Alta matematica-Nichita-Stănescu
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Transcript of Alta matematica-Nichita-Stănescu
5/17/2018 Alta matematica-Nichita-Stănescu - slidepdf.com
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Eagoaph Wpćefwgv ‗ veasf~wvd zbfpag
‑‑Dbkagh zbf}afa fwpf efdbkagć dh lb`vd wvndal‑‑
Hdfxhe`~v Lhgf`bewia
/78 lh~% 8177, `% 87 `fg% 81?7!
Eagoaph Wpćefwgv fwpf vevd `aep~f gfa lha lh~a g~fhpb~a `f dapf~hpv~ć ha ^blîeafa% Ećwgvp dh
ZdbafŒpa åep~ ,b chladaf `f aepfdfgpvhda$ v~lfh}ć dagfvd `ae dbghdaphpfh ehphdć zfep~v gh lha hzba wć pf~laef
Chgvdphpfh `f Cadbdbkaf `ae Nvgv~fŒpa% @fnvpfh}ć f`apb~ahd gv sbdvlvd `f zbf}aa —Wfewvd avna~aa“ åe 813:
z~ae aepf~lf`avd gć~vah ~fvŒfŒpf wć ~fhda}f}f b wzfgphgvdbhwć lapbdbka}h~f h dvlaa%
@fnvpvd fwpf v~lhp `f hdpf sbdvlf= B sa}avef h wfepalfepfdb~ /8130!$ 88 fdfkaa /8133!$ Bvd wa
wcf~h /813?!$ Dhvw Zpbdflhfa /813?!$ Efgvsaepfdf /8131!$ Åe `vdgfdf wpad gdhwag /81;:!$ Fzagh lhkeh
/81;?!$ Bzf~fdf alzf~cfgpf /81;1!$ Eb`v~a Œa wflef /81?>!%
Eagoaph Wpćefwgv w,h alzvw gh vevd `aep~f gfa lha aepf~fwheŧa Œa shdb~bŒa g~fhpb~a `f zbf}af
lb`f~eć9 gh b ~fgvebhŒpf~f h shdb~aa g~fhŧafa whdf$ a w,h gbecf~ap åe 81;6 dh Safeh$ z~flavd Of~`f~$ ah~ åe
81?>$ åe Avkbwdhsah$ z~flavd ‑‑Gveveh `f Hv~‑‑ %
Ghdfh gh~f wć,a whpawchgh efsbah `f wakv~heŧć `vgfh wz~f `awgazdaefdf ~akv~bhwf hdf wza~apvdva$ wz~f
kfblfp~af wa ca}agć% G~apagh dapf~h~ć gbewa`f~h gć `f dh Abe Nh~nv$ zbf}ah ~blîefhwgć ev h lha gvebwgvp b
hwflfefh ghzhgaphpf ̀ f hnwp~hgpa}h~f gh åe gh}vd dva Eagoaph Wpćefwgv%
Zbfpvd w,h wapvhp åep~,b hwgfe`feŧć .ebnadć.= z~ae pfe`aeŧh `f h hmvekf zîeć dh fwfeŧh da~agvdva$ fd wf
hz~bzaf `f Flaefwgv Œa Ndhkh9 z~ae cb~ŧh aebshŧafa dh easfdvdva dalnhmvdva h~pawpag$ wf hz~bzaf `f H~kof}a9
z~ae ghzhgaphpfh `f åegac~h~f h lfwhmvdva åe cb~lvdf `f lhxalć hnwp~hgpa}h~f$ wf hz~bzaf `f Abe Nh~nv%
G~apagvd Fvkfe Walabe `fcaefŒpf zbf}ah dva Eagoaph Wpćefwgv .zbf}ah zbf}afa.% Wpćefwgv åewvŒa hca~lć=
.zbf}ah ev fwpf evlha h~pć$ fwpf aewćwa sahŧh$ aewvŒa wvcdfpvd safŧaa.%
@f,h dvekvd åep~fkaa whdf g~fhŧaa wf `awpaek åegf~gć~a `f h `fcaea gbegfzŧah wh `fwz~f gvebhŒpf~f z~ae h~pć$
