' Lf x 4 y 2 ? 4 5 tma vh`ua kf x3y * xy3 4 x? * y? ls h' 8? 4 83 c' 83 4 8? o' 38 4 ?8 e' gkga kf tmasa _k`utlkg4 Flrst batmke 4 x 4 y ? 4 5 x y xy 3
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?6 5= 39 65
=5 <8
83 8?
3' Lf h 4 c 2 ? 4 5 tma vh`ua kf -3h*?c, 4 -?h*5c, ls h' 15 4 31 c' = 4 31 o' 89 4 35 e' 8= 4 31 ?' Lf x 4 y ? 4 5 tma vh`ua kf ? x 1 y 4 1 x ? y ls …'' h' > 39 c' 3<
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SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ o'
OMH^UAS " 8
3< 39
e' 8 5' Lf x 4 y ? 4 5 tma vh`ua kf ls …' ? y x 4 3 x y 8 h' 3 < c' 8> < o' 88 8? e' 8> 1' Lf H4 C2341 tmag -8>H*?C, 1H*|3C 2 …'' h' 945 c' 94? o' 641 e' 94< 6' Lf p 4 q 2 3 4 ? hge x 4 y 2 5 4 1 tmag tma vh`ua kf 1p x * ?qy 4 8>p x * 5qy ls h' 98 4 =3 c' 39 4 3= o' 89 4 3= e' gkga kf tmasa 9' 3hx ? yx Lf h 4 c ? 4 5 hge x 4 y 1 4 3 tma vh`ua kf ''''' 5hy cx h' c' o' e'
?43 ?48? 394?5 35488
Uypa ‟94 _lbp`a prkc`ab subs kg rhtlk ' 8' Hg h``ky ls tk okgthlg okppar hge zlgo lg tma rhtlk kf <45' Uma zlgo raqulrae tk ba`t wltm 35 ji kf okppar ls …'' ji 3 h' 8> ? 8 c' 8> ? 3 o' < ? e' < Yalim Yalimtt kf okpp okppar ar < _k`utlkg4 Yalim limt kf zlgo lgo 5 35 < <6 ?3 3 8> ji' <6 2 < x x 5 x < ? ? 3' Alimt pakp`a hra p`hgglgi tk smhra aquh``y tma okst kf h ragth` ohr' Lf kga parskg wltmerhws frkb tma hrrhgiabagt hge tma ktmars smhra aquh``y agtlra okst kf tma ohr tmag tma smhra kf ahom kf tma rabhlglgi parskgs lgorahsae cy …' -Bhrom 3>>9'
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ h' c' o' e'
8 < 8 = 8
9 9 =
_k`utlkg4 @at tma ragt ca = x 9 2 Ss'16' Sagt smhra fkr ahom kf 9 pakp`a 2
16 9
Sagt smhra fkr ahom kf = pakp`a 2
2Ss'=
Sagt smhra kf ahom parskg lgorahsae 2
16 =
2Ss'9
Sagt smhra lgorahsae cy 2 = ‟9 2 Ss'8'
8
9 ?' Shtlk kf ahrglgi kf H hge C ls 549' Lf tma ahrglgis kf H lgorahsa cy 1>! hge tmksa kf C eaorahsa cy 31!, tma gaw rhtlk kf tmalr ahrglgi caokbas =49' Ymht ls H“s ahrglgi; -Huiust 3>>9,' h' Ss'38>>> c' Ss'36>>> o' Ss'3=>>> e' Ehth lgheaquhta
