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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 39 "Nekonecne ciselne rady - \+
zakladni pojmy" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Soucet rady" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "
" {TEXT -1 19 "Urcete soucet rady " }{XPPEDIT 18 0 "sum(1/(n*(n+1)),n \+
= 1 .. infinity);" "6#-%$sumG6$*&\"\"\"\"\"\"*&%\"nGF(,&F*F(\"\"\"F(F(
!\"\"/F*;\"\"\"%)infinityG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "rada:=Sum(1/(n*(n+1)), n=1..infinity);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%radaG-%$SumG6$*&\"\"\"F)*&%\"nG\"\"\",&F+\"
\"\"F.F.\"\"\"!\"\"/F+;F.%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "convert(op(1,rada),'parfrac',n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&\"\"\"F%%\"nG!\"\"\"\"\"*&F%F%,&F&F(F(F(F'!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s[n]:=1-1/(n+1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#%\"nG,&\"\"\"F)*&\"\"\"F+,&F'F)F)F)
!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit(s[n],n=
infinity):%=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$,
&\"\"\"F(*&\"\"\"F*,&%\"nGF(F(F(!\"\"!\"\"/F,%)infinityGF(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "poslcass := proc(a, b, d)\nlocal i
, j, s, e;\n    s := 0;\n    j := 0;\n    for i from b to d + b - 1 do
\n        j := j + 1;\n        s := s + eval(subs(n = i, a));\n       \+
 lprint(evaln(s[j]) = s)\n    od;\n    e := sum(a, n = b .. n);\n    l
print(evaln(s[n]) = e);\n    Sum(a, n = b .. infinity) = limit(e, n = \+
infinity)\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "poslca
ss(op(1,rada),1,5);" }}{PARA 6 "" 1 "" {TEXT -1 10 "s[1] = 1/2" }}
{PARA 6 "" 1 "" {TEXT -1 10 "s[2] = 2/3" }}{PARA 6 "" 1 "" {TEXT -1 
10 "s[3] = 3/4" }}{PARA 6 "" 1 "" {TEXT -1 10 "s[4] = 4/5" }}{PARA 6 "
" 1 "" {TEXT -1 10 "s[5] = 5/6" }}{PARA 6 "" 1 "" {TEXT -1 16 "s[n] = \+
1-1/(n+1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*&%\"
nG\"\"\",&F*\"\"\"F-F-\"\"\"!\"\"/F*;F-%)infinityGF-" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(1/(n*(n+1)), n=1..infinity)=sum(1/(n*
(n+1)), n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*
&\"\"\"F(*&%\"nG\"\"\",&F*\"\"\"F-F-\"\"\"!\"\"/F*;F-%)infinityGF-" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 766 "sumplots := proc(rada)\nlocal term
, n, a, b, psum, m, points, i, c, sn, p1, p2;\n    if nargs = 2 then c
 := args[2] else c := 0 fi;\n    if typematch(rada, 'Sum'(term::algebr
aic,\n    n::name = (a::integer) .. (b::integer))) then\n        psum \+
:=\n            evalf@(unapply(Sum(term, n = a .. a + m - 1), m))\n   \+
         ;\n        points := [seq(\n            [[i, psum(i) + c], [i
 + 1, psum(i) + c]],\n            i = 1 .. b - a + 1)];\n        point
s := map(op, points);\n        p1 := PLOT(CURVES(points), AXESLABELS(n
, \"s[n]\"));\n        sn := evalf(c + sum(term, n = a .. infinity))\n
    else ERROR(\"expecting a Sum structure as input\")\n    fi;\n    i
f sn < infinity then\n        p2 := plot(sn, n = a .. b, linestyle = 4
);\n        display(\{p1, p2\})\n    else p1\n    fi\nend:\n" }}}
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0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Urcete soucet rady " 
}{XPPEDIT 18 0 "sum(1/((3*n-2)*(3*n+1)),n = 1 .. infinity);" "6#-%$sum
G6$*&\"\"\"\"\"\"*&,&*&\"\"$F(%\"nGF(F(\"\"#!\"\"F(,&*&\"\"$F(F-F(F(\"
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" 0 "" {MPLTEXT 1 0 46 "rada:=Sum(1/((3*n-2)*(3*n+1)), n=1..infinity);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%radaG-%$SumG6$*&\"\"\"F)*&,&%\"
nG\"\"$!\"#\"\"\"\"\"\",&F,F-F/F/\"\"\"!\"\"/F,;F/%)infinityG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "convert(op(1,rada),'parfrac'