`fwz~f fgoadan~vd `aep~f gbeŧaevpvd Œa cb~lh zbf}afa$ `fwz~f ~fdhŧah zbfpvdva gv gvsîepvd% Zbf}aa zf hgfhwpć
pflć= .H~w zbfpagh.$ .H~ph zbf}afa.$ .H~ph zbfpagć.$ .H~ph wg~awvdva.$ .Hvpbzb~p~fp.$ .Zbf}ah.$ .Zbfpvd gh
Œa wbd`hpvd.$ .Gćvph~fh pbevdva.$ .Zbfpvd.$ .Pbevd.$ .Paln~vd.%
Eagoaph Wpćefwgv `flbepfh}ć fdflfepfdf h~pfa$ zf gh~f df ~fhŒfh}ć `vzć b b~`aef z~bz~af% Åe zbflvd.Eb` 77.$ zbfpvd åegfh~gć wć dćlv~fhwgć$ wć `fwgac~f}f hgfhwpć b~`aef z~bz~af= .Hl kîe`ap ve lb` hpîph
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`f `vdgf " @f h a}na `bvć gvsaepf " @f zh~gć ah~nh sf~`f h~ aecdb~a " Ah~ cdb~adf w,h~ af~na%.%
Z~alh fphzć , Da~agh wfepalfepfdb~$ h`bdfwgfepaeć$ åe gh~f z~f`blaeć a`fhdv~adf ~blhepagf$
wvnafgpvd gvebhŒpf~aa fwpf fvd da~ag åe mv~vd gć~vah wf ga~gvlwg~af dvlfh ~fhdć%
Åe z~alfdf `bvć sbdvlf /.Wfewvd avna~aa.$ .B sa}avef h wfepalfepfdb~.!$ lhmb~aphpfh zbf}aadb~ wvep `f
`~hkbwpf$ f~bwvd fwpf ve z~adfm `f h gblveagh c~ćlîephpfdf åephlzdć~a hdf caaeŧfa aepf~abh~f$ gh~f wvep lha
zvpf~eagf `fgîp hdf gbe`aŧafa ca}agf `hpf%
Avna~fh z~b`vgf b wfe}hŧaf `f zdvpa~f$ }nb~vd wf ehŒpf `ae `~hkbwpf gh b gbdbheć h aecaeapvdva åep~f zćlîep
Œa gf~%
Åe zbf}ah .Gîepfg.= .@v,lć$ cf~aga~f$ åe wvw Œa a}nfŒpf,la " Pîlzdh `f wpfdf$ zîeć gîe` " Dvlfh lfh
z~fdvekć Œa åe efwcî~Œa~f " Wf chgf gbdbheć whv hdpgfsh " Lvdp lha åehdp Œa lvdp lha gv~îe`%.$ wph~fh `f
avna~f f wph~fh `ae hch~h aegf~papv`aeaa$ fwpf .gîepfgvd `f a}nîe`ć h wvefpvdva Œa h dvlaeaa.% Avna~fh fwpf b
wph~f da~agć gblzdfxć$ fwpf b afŒa~f `ae wble$ lbpasvd cve`hlfephd caae` gfd hd ~hwć~apvdva= .Wbh~fdf whdpć
`ae dvg~v~a wp~akîe` " Gdhpaeć lvgoaadf wv~`f Œa k~hsf%. /.B ghdhpb~af åe }b~a.!%
H `bvh fphzć , Da~agh efgvsaepfdb~$ lhpv~apćŧaa g~fhpbh~f åegfzf gv sbdvlvd .@~fzpvd dh palz.%
Sa}abehdawlvd dva Eagoaph Wpćefwgv wf hnwp~hgpa}fh}ć$ sa}avefh hwvz~h wzhŧavdva Œa palzvdva wf wgoalnć%
Sbdvlvd .88 Fdfkaa. fwpf gbewa`f~hp gfh lha nveć gh~pf h dva Eagoaph Wpćefwgv% Zbfpvd ̀ f}sbdpć
ve evlć~ `f ~hzb~pv~a gh~f `fdalapfh}ć zbf}ah Œa fxawpfeŧh zbfpvdva% Zvegpvd `f zdfgh~f fwpf g~a}h `f