_k`utlkg4 H4C 2 549
C“s gaw ahrglgi 2 9 x x
@at tmalr ahrglgis ca 5 x 9 x '
91 8>>
38 x
H“s gaw ahrglgi 2 5 x x
Gkw rhtlk kf gaw ahrglgis 2 6 x x
5
6 38 x
81> 8>>
6 x '
= 9
Hs ehth ls lgheaquhta, H“s ahrglgi ohggkt ca fkuge'
Uypa ‟=4 ^rkc`abs kg elstrlcutlkg' 8' Elvlslkg kf Ss' ?35 catwaag ] hge \ ls lg tma rhtlk 88 4 9' ] & \ wku`e iat Supaas h' -3>5 83>, c' -3>> 835, o' -8=> 855, e' gkga kf tmasa _k`utlkg4 Flrst batmke 4 @at x4y 2 8849
88j 9j ?35
x 2 88j hge y 2 9j wmara j ls j ls okbbkg bu`tlp`a…
8=j ?35
y 9j 9x8 9x8= Ss'836
j
?35 8=
8=
Gkw x*y2?35'
Frkb -8, x 88j 88x 88x8= Ss' Ss'8<= 8<= hge hge
88 9 4 -Elvlelgi cy 88*928=,' Gkw 8= 8= 88 9 x?3529x=2Ss'836 Varlflohtlkg4 Kg varlfylgi lg x x?35 x?35 88x 88x8= Ss'8<= 8<= hge hge y 8= 8= kptlkg -h, -c, -o, gkga ls elvlslc`a cy 88'
_mkrtout4 x 4 y 88 4 9
3' Hghge ahrgs Ss' => lg 9 mkurs hge ^rkbkea Ss' <> lg 83 mkurs' Uma rhtlk kf tmalr ahrglgis ls h' ?3 4 38
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
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5'
1'
6'
OMH^UAS " 8
c' 3? 4 83 o' = 4 < e' gkga kf tmasa Uma hgi`as kf h trlhgi`a hra lg rhtlk 34 94 88' Uma hgi`as hra h' -3>¾ 9>¾ <>¾, c' -?>¾ 9>¾ =>¾, o' -8=¾ 6?¾ <<¾, e' gkga kf tmasa Uma rhtlk kf twk gubcars ls 94 8> hge tmalr elffaragoa ls 8>1' Uma gubcars hra h' -3>> ?>1, c' -8=1 3<>, o' -351 ?1>, e' gkga kf tmasa Uma rhtlk kf twk gubcars ls <41 hge tmalr sub ls 335' Uma gubcars hra…' h' 8>> 835 c' 81>95 o' 855=> e' 3>>35 Uwk gubcars hra lg tma rhtlk kf 34? hge tma elffaragoa kf tmalr squhras sl ?3>' Uma gubcars hra…' h' 838= c' 8635 o' 8538 e' Gkga kf tmasa
Uypa ‟<4 Uk flge wmht smku`e ca heeae kr suctrhotae tk gubarhtkr hge eagkblghtkr tk bhja skba rhtlk kr vloa varsh' 8'
Uma gubcar wmlom wmag suctrhotae frkb ahom kf tma tarbs kf tma rhtlk 8< 4 ?8 raeuolgi lt tk 8 4 5 ls h' 81 c' 1 o' 8 e' gkga kf tmasa _k`utlkg4 @at x ca suctrhotae frkb gubarhtkr hge eagkblghtkr' 8< x
3'
?'