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\"\"!\"\"#F*F(*&F%F%,&F'F(F*F*F+#!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "poslcass(op(1,rada),1,5);" }}{PARA 6 "" 1 "" {TEXT 
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6 "" 1 "" {TEXT -1 11 "s[3] = 3/10" }}{PARA 6 "" 1 "" {TEXT -1 11 "s[4
] = 4/13" }}{PARA 6 "" 1 "" {TEXT -1 11 "s[5] = 5/16" }}{PARA 6 "" 1 "
" {TEXT -1 23 "s[n] = -1/3/(3*n+1)+1/3" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$SumG6$*&\"\"\"F(*&,&%\"nG\"\"$!\"#\"\"\"\"\"\",&F+F,F.F.\"\"
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0 5 "s[n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&\"\"\"F&,&%
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1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "s
ave poslcass, sumplots, `cisrady.txt`;" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Sierpinskeho koberec" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 464 "Vypoctet obsah Sierpinsk\351ho ko
berce, ktery se konstruuje takto: jednotkovy ctverec rozdelime na deve
t shodnych ctvercu a odstranime vnitrek prostredniho ctverce. Kazdy ze
 zbyvajicich osmi ctvercu rozdelime opet na devet shodnych ctvercu a z
novu odstranime vnitrek prostredniho ctverce. Kdyz tuto operaci prodlo
uzime do nekonecna dostaneme utvar, ktery se nazyva Sierpinskeho kober
ec. Jednotlive iterace sierpinskeho koberce muzeme kreslit v Maplu pom
oci procedury " }{TEXT 259 11 "sierpkob(n)" }{TEXT -1 1 ":" }}}{EXCHG 
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[l7$FievFbf[l7$F]fvFbf[l7$F]fvF_f[lF][r-F'6#7&7$FghvF]dq7$FghvF`dq7$F[
ivF`dq7$F[ivF]dq-F'6#7&7$FbivF]dq7$FbivF`dq7$FfivF`dq7$FfivF]dq-F'6#7&
7$FehpF]e[l7$FehpF`e[l7$FihpF`e[l7$FihpF]e[l-F'6#7&7$FbivF]e[l7$FbivF`
e[l7$FfivF`e[l7$FfivF]e[l-F'6#7&7$FehpF_f[l7$FehpFbf[l7$FihpFbf[l7$Fih
pF_f[l-F'6#7&7$FghvF_f[l7$FghvFbf[l7$F[ivFbf[l7$F[ivF_f[l-F'6#7&7$Fbiv
F_f[l7$FbivFbf[l7$FfivFbf[l7$FfivF_f[l-F'6$7&7$FchmF[cm7$FehmF[cm7$Feh
mF^cm7$FchmF^cmF:-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 2 
400 300 300 2 0 1 0 2 9 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 457 354 0 0 0 0 0 0 }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 7 "Reseni:" }}{PARA 0 "" 0 "" {TEXT -1 113 "Obsah j
ednotkoveho ctverce je roven jedne a od tohoto obsahu budeme odecitat \+
obsah odstranenych ctvercu. Oznacme " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG
6#%\"nG" }{TEXT -1 82 " obsah odstranenych ctvercu v n-te iteraci a P \+
hledany obsah Sierpinskeho koberce." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 45 "V n-te iteraci odstranujeme ctverce o st
rane " }{XPPEDIT 18 0 "(1/3)^n;" "6#)*&\"\"\"\"\"\"\"\"$!\"\"%\"nG" }
{TEXT -1 33 ", tedy obsah takoveho ctverce je " }{XPPEDIT 18 0 "(1/9)^
n;" "6#)*&\"\"\"\"\"\"\"\"*!\"\"%\"nG" }{TEXT -1 34 " a jejich pocet v
 n-te iteraci je " }{XPPEDIT 18 0 "8^(n-1);" "6#)\"\"),&%\"nG\"\"\"\"
\"\"!\"\"" }{TEXT -1 10 ". Pak tedy" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "a[n]:=8^(n-1)/9^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%\"aG6#%\"nG*&)\"\"),&F'\"\"\"!\"\"F,\"\"\")\"\"*F'!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Celkovy obsah Sierpinskeho koberce
 pak spocteme:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "P:=1-Sum(
a[n], n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,&\"\"
\"F&-%$SumG6$*&)\"\"),&%\"nGF&!\"\"F&\"\"\")\"\"*F.!\"\"/F.;F&%)infini
tyGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(P);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 
"Rada" }{XPPEDIT 18 0 "Sum(a[n],n = 1 .. infinity);" "6#-%$SumG6$&%\"a