ehpv~ć fxawpfeŧahdć zf gh~f zbfpvd åegfh~gć wć b `fzhŒfhwgć z~ae lf`aphŧaf% Wpadvd hgfwpb~ zbf}aa fwpf
wbdfle wa f~lfpag$ fx% .Fdfkah aephah.= .Pbpvd fwpf aesf~wvd pbpvdva " Wzvef Ev `bh~ hgfdh " Gh~f åd Œpaf zf
@h " Åewh fd gh~f Œpaf pbpvd " Dh Ev Œa dh @h h~f cbadf ~vzpf. /Ev zbpa efkh gfsh gf ev gvebŒpa9 zf lhwv~ć
gf gvebŒpa lha lvdp fŒpa lha gbeŒpafep `f ~fdhpasaphpfh dvg~v~adb~ whdf!%
@f chzp$ fdfkaadf wvep .`fcaeaŧaa. cadbwbcagb,da~agf hdf veb~ gbegfzpf cve`hlfephdf hdf fxawpfeŧfa
zbfpvdva% Åe fdfkaa pflh fwpf wvcf~aeŧh ̀ f `asa}avef$ pîema~fh `vzh veaphpf= .@v~f~f h ~vzf~aa åe `bvć h
dvlaa " Gh wć,la zćp~ve`ć z~ae bgoaa$ `ba " @v~f~f h ~vzf~aa åe `bvć h wvefpfdb~ " Dvlaa$ " Gh wć,la
dbsfhwgć palzhefdf$ `bvć. /.Fdfkah 0.!% ^fchgf~fh veaphŧaa z~alb~`ahdf ev fwpf zbwanadć zîeć gîe` zbfpvd
ev sh Œpa dalnh af~naa$ h wîlnv~adb~$ h wpfdfdb~%
Åe vdpalh fphzć$ evlapć da~agh c~akvdva$ pflh cve`hlfephdć fwpf lbh~pfh% Sbdvlfdf `f zbf}aa `ae
hgfhwpć fphzć wvep .Lć~fŧah c~akvdva.$ .Eb`v~a Œa wflef.$ .Fzagh Lhkeh.$ .Bzf~fdf alzf~cfgpf.%
Lbh~pfh fwpf b fxzf~afeŧć dalapć$ åewfleće` åepbh~gf~fh åe lap$ åe efgvsîep% Lbh~pfh fwpf b åegofaf~f
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lfphcb~agć h hsfepv~aa wza~apvhdf h fvdva zbfpag åe wzhŧavd gvebhŒpf~aa%
G~apagvd dapf~h~ Hv~fd Lh~pae gbewfleh gć veasf~wvd da~agaa dva Eagoaph Wpćefwgv fwpf .b zh`v~f `f
walnbdv~a.% Gh pbŧa lh~aa zbfŧa$ Eagoaph Wpćefwgv h cbwp chwgaehp `f fwfeŧf$ `f gffh gf fxz~alć whv
hwgve`f cfeblfehdvd% Zbfpvd h åegf~ghp wć zhp~ve`ć `aegbdb `f dalaph gvebhŒpf~aa$ `fwgbzf~ae` gć phaefdf
ev wf dhwć åesaewf$ ev,Œa `asvdkć evgdfvd aepal%
Zbfpvd h hsvp åewć sbghŧah `f h zćp~ve`f alzfefp~hnadvd$ `f h,Œa åegoazva ~fhdapćŧa ae`fcaeanadf% @f haga
c~fgsfeŧh åe sbghnvdh~vd wćv h gvsaepfdb~ gbewp~vapf gv z~fcaxvd ef$ hgfwpfh shdb~acagîe` walvdphe b
hca~lhŧaf Œa b efkhŧaf%
Åep~fhkh wh da~agć zfe`vdfh}ć åep~f zbdaa gfep~hda}hpb~a$ åep~f `h wa ev$ åep~f dvlaeh Œa åepvef~ag$ åep~f adv}af
Œa ~fhd$ åep~f wph~f `f sfkof wa wph~f `f saw%
Lb`f~eaphpfh Œa shdbh~fh da~agaa whdf ~f}a`ć åe sa}avefh zbfpagć b~akaehdć Œa åe dalnhmvd zbfpag% Zbfpvd