?8 x x 81'
8 5
96 5 x ?8 x
Varlflohtlkg 4
8< 81 ?8 81
96 ?8 5 x x 5
86
51 ? x
8 5
Uma gubcar wmlom wmag suctrhotae frkb ahom kf tma tarbs kf tma rhtlk 3<4?9 raeuolgi lt tk ?45l' h' 3 c' ? o' 5 e' 1 Uma gubcar wmlom wmag heeae tk ahom kf tma tarbs kf tma rhtlk 3<4?9 raeuolgi lt tk 541 ls… h' 3
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
5'
1'
6'
9'
c' ? o' 5 e' 1 Uwk gubcars hra lg tma rhtlk 34 ?' Lf 5 ca suctrhotae frkb ahom tmay hra lg tma rhtlk ? 4 1' Uma gubcars hra h' -8635, c' -56, o' -3?, e' gkga kf tmasa Uwk gubcars hra lg tma rhtlk kf ?41' Lf 83 ls heeae tk ahom tmay hra lg tma rhtlk 34?' Uma gubcars hra h' 8131 c' 8=?> o' ?66> e' 355> Uwk gubcars hra lg tma rhtlk kf 94=' Lf ? ls heelgi tk ahom kf tmab tmag tmalr rhtlk caokbas =4<' Uma gubcars hra …' h' 8586 c' 3539 o' 3835 e' 868= Lg 5> `ltras blxtura kf i`yoarlga hge whtar tma rhtlk kf i`yoarlga hge whtar ls ?48' Uma quhgtlty kf whtar heeae lg krear tk bhja tmls rhtlk 348 ls…' @ltras h' 81 c' 8> o' = e' 1
_k`utlkg4 I`yoarlga4 whtar 2 ?48 2 ?>48>' @at x `ltras `ltras kf whtar ca heeae' 2 3> * 3 x
='
8> 2 3 x
?> 8> x
3 8
?>
x 21'
Uma hias kf twk parskgs hra lg tma rhtlk 149' Alimtaag yahrs hik tmalr hias wara lg tma rhtlk kf =48?, tmalr prasagt hias -lg yahrs hra …' h' 1>9> c' 9>1> o' 5>16 e' gkga kf tmasa <' Ymht bust ca heeae tk ahom tarb kf tma rhtlk 5<46= sk tmht lt caokbas ?45; h' ? c' 1 o' = e' < 8>' Uma stueagts kf twk o`hssas hra lg tma rhtlk 149 lf 8> stueagts `aft frkb ahom o`hss tma rabhlglgi stueagts hra lg tma rhtlk kf 546, tmag tma gubcar kf stueagts lg ahom o`hss ls … h' ?>5> c' 3135 o' 5>6> e' 1>9>
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ Uypa ‟8>4 Blsoa``hgakus subs4 8'
Lf h'
3'
?'
x x 3 y 8 ?
y
3x y
tmag rhtlk …''
"8
8 c' 8 ? o' ?8 e' "?"8 Hghge ahrgs Ss' => lg 9 mkurs hge ^rkbkea Ss' <> lg 83 mkurs' Uma rhtlk kf tmalr ahrglgis ls h' ?3 4 38 c' 3? 4 83 o' = 4 < e' gkga kf tmasa => Ss'+mr Hghge)s ahrglgi => 83 ?3 9 _k`utlkg4 2 2 x 2 ^rkbkea)s ahrglgi <> 9 <> 38 Ss'+mr 83 Uma rhtlk catwaag tma spaaes kf twk trhlgs ls 94 =' Lf tma saokge trhlg rugs 5>> Jbs' lg 1 mkurs tma spaae kf tma flrst trhlg ls h' 8> Jb+mr c' 1> Jb+mr o' 9> Jb+mr e' gkga kf tmasa
_k`utlkg4
_paae kf flrst trhlg _paae kf saokge trhlg
2
x jb
5>> jb 1
5'
mr
9 =
x
=>
x
9 =
x=>29> jb+mr
mr