G6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 71 " je geometrickou radou, p
ro urceni jejiho souctu muzeme pouzit procedur" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 432 "kvocgeom := proc (rada) local i; global opera
t, kvoc, b, a; b := expand(rada); if type(b,`+`) then operat := nops(b
); for i to operat do kvoc[i] := simplify(subs(n = n+1,op(i,b))/op(i,b
)); a[i] := simplify(op(i,b)/(kvoc[i]^(n-1))); lprint(evaln(kvoc[i]) =
 kvoc[i]); lprint(evaln(a[i]) = a[i]) od else operat := 1; kvoc := sim
plify(subs(n = n+1,b)/b); a := simplify(b/(kvoc^(n-1))); lprint(('kvoc
') = kvoc); lprint(('a') = a) fi end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "kvocgeom(a[n]);" }}{PARA 6 "" 1 "" {TEXT -1 10 "kvoc \+
= 8/9" }}{PARA 6 "" 1 "" {TEXT -1 7 "a = 1/9" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 139 "geom := proc (k, fclen) local s, i; s := 0; if \+
1 < operat then for i to operat\ndo s := s+fclen[i]/(1-k[i]) od else s
 := fclen/(1-k) fi end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
geom(kvoc,a);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Dostali jsme tedy P=1-1=0 a o
bsah Sierpinskeho koberce je roven nule." }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 35 "Sierpinskeho trojuhelnik a pyramida" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "restart;with(plots):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 636 "serp1 := proc(\nL::algebraic, lev::integer, x0:
:algebraic, y0::algebraic)\nglobal p, s;\noption remember;\n    if s =
 0 then p[0] := plots[polygonplot](\n        [[x0, y0], [x0 + 2*L, y0]
, [x0, y0 + 2*L]],\n        style = line)\n    fi;\n    s := s + 1;\n \+
   p[s] := plots[polygonplot](\n        [[x0, y0 + L], [x0 + L, y0 + L
], [x0 + L, y0]],\n        color = green);\n    if 1 < lev then\n     \+
   serp1(1/2*L, lev - 1, x0, y0);\n        serp1(1/2*L, lev - 1, x0 + \+
L, y0);\n        serp1(1/2*L, lev - 1, x0, y0 + L)\n    fi;\n    RETUR
N(plots[display](\n        [seq(p[i], i = 0 .. 1/2*3^lev - 1/2)],\n   \+
     scaling = constrained, axes = none))\nend:\n\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "tr:=array(1..2,1..2):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "s:=0:tr[1,1]:=serp1(100,1,0,0):" }}}{EXCHG 
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]; \n       c := verts[3];\n       d := verts[4];\n        RETURN( [[a
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{PARA 3 "" 0 "" {TEXT -1 16 "Mengerova krivka" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 759 "menger1 := proc(L::algebraic, lev::integer, x0,
 y0, z0)\nlocal i, j, k, f;\nglobal p, s;\noption remember;\n    s := \+
s + 1;\n    f := 0.4;\n    p[s] := cutout(hexahedron([x0, y0, z0], L),
 f);\n    if s = 2 then p[1] := p[2] fi;\n    if 1 < lev then for i fr
om -1 to 1 do for j from -1 to 1\n            do for k from -1 to 1 do
\n                    if k = 0 and 1 < abs(i) + abs(j) or\n           \+
         k <> 0 and 0 < abs(i) + abs(j) then\n                        \+
menger1(1/3*L, lev - 1, x0 + 2/3*i*L,\n                        y0 + 2/
3*j*L, z0 + 2/3*k*L)\n                    fi\n                od\n    \+
        od\n        od\n    fi;\n    RETURN(plots[display](\n        [
seq(p[i], i = 1 .. 1/19*20^lev - 1/19)],\n        orientation = [35, 7
0], scaling = constrained))\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "with(plottools): s:=0: menger1(100, 2, 0, 0, 0);" }}
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w7&Fe_pFd_pFi]p7%F\\wFduFdv7&Fg_pFi]pFh]pFa_pFB-F$6'7&Fe\\nFi]pFd_pF\\
]n7&F\\]nFd_p7%FivFivFivFi\\n7&Fi\\nF]`pF\\^pFf\\n7&Ff\\nF\\^pFi]pFe\\
nFB-F$6'7&7%FhbnF\\wFjpF]`pFd_p7%FhbnFdvFjp7&Fd`pFd_pF[_p7%F]cnFdvFjp7