gbecf~ć gvsîepvdva ae`fzfe`feŧh$ gbewa`f~îe`v,d g~fhpb~ `f dvla= .Evlha gvsfepfdf }nv~hv åep~f eba "
Aehaepf Œa aehzba$ " Sh~pfmvd db~ zvpfh ca hz~bhzf }ć~ap " Y%%%\ " Gh wć z~asfwg ah~nh åegdaehpć `f " gć`f~fh
veva gvsîep.% Dalnhmvd wćv zbfpag fwpf hnwp~hgp$ nh}hp zf lfphcb~ć wa zh~hnbdć%
Åe sbdvlvd .Gh~pfh `f ~fgapa~f.$ zbfpvd `awpaekf p~fa lb`v~a `f h g~fh zbf}ah=
, cbefpag , nh}hp zf wbeb~aphpfh wvefpfdb~$ zf hdapf~hŧaf
, lb~cbdbkag , nh}hp zf lbnadaphpfh gvsîepvdva$ dh cfd gh dh H~kof}a
, waephgpag , nh}hp zf gbewp~vgŧaadf waephgpagf gh ae gh}vd dva Nh~nv%
Eagoaph Wphefwgv fxzf~alfepfh}h ve da~awl wflhepag z~ae gvdpash~fh walnbdv~adb~ fxawpfeŧahdf% Fd
~fhda}fh}ć lvphŧaa `f wfew z~ae gbewp~vgŧaa waephgpagf hzh~fep hnwv~`f$ z~ae wgoalnh~fh shdb~aa k~hlhpaghdf
h gvsaepfdb~ /sć}vd lhaeadb~!%
Dh easfdvd fxz~fwafa zbfpagf$ zbf}ah dva Eagoaph Wpćefwgv fwpf b ŧfwćpv~ć `f eb`v~a Œa wflef% Wflevd fwpf
wfewvd gbesfeŧabehd hd gvsîepvdva$ ah~ eb`vd fwpf evgdfvd wflhepag hd gvsîepvdva zbfpag%
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Hdpć Lhpflhpagć, Eagoaph Wphefwgv
Pfxpfdf dva Eagoaph Wpćefwgv zbp zvef åe chŧh gapapb~vdva wfewv~a gh~f ev wf dhwć vŒb~
`fwgac~hpf% Fwpf Œa gh}vd zbf}afa ‑‑Hdpć lhpflhpagć‑‑%
Åe z~alvd ~îe` papdvd hgfwpfa zbf}aa h~f b wfleacaghŧaf z~bcve`ć% @ae wph~p zbfpvd `fwdvŒfŒpf
chzpvd gć ev fwpf sb~nh `f lhpflhpagh zf gh~f cafgh~f `aep~f eba b gvebhŒpf /lha lvdp whv lha
zvŧae!9 wf bnwf~sć ̀ b~aeŧh fvdva da~ag `f h h~ćph Œa ve hdp lb` åe gh~f wf zbhpf sf`fh dvlfh%
Lhpflhpagh fwpf zf~gfzvpć gh b vefhdpć saphdć zfep~v cafgh~f zf~wbheć$ z~ae aepf~lf`avd fa dvlfh åŒap~ćafŒpf `fwpaevd$ w,h~ zvpfh wzvef gć fh h~f ve ~bd `f aepf~chŧć h caaeŧfa vlhef gv pbp gffh gf fxawpć åe
mv~% B hwpcfd `f lhpflhpagć$ ~aka`ć åe cb~lfdf fa `fcaeapb~aa$ p~fnvaf wgoalnhpć$ `ae zvegpvd `f sf`f~f
hd zbfpvdva% Dvlfh h~f efsbaf `f fdflfepf hnwp~hgpf zfep~v h walŧa gv h`fsć~hp sahŧh% Avna~fh p~fnvaf
hŒf}hpć wvn wflevd veva zbwpvdhp gf p~fnvaf z~bpfmhp z~ae aepf~lf`avd vefa hnb~`ć~a gbkeapas,
wfe}b~ahdf z~bcve` wfepalfephdf% Fvd da~ag `b~fŒpf wgoalnh~fh zf~wzfgpasfa åe gh~f wf hcdć Œa b `hpć gv