Ehl`y ahrglgis kf twk parskgs hra lg tma rhtlk 541 hge tmalr ehl`y axpagsas hra lg tma rhtlk 94 <' Lf ahom shvas Ss' 1> par ehy tmalr ehl`y lgokbas lg Ss' hra h' -5> 1>, c' -1> 5>, o' -5>> 1>>, e' gkga kf tmasa _k`utlkg4 Ahrglgis kf H4C 2 541 2 5 x 4 1 x Axpagsas kf H4C 2 94< 2 954<5 Gkw ahrglgis ‟ axpagsas 2 shvlgis Fkr C 1 x < y 1> …
-?,
… -8 - x x okbbkg bu`tlp`a, …
-3, -y okbbkg bu`tlp`a,
Fkr H 5 x 9 y 1>
…
-3,
Bu`tlp`ylgi -3, cy < hge -?, cy 9
?6 x 6? y 51> ?1 x 6? y ?1>
x 8>> Frkb -8, tmalr lgokbas hra 5 x 2 5>> hge 1 x 2 1>>' _mkrt out4 Kg Kg varlflohtlkg 5>>41>> hra lg tma rhtlk kf 541' Umalr axpagsas -5>>"1>, 4 -1>>"1>, 2 ?1>451> 2 94<'
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ 1'
6'
OMH^UAS " 8
Bkgtm`y lgokbas kf H hge C hra lg tma rhtlk kf 54? hge tmalr axpagsas hra lg tma rhtlk kf ?43' Lf ahom kf tmab shvas Ss'6>> avary bkgtm, tmag tmalr hgguh` lgokbas hra … h' Ss'8=>> Ss'35>> c' Ss'35>> Ss'8=>> o' Ss'83>> Ss'<>> e' Ss'86>> Ss'83>> ^ X hge S hra tmraa oltlas' Uma rhtlk kf hvarhia tabparhtura catwaag ^ hge X ls 884 83 hge tmht catwaag ^ hge S ls <4 =' Uma rhtlk catwaag tma hvarhia tabparhtura tabparhtura kf X hge S ls h' 33 4 39 c' 39 4 33 o' ?3 4 ?? e' gkga kf tmasa _k`utlkg4 4X 2 88483 2 <<48>= -Bu`tlp`ylgi cy < 4 ^4S2 < 4 =2 << 4 == -Bu`tlp`ylgi cy 88,
9'
^4 X4 S 2 << 4 8>=4==
X4S 2 8>=4== 2 39433'
Lg h fl`b smkktlgi H hge C raoalvae bkgay lg h oarthlg rhtlk hge C hge O h`sk raoalvae tma bkgay lg tma shba rhtlk' Lf H iats Ss'8 6>>>> hge O iats Ss'3 1>>>>' Flge tma hbkugt raoalvae cy C' h' Ss'3>>> c' Ss'31>>>> o' Ss'8>>>>> e' S_'81>>>> _k`utlkg4 Mara H4C 2 h 4 c h3 4 hc -bu`tlp`ylgi cy h, C4O 2 h 4 c hc 4 c 3 -bu`tlp`ylgi cy c,
='
H4C4O 2 h3 4 hc 4 c3 Mara h3 86>>>> hge c3 31>>>>
h 2 5>> hge c 2 1>>
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SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
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8 2h
g b
wmara g b a'i' x1 x
='
9
8 x
91
8 3
x
-h b ,g hbg a'i' ?
3 ? 8 x ? 5 ? 5 x x x 3 3
b b b <' -hc, h c a'i'
-3 x,1 31 x1 ?3 x1 ' b
5
5 h bcb =8 h ? ? =8 8>' a'i' 5 5 x x c x
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ 88' Lf h x h y tmag x y a'i' 3 x 35 tmag x 2 5' 83' Lf hg cg hge h c tmag g2> a'i' lf 3 2 ? tmag g2>' g
g
3
3
‟3, 8?' Lf xh yh tmag lt ls gkt gaoasshry tmht x y a'i' -3, 2 - ‟ 8
85' Lf x y tmag x y g g
Uypa ‟84 Chsae kg slbp`a axprasslkg4
Bu`tlp`a Omkloa Xuastlkgs 8'
5x ‟8+5 ls axprassae hs h' ‟5x 8+5 c' x ‟8 o' 5+x8+5 e' gkga kf tmasa 8
5 x 5 _k`utlkg4
3'
Uma vh`ua kf h' c' o' e'
5 8
x 5
> 3+? 8 gkga kf tmasa
>
3 p 3q? _k`utlkg4 8 xy ? ?'