&Ff`pF[_pFh^p7%F]cnF\\wFjp7&Fh`pFh^pF]`pFc`pFB-F$6'7&7%FhbnFjpFdvF\\^p
F]`p7%FhbnFjpF\\w7&F^apF]`pFh^p7%F]cnFjpF\\w7&F`apFh^pF_^p7%F]cnFjpFdv
7&FbapF_^pF\\^pF]apFB-%'COLOURG6#%*XYSHADINGG-%*AXESSTYLEG6#%%NONEG-%(
SCALINGG6#%,CONSTRAINEDG-FC6#%&PATCHG-%+PROJECTIONG6%$\"#N\"\"!$\"#qFh
bp\"\"\"" 3 400 300 300 2 0 1 0 2 2 0 1 1 1.000000 70.000000 
35.000000 0 0 0 0 0 0 0 0 0 0 0 222 85 66 222 85 66 1 1 0 0 0 2366 
-32304 0 0 0 0 0 1 }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Kochova kr
ivka" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots)
:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Vypoctete d\351lku " }{TEXT 
256 14 "Kochovy krivky" }{TEXT -1 40 ". Kochova krivka vznikne timto p
ostupem:" }}{PARA 0 "" 0 "" {TEXT -1 125 "V prvni iteraci si nakreslim
e usecku jednotkove delky. V druhe iteraci si usecku rozdelime na tri \+
shodne dily, k prostrednimu" }}{PARA 0 "" 0 "" {TEXT -1 121 "dilu dokr
eslime rovnostranny trouhelnik tak, ze tento dil bude jednou z jeho st
ran a pak uvedeny prostredni dil vymazeme." }}{PARA 0 "" 0 "" {TEXT 
-1 296 "Dostaneme tak lomenou caru skladajici se ze ctyr shodnych usec
ek. Ve treti iteraci aplikujeme uvedeny postup na vsechny ctyri usecky
 z predchozi iterace. Prodlouzime-li tento postup do nekonecna, dostan
em utvar, ktery se nazyva Kochova krivka. Kochovy krivky lze v Maplu k
reslit pomoci procedury " }{TEXT 257 12 "kochkriv(n)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 861 "kochkriv := proc(n)\nlocal i, j, x
, y, dx, dy, uy;\nglobal kres, s, pomsez;\n    uy := evalhf(sqrt(1/12)
);\n    s := [x, y], [x + 1/3*dx, y + 1/3*dy],\n        [x + 1/2*dx - \+
uy*dy, y + 1/2*dy + uy*dx],\n        [x + 2/3*dx, y + 2/3*dy];\n    kr
es[1] := [0, 0], [1, 0];\n    pomsez := [0, 0], [1, 0];\n    for i fro
m 2 to n do\n        dx := pomsez[2, 1] - pomsez[1, 1];\n        dy :=
 pomsez[2, 2] - pomsez[1, 2];\n        x := pomsez[1, 1];\n        y :
= pomsez[1, 2];\n        kres[i] := s;\n        for j from 2 to nops([
pomsez]) - 1 do\n            x := pomsez[j, 1];\n            y := poms
ez[j, 2];\n            dx := pomsez[j + 1, 1] - x;\n            dy := \+
pomsez[j + 1, 2] - y;\n            kres[i] := kres[i], s\n        od;
\n        pomsez := kres[i], [1, 0];\n        kres[i] := pomsez\n    o
d;\n    PLOT(CURVES([kres[n]]), SCALING(CONSTRAINED),\n        AXESSTY
LE(NONE))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v:=array
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42 "K animaci Kochovy krivky slouzi procedura " }{TEXT 258 13 "kochkra
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"kochkranim := proc(n)\nlocal x, y, dx, dy, uy, i, j;\nglobal kres, pl
, pomsez, s, kresl;\n    uy := evalhf(sqrt(1/12));\n    s := [x, y], [
x + 1/3*dx, y + 1/3*dy],\n        [x + 1/2*dx - uy*dy, y + 1/2*dy + uy
*dx],\n        [x + 2/3*dx, y + 2/3*dy];\n    kres[1] := [0, 0], [1, 0
];\n    pl[1] := PLOT(CURVES([kres[1]]));\n    pomsez := [0, 0], [1, 0
];\n    for i from 2 to n do\n        dx := pomsez[2, 1] - pomsez[1, 1
];\n        dy := pomsez[2, 2] - pomsez[1, 2];\n        x := pomsez[1,
 1];\n        y := pomsez[1, 2];\n        kres[i] := s;\n        for j
 from 2 to nops([pomsez]) - 1 do\n            x := pomsez[j, 1];\n    \+
        y := pomsez[j, 2];\n            dx := pomsez[j + 1, 1] - x;\n \+
           dy := pomsez[j + 1, 2] - y;\n            kres[i] := kres[i]
, s\n        od;\n        pomsez := kres[i], [1, 0];\n        kres[i] \+
:= PLOT(CURVES([pomsez]));\n        pl[i] := kres[i]\n    od;\n    for
 i to n do kresl[i] := plots[display]([pl[i]]) od;\n    i := 'i';\n   \+
 plots[display]([kresl[i] $ (i = 1 .. n)], axes = none,\ninsequence=tr
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