hgfhwph h`vgf h~kvlfepf zf~cfgp shdhnadf åe lhpflhpagh wh$ `h~ Œa gbep~hh~kvlfepf åe `fchsbh~fh
lhpflhpagaa zf gh~f cafgh~f `aep~f eba b hzdagćl sbdveph~ whv efaepfeŧabehp z~bz~afa ebhwp~ffxawpfeŧf% Gbewa`f~ gć Eagoaph Wpćefwgv ev `b~fŒpf wć alz~alf gapapb~vdva z~bz~ah wh lhpflhpagć$ ga
lha `fk~hnć aewpakć wzapapvd vlhe `ae cafgh~f åe h,Œa wgoalnh phgpagh z~ae gh~f fd fxawpć gh fdflfepf
wphŧabeh~ whv `aehlag% Z~ae papdvd zbf}afa wf `b~fŒpf b ~fsbdpć h ebhwp~ć ålzbp~ash lhpflhpagaa
hnwv~`f åe gh~f p~ćal9 eba wvepfl ghzhnada `f h ef g~fh ve wawpfl z~bz~av z~ae gh~f zvpfl wfwa}h
dvlfh$ h`agć b hdpć lhpflhpagć zf~cfgp lvdhpć gbewp~vgŧafa ebhwp~f wza~apvhdf$ wfepalfephdf whv gh~f
ŧaef `f fwfeŧć%
Z~alh wp~bcć fwpf ve cfd `f zdf`bh~af h papdvdva% Fvd da~ag ef `fwdvŒfŒpf åekvwpfdf gć~ć~a hdf
lhpflhpagaa åe gh~f fxawpćl$ åe gh~f p~ćal gdazfdf% Wvep h`vwf `~fzp h~kvlfepf h`fsć~v~a gh~f `ae
zćghpf ev wf wapvfh}ć zf hgffhŒa dvekalf `f ve`ć gv z~bz~ah ebhwp~ć caaeŧć/—Eba Œpal gć vev b~a vev
chg vev$" `h~ ve aeb~bk b~a b zh~ć" ev Œpal gîp chgf“!% Ål hgfwp lblfep hd zbf}afa hzh~ fdflfepfdf
`aep~f gfdf lha `awpaegpf$ hzh~ŧaeîe` veb~ `blfeaa cbh~pf `asf~wf hdf safŧaa ~fhdf Œa alhkaeh~f$ hl
zvpfh wzvef$ whv hdf gblzb}aŧafa h~pawpagf `hgć fwpf wć ef ~fcf~al dh wzfgaadf kfev~adb~ dapf~h~f% Wz~f
fxflzdv ―aeb~bkvd“ /`fhdpcfd evlap Œa dagb~e whv veagb~e!$ fwpf ve healhd lapbdbkag$ caae`
~fzf~f}fephp `f ve ghd hdn gf h~f ve gb~e åe lamdbgvd c~veŧaa gć~vah a wf hp~anvaf zvpf~a la~ hgvdbhwf
z~fgvl zv~acagh~fh p~vzvdva whv a}sb~ `f sahŧć% Hgfhwpć ebvć lhpflhpagć wphnadfŒpf cć~ć eaga ve wfle
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`f åe`bahdć b ~fdhŧaf dbkagć$ gf,a `~fzp efbnaŒevapć$ `h~ shda`ć$ `fbh~fgf fwpf z~bz~ah lhpflhpagć% Åe
fh wf zbp åelvdŧaa aeb~bkaa gv zf~fdf$ åewć åe gfh åe gh~f p~ćafwg lhmb~aphpfh bhlfeadb~ b hwpcfd `f
åelvdŧa~f fwpf gbewa`f~hpć gh b hnf~hŧaf% B hdpć hwbgaf~f aef`apć h gvsaepfdb~ wf kćwfŒpf åep~ ,b
zvpf~eagć alhkaef wvkfwpasć ~fz~f}fephpć `f ve eb~ Œa b gb~hnaf9 `f hgfhwpć `hpć fwpf sb~nh `fwz~f
wgć`f~f% Eb~aa åe aecaeaphpfh db~ zf gf~ zbp ~fz~f}feph b lh~f z~ae gh~f wf zbhpf hmvekf dh z~bz~ah