Ymlom ls Urua ; > > h' 3 7 -8+3, > > c' 3 0 -8+3, > > o' 3 2 -8+3, e' gkga kf tmasa
5'
?8 x 3
h' c'
'''' x 3 ? 8
? x 3 3 o' ? x ? e' x 3
wmara p q x y ≠ > ls aquh` tk
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
OMH^UAS " 8
Uypa ‟34 Avh`uhta 8'
‟8+=
Uma vh`ua kf 3-316, h' 8 c' 3 o' 8+3 e' gkga kf tmasa _k`utlkg4 3-316,
3'
8=
ls
1'
6'
9'
= x
3-3 , 3x 3
_k`utlkg4 3 '5 3 -3 ,
5'
8 =
8 =
3x 38
3 3
8
3² '5· ls aquh` tk h' h frhotlkg c' h pksltlva lgtaiar o' h gaihtlva lgtaiar e' gkga kf tmasa 8
?'
=
3
?
5
8
3
3
?
8 5
3 x3 3
3x
? 5
8
?
3 3'3 3 3
8 ? 3 3
8+?
Uma vh`ua kf = ls h' ? 3 c' 5 o' 3 e' gkga kf tmasa 8+1 Uma vh`ua kf 3 Ù -?3, ls h' 3 c' 8> o' 5 e' gkga kf tmasa Uma vh`ua kf 5+-?3,8+1 ls h' = c' 3 o' 5 e' gkga kf tmasa 8+? Uma vh`ua kf -=+39, ls h' 3+? c' ?+3 o' 3+< e' gkga kf tmasa
x,y,z; Lf 3 x x ? y x1z ?6> tmag wmht hra tma vh`uas kf x,y,z;
h' c' o' e'
?38 83? 3?8 8?3 3 ?6> 3 8=>
_k`utlkg4
3 <>
?6> 3? x?3x18 3 x x? y x1 z ? 51 x ? y 3 z 8 ? 81 1 1 8
33 5 ' Ymlom ls hg lgtaiar'
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ Uypa ‟ ?4 _lbp`a subs kg slbp`lflohtlkg 8'
Uma slbp`lflae vh`ua kf 86x ‟?y3 Ù = ‟8x?y ‟3 ls h' 3xy c' xy+3 o' 3 e' gkga kf tmasa ? 3
8 ?
_k`utlkg4 86 x y x= x y 3'
_lbp`lflohtlkg kf h' c' o' e'
x
x b?g x x 5bhg x
h' c' o' e'
x x 3 3 = y
ls …''
6 b6 g
x b?g5 b< g
6 b6 g
x1 b6 g 6 b6 g
3 ?
? 3
-? , x-5 x-5 , x-1 ,
-?3 ,? x-5? ,3 xx-13 ,?
x1b6 g6b6 g x b
6
2
6
28
?6 x56 x16
3
xy 3 xy 3
8
=8 x 5 ? x x _k`utlkg4 = 8 y - y = , 5 h ‟c
Uma vh`ua kf y h' h*c y c' y o' 8 h*c e' 8+y
c ‟o
Ùy
5
o ‟h
Ùy
‟c
Ù xc o Ù xo h ls aquh` tk ] 8 > gkga kf tmasa ‟
‟
8 5
h ‟c
Ù y ‟
_k`utlkg4 y hc x yco xyoh xyhc xh a' f' i' m'
6
? x5 x1
mhs slbp`lflae vh`ua aquh` tk
5
6'
?
x x {-??,3 Ù -53,? Ù -1?,3} + {-?3,? Ù -5?,3 Ù -13,?} ls h' ?+5 c' 5+1 o' 5+9 e' 8
_k`utlkg4
1'
x
x?
8
x b x g x g
? 3
5'
86 x
y 3
b
_k`utlkg4 ?'
3
?x y
3
? xy3
ls
2 y hccooh x
8 y
hc
2 y > x
8 y
hc
8 y
h c
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ 9'
raeuoas tk
a' f' i' m'
='
x
_mkw tmht x 8 > ‟ 8 Gkga
_lbp`lflohtlkg kf h' c' o' e'
x
b 3 g ? b= g
x
x
ls …''
1 b6 g
b
x x g
x b g x
Uypa ‟ 54 Mhrear subs kg slbp`lflohtlkg 8'
Uma vh`ua kf h' c' o' e'
3'
8 > 3 gkga kf tmasa
_mkw tmht h' c' o' e'
> ‟ 8 ? 8
x8 _k`utlkg4 _k`utlkg4 c x
h c
2 x- hc,-hc , x x-co,-,-co , x x -oh ,-, -o h ,
?'
Kg slbp`lfl slbp`lflohtlkg ohtlkg h' ? c' ‟ ? o' ‟8 ? 8 e' ?
5'
_mkw tmht h' c' o' e'
1'
8 > ‟ 8 Gkga
_mkw tmht h' "8 c' > o' 8
co
xc x o x
o h
xo x h x
h c h c x-xco ,co xx-xo h ,o h 2 - x , x-
2 h3 c3 c3 o3 o 3 h 3
x*y
x
y*z
c*o*h
o*h*c
x
h*c*o
x*z
x
raeuoas tk
>
2 x 2 x 28
b raeuoas tk
ls ilvag cy
x
ls ilvag cy
OMH^UAS " 8
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ e' Gkga
6'
8
_mkw tmht x hc
h' c' o' e'
‟ 2 x ‟
='
<'
8 ? ‟ 8 Gkga
h' 8
e'
8>'
o' e'
o' e'
83'
ls ilvag cy
8
8
8
x
8 -c o,-h c,
8 -o h,-c o,
>
2 x
2 x 2 8
x
raeuoas tk
x
x
x
x
---,, - - , ---,, ---,,
88' Kg slbp`lfloh slbp`lflohtlkg tlkg h' c'
x x o h
8 oc
-h c,-o h,
Kg slbp`lfl slbp`lflohtlkg ohtlkg h' c'
8
8 > "8 gkga kf tmasa
_mkw tmht
o'
8 c h
- hc ,- ho , -co ,-ch , -o h ,-o c ,
-h c,-c o,-o h ,
Uma vh`ua kf
c'
c o o h h c
_mkw tmht
h' c' o' e'
8 c o
> 8 ‟ 8 Gkga
_k`utlkg4 2 x
h' c' o' e'
x x
8
9'
8 ho
1
9
h*c
x
c*o
x
h*c
x
c*o
x
? 3 < 3 x< ls aquh` tk… < ? ? h' 8 c' o' ?
? ?
raeuoas tk
raeuoas tk
ls raeuoas tk
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ e'
? < ? 1
9
3
3
1
? < _k`utlkg4 < ? ? x < 2
8 9 9 1
?3
9 31
?+5
8?' Z{-3, -5, ' -=, h' H frhotlkg c' hg lgtaiar o' 8 e' gkga kf tmasa
1+6
9+=
' -86,
3
3
5
8 5 91
?
5 5
8+3 '
8
2
x?
9
? ? 3 3 2 3 x? 8+ 3 ? ?x? 8+ 3
2
x ?
5
<+8>
' -?3,
?+31
}5R
?5 5
?
2
?1+ 5 1
?
x
?9 9+ 3
9+ 5
? x?
x?3
8
ls
1 9 < ? 83 83 3 ? 5 ? 1 6 5 9 = 1 <8> 5 6 = 8> 5 _k`utlkg4 -3, '-5, '-=, '-86, '-?3, 2 3 '-3 , '-3 , '-3 , '-3 ,
8 ? 1 9 < 2 3 3' 3 3' 3 3' 3 3' 3 3 5
?
31
8 ? 1 9 < 5 2 3 3 3 3 3 3
?
31
23
31 ? x5x 3 31
236 Ymlom ls hg lgtaiar
85' Uma Urua kptlkg ls 3+?
3
h' x 2 ? x 3+? ? c' x 2 x 3+? 3 o' x 7 ? x 3+? 3 e' x 0 ? x ‟ ‟ 81' Uma vh`ua kf -=+39, 8+? Ù -?3+35?, 8+1 ls h' <+5 c' 5+< o' 3+? e' gkga kf tmasa 86' _lbp`lflae vh`ua kf -831,3+? Ù 31 31 Ù ? 1? Ù 18+3 ls h' c' o' e'
1 8+1 8 gkga kf tmasa
--x x ‟y, } ls
6
3
89' Uma vh`ua kf {-x*y,3+? -x ‟y,?+3+ x*y x*y Ù 3 h' -x*y, c' -x ‟y, o' x*y e' gkga kf tmasa
? 6
?