phnfdć lhpflhpagć$ gblzhpanadć `fwpaevdva fvdva da~ag% Gb~hnah zbhpf ~fz~f}feph b efgvebwgvpć$ b
f~bh~f gf h hsvp dbg åe lhpflhpagh ~aka`ć åe gh~f zbfpvd w,h walŧap ghzpas$ b alzf~cfgŧavef gf w,h
ećwgvp z~ae kfef}h z~bz~avdva gb~z åe gh~f wf `fwcćŒbh~ć% Hgfhwpć gb~hnaf zbhpf ca ghdfh wz~f
h`fsć~hph åepîdea~f gv waefdf$ fh zvpîe` ca ~fz~f}fephpć cbhp~f walzdv dh easfd wfe}b~ahd z~aep~ ,ve
kîe`% Hgfwph fwpf Œa lbpasvd zfep~v gh~f wf `awgvpć `fwz~f b wgć`f~f$ cć~ć gb~hnaf ev zbŧa gvp~faf~h
lh~fh$ cć~ć kîe` ev wf zbhpf }ćlawda a`ffh% Gîe` saef sb~nh `fwz~f ålzć~ŧa~f$ Eagoaph Wpćefwgv
z~fah `bvć fdflfepf bh~fgvl hwflćećpbh~f z~ae lb`vd åe gh~f wf z~asfŒpf dapf~hpv~h zbzvdh~ć$
nhdh`h åe z~aegazhd% Lhpflhpagh gdhwagć ev zf~lapf hgfhwpć ålzć~ŧa~f$ åewć hl zvpfh wzvef gć
lhpflhpagh ~fdhpasć h zbfpvdva zbhpf Œa goah~ fwpf åe wph~f wć z~f}agć ~f}vdphpvd% Hgfwpf `bvć
fdflfepf /lvepfdf Œa ghz~h! `vg kîe`vd lha ~fzf`f whv lha pî~}av/åe cvegŧaf `f zf~wbheć! dh
—Lab~aŧh“ åe gh~f fxawph ve ~hzb~p `f hcaeaphpf åep~f fdf$ `fga ~f}vdphpvd hgfwpfa bzf~hŧaa `f ålzć~ŧa~f
zbhpf ca gbewa`f~hp zf~cfgŧavefh$ ebecakv~hpasvd eflhahsîe` dbg åe hgfhwpć bzf~hŧaf `fbh~fgf ev lha
fxawpć$ ~f}vdphpvd fwpf wvndal z~ae kfef}h gblveć h h~kvlfepfdb~%
Åe gfdf `ae v~lć ~fsaef h~kvlfepvd cb~pf hd z~alfa wp~bcf gf gbewpć åe aeghzhgaphpfh
`fwćsî~Œa~aa avna~aa åegh~ehpf åe hgfhwpć caaeŧć gdhvwp~bcbnagć zfep~v lhpflhpagh `f ~îe`% Fwpf ~fwalŧap
vŒb~vd gh~hgpf~ a~beag hd hgfwpva hwzfgp hd safŧaa gvzdvdva$ wvkf~hp z~ae aepf~mfgŧah —sha“-/—`h~ fv Œa gv
paef" ev Œpal$ sha ev Œpal gîp chgfl“!% Hgfwpf sf~wv~a wf ~ćwc~îek hwvz~h aepf~dbgvpb~vdva Œa wvn
cb~lh vefa awzapf$ zbfpvd `b~ae` wć alzvdwabef}f avnaph wz~f h chgf z~alvd zhw gćp~f lhpflhpagh
~fdhpasć$ b~hgbd hd z~bzafa db~ fxawpfeŧf `vhdf%
Åe gfh `f,h `bvh wp~bcć Eagoaph Wpćefwgv gbzdfŒfŒpf gapapb~vd z~ae gvebŒpaeŧfdf whdf aef`apf hdf
ebvdva paz `f lhpflhpagć% Dfxflvd —`fwakv~“ zbhpf ca gbewa`f~hp gh ve h~kvlfep åe zdvw `îe` g~f`ap
wzvwfdb~ zbfpvdva% ^fchgf~fh ~fhdapćŧaa z~ae aepf~lf`avd gvsîepvdva wf ~fhda}fh}ć gv hmvpb~vd veb~
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