- x y , - x y, ? 3 - x y , ? x6 - x y , 3 x6 _k`utlkg4 8 ? ? x y x y , - x y , 3 x 6 - x y , 3 x 6 3
8=' Uma vh`ua kf h' > c' 313 o' 31>
?
x
ls
2
- x y,5 - x y,
x y
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ e' 35=
8<' Uma vh`ua kf h' c' o' e'
8 "8 > Gkga
x
x
x
ls
Uypa ‟ 14 Chsae kg j batmke' 8'
Lf x8+p 2 y8+q 2 z8+r hge xyz 2 8 tmag tma vh`ua kf p*q*r ls h' 8 c' > o' 8+3 e' gkga kf tmasa _k`utlkg4 @at x
3'
8 p
y
8
z
q
8
r
j
x
8 p
j y
x j p y j 3 hge z j r Gkw xyz 8
8
q
j hge z
j p 'j q 'j r 8
8
r
j j pqr j >
p q r >
Lf hx 2 c cy 2 o oz 2 h tmag xyz ls h' 8 c' 3 o' ? e' gkga kf tmasa _k`utlkg4 h x c c y o oz h
-h x , y o
h xy o
z xy z xyz 8 o h h h h h xyz 8 , Gkw
?'
Lf 3x 2 ?y 2 6"z h' c' o' e'
ls
8 > 3 gkga kf tmasa
_k`utlkg4 @at 3 x ?y 6 z j 8 x
8
Gkw 3 x ? 2 6 j 'j
5'
8
8
x
y
8
8
8
z
x
y
8
j
8 8 3
j
x y
Lf 3h 2 ?c2 -83,o tmag
8
> z
8
j z
xf '5 x 1 y 3>z tmag z ls aquh` tk h' xy x y c' xy 8 o' xy xy e' x y
1'
y
8
3 x j ?y j hge 6 z j 3 j x ? j
raeuoas tk
8 y
8
hge 62j 3
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
6'
9'
='
h' 8 c' > o' 3 e' Gkga h c o Lf ? 21 2-91, tmag tma vh`ua kf hc"o-3h*c, raeuoas tk h' 8 c' > o' ? e' 1 h c o Lf 3 2? 2-83, tmag tma vh`ua kf hc"o-h*3c, raeuoas tk h' > c' 8 o' 3 e' ? p
q
r
s
Lf h 2c 2o 2e hge hc 2 oe tmag tma vh`ua kf h' c'
o' > e' 8
<'
Lf 3h25c2=o hge hco23== tmag tma vh`ua h' c' o' e'
raeuoas tk
ls ilvag cy
_k`utlkg4 3h 5c =o h 33c 3?o `at h 3c ?o j h j 3c j hge ?o j j j j j Gkw hco hco 3== j ' ' 3== j ? 893 893= 83? j 83 c hge o 3 ? 3 ? 8 8 8 8 8 8 j 83 j 83 h j 83 c * * 6 hge o 5 Gkw 3 3 ? ? 3h 5c =o 3x83 5x 5x6 =x5 5 5 ? 88
<6 <6 x 8>' Lf-1'69=, 2->'169=,y28>z tmag h' c' o'
e' Gkga
x
_k`utlkg4 1'69= ->'169=>, y 8> z -1'69=, x 8>z hge ->'169=, y 8>z
z
z
x
y
1'69= 2 8>
8 z
8 x
8 y
hge >'169= 2 8>
8 x
8 y
8
x
1'69= >'169=
8> z z y
8>
> - elvlelgi cy z,' z
z
8> 8>
z
x y
8
z x
z y
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
OMH^UAS " 8
Uypa ‟ 64 Chsae kg axphgslkg + fhotkrlzhtlkg4 8'
8 8 8 8 8 8 8 8= = = = 5 5 3 Vh`ua kf h h h h h h h h 3 ls
h
h'
8 h 8
c' h h h3
o'
8 h3 8
3 e' h h 3
8 8 8 8 8 8 8 8= = = = 5 5 3 3 _k`utlkg4 h h h h h h h h 8 8 8 8 8 8 8 8 8 3 8 8 85 = = 5 5 3 3 5 5 5 3 h h h h h h h h h h h h 3 3 3 8 8 8 8 8 8 8 8 8 3 8 3 8 8 3 3 3 3 5 5 3 3 h 3 3 h h h h h h h h h h h h h
3'
x
g
3
g
g8
g 8
y 3 x3 y3
'''
h' x3 y 3 c' x3 y 3 o' x3 y 3 e' x3 y 3
?'
g g g x 8
h' c' o' e'
5'
8 g8
'''
x g g x 8 x g8 Gkga kf tmasa
c hhc c h x x Kg slbp`lflohtlkg h x c x hc x ch
h' c' o' e'
h c
raeuoas tk
8 ‟ 8 > Gkga
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h c
hhc hcc x h c x hc hc
h c
hhcc x hc x hc
h c
x x
hc
8hc 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
OMH^UAS " 8
? ? ? ?hco h c o h c o > tmag Uypa ‟ 94 Chsae kg lf
8'
Lf x 2 ? * ? " tmag h' 81
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c' 8> o' 83 e' gkga kf tmasa 8
? _k`utlkg4 x ? ?
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8
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h' c' o' e'
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8
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Lf h' c' o' e'
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prkva tmht
83 8? 81 89
"83x ls ilvag cy
prkva tmht 1x? "81x ls ilvag cy
tmag h?"?h ls
1'
Lf x2 h' 31 c' 36 o' 39 e' ?>
6'
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9'
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∔ ∔
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tmag tma vh`ua kf h *?h"3 ls
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ Uypa ‟ =4 Blsoa``hgakus axhbp`as4 8'
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8 x 8 x3 3
3'
8
8
3 8 8 3
8 8 8 3 8 x
8
x 8 x 8 3 x3 x 3
3
3
3
8
8 3
8 x 3 8 8 3 8 x
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8 3
8 3
8
3x
8
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8
z c
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h3
co
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SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ 88' Uma vh`ua kf z ls ilvag cy tma fk``kwlgi lf z h' 3
c' o' e'
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8
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8?'
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x ?
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2 39 raeuoas tk
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SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ e' Gkga
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SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ Uypa ‟ 84 Chsae kg rhtlkgh`lzhtlkg4
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2
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OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ o' e'
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OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ @L_U KF FKSBQ@HA 8' h x g `kih g x a'i' -l, ? 2 39 tmag `ki ? 392? ?
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SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
OMH^UAS " 8
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xy x y ?'
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SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ Uypa ‟ 34 @hw kf Xuktlagt 8'
3'
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8'
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8 5
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8'
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3'
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8
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8 8>
OMH^UAS " 8
OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ o' ‟3 e' gkga kf tmasa
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sk`utlkg4 `ki< ?'
5'
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Uypa ‟ 14 Chsae kg 8'
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`ki y x
3 ? 3 6 gkga kf tmasa
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Uma `kihrltmb kf 38<13 tk tma chsa kf 3 h' Aquh` c' Gkt aquh` o' Mhva h elffaragoa kf 336<
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SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_ 6'
e' Gkga Uma sub kf tma sarlas h' c' o' e' Gkga
ls ilvag cy
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8'
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3'
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OMH^UAS " 8
SHULK HGE ^SK^KSULKG LGELOA_ @KIHSLUMB_
OMH^UAS " 8
Uypa ‟ =4 Axprasslgi `ki lg xybg 8'
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3'
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8 3
8
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3'
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