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\deflang1031\pard\ri4\plain\f3\fs22\cf0 01.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}GradSchranke:=proc(A,B,C,k)
\par \pard\li600\ri1\fi-300\plain\f4\fs22\cf2 local pol1,pol2,deg1,deg2,a,b;
\par begin
\par pol1:=collect(A-B,k);
\par pol2:=collect(A+B,k);
\par if pol1=0
\par then deg1:=-1
\par else deg1:=degree(pol1,k)
\par end_if;
\par if pol2=0
\par then deg2:=-1
\par else deg2:=degree(pol2,k)
\par end_if;
\par if deg1<=deg2
\par then print("Teil 1");
\par return(degree(C,k)-deg2);
\par end_if;
\par a:=coeff(pol1,k,deg1);
\par if deg2=0)
\par then print("Teil 2a");
\par return(degree(C,k)-deg1+1);
\par else print("Teil 2b");
\par return(max(-2*b/a,degree(C,k)-deg1+1));
\par end_if;
\par end_proc:
\par \pard\ri4\plain\f3\fs22\cf0 02.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}GradSchranke(k-n,-(k+1),k+1,k)
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect "Teil 2a"
\par {\pict\wmetafile8\picw564\pich827\picwgoal319\pichgoal468
0100090000031701000005001C0000000000050000000B0200000000050000000C023B03340203
0000001E00030000001E00050000000C02D5014001050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02
0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000
00417269616C00F47740000000ED070ADF7DA8F47786A8F4770100000000003000040000002D01
020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200
00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65
73204E657720526F6D616E007DA8F47786A8F4770100000000003000040000002D0103000B0000
0026060F000C004D6174685479706500007B000500000009020000FF0007000000210501003100
3B01640008000000FA0200000000000000000000040000002D01040004000000F0010000070000
00FC020000FFFFFF000000040000002D01000004000000F00101001C000000FB02100007000000
0000BC02000000000102022253797374656D0077400000002C060AD87DA8F47786A8F477010000
0000003000040000002D010100040000002701FFFF04000000F001020004000000F00103000400
00002701FFFF030000000000
}\plain\f4\fs22\cf1\protect
\par
\par \pard\ri4\plain\f3\fs22\cf0 03.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}GradSchranke(k*(k+1)^2,-k*(k+1),k^2*(k+1),k);
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect "Teil 1"
\par {\pict\wmetafile8\picw564\pich827\picwgoal319\pichgoal468
0100090000031701000005001C0000000000050000000B0200000000050000000C023B03340203
0000001E00030000001E00050000000C02D5014001050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02
0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000
00417269616C00F477400000004E080A4C7DA8F47786A8F4770100000000003000040000002D01
020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200
00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65
73204E657720526F6D616E007DA8F47786A8F4770100000000003000040000002D0103000B0000
0026060F000C004D6174685479706500007B000500000009020000FF0007000000210501003000
3B01640008000000FA0200000000000000000000040000002D01040004000000F0010000070000
00FC020000FFFFFF000000040000002D01000004000000F00101001C000000FB02100007000000
0000BC02000000000102022253797374656D0077400000002C060AD97DA8F47786A8F477010000
0000003000040000002D010100040000002701FFFF04000000F001020004000000F00103000400
00002701FFFF030000000000
}\plain\f4\fs22\cf1\protect
\par
\par \pard\ri4\plain\f3\fs22\cf0 04.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}GradSchranke(k+1,-1,1,k)
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect "Teil 1"
\par {\pict\wmetafile8\picw945\pich827\picwgoal535\pichgoal468
0100090000039E01000008001C0000000000050000000B0200000000050000000C023B03B10303
0000001E00030000001E00050000000C02D5011802050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02
0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000
00417269616C00F4774000000088070AD57DA8F47786A8F4770100000000003000040000002D01
020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200
00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65
73204E657720526F6D616E007DA8F47786A8F4770100000000003000040000002D0103000B0000
0026060F000C004D6174685479706500007B001C000000FB0210FF000000000000900100000002
0700000053796D626F6C0077400000002C060ADA7DA8F47786A8F4770100000000003000040000
002D0104001C000000FB0210FF0000000000009001000000020700000053796D626F6C00774000
0000E1070AAC7DA8F47786A8F4770100000000003000040000002D0105001C000000FB0210FF00
00000000009001000000020700000054696D6573204E657720526F6D616E007DA8F47786A8F477
0100000000003000040000002D010600040000002D010300040000002D010400040000002D0105
000500000009020000FF0007000000210501002D003B018800040000002D010600040000002D01
0300070000002105010031003B013C0108000000FA0200000000000000000000040000002D0107
0004000000F001000007000000FC020000FFFFFF000000040000002D01000004000000F0010100
1C000000FB021000070000000000BC02000000000102022253797374656D00774000000063080A
D27DA8F47786A8F4770100000000003000040000002D010100040000002701FFFF04000000F001
020004000000F001030004000000F001040004000000F001050004000000F00106000400000027
01FFFF030000000000
}\plain\f4\fs22\cf1\protect
\par
\par \pard\ri4\plain\f3\fs22\cf0 05.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}REtoPol:=proc(A,B,C,k)
\par \pard\li600\ri1\fi-300\plain\f4\fs22\cf2 local deg,g,j,rec,sol;
\par begin
\par deg:=GradSchranke(A,B,C,k);
\par if deg<0
\par then return("keine Polynoml\'f6sung")
\par end_if;
\par g:=_plus(aa[j]*k^j $ j=0..deg);
\par rec:=collect(A*subs(g,k=k+1)+B*g-C,k);
\par sol:=solve(\{coeff(rec,k,i) $ i=0..degree(rec,k)\},\{aa[j] $ j=0..deg\},IgnoreSpecialCases);
\par if sol=NIL or sol=\{\}
\par then return("keine Polynoml\'f6sung")
\par else subs(g,sol[1]);
\par end_if;
\par end_proc:
\par \pard\ri4\plain\f3\fs22\cf0 06.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}REtoPol(k-n,-(k+1),k+1,k)
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect "Teil 2a"
\par {\pict\wmetafile8\picw1048\pich1141\picwgoal594\pichgoal646
0100090000032F0200000B001C0000000000050000000B0200000000050000000C027504180403
0000001E00030000001E00050000000C0287025202050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02
0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000
00417269616C00F47740000000ED070AE37DA8F47786A8F4770100000000003000040000002D01
020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200
00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65
73204E657720526F6D616E007DA8F47786A8F4770100000000003000040000002D0103000B0000
0026060F000C004D617468547970650000BF001C000000FB0210FF000000000000900101000001
0700000054696D6573204E657720526F6D616E007DA8F47786A8F4770100000000003000040000
002D010400040000002D010300040000002D010400040000002D0103001C000000FB0210FF0000
000000009001000000020700000053796D626F6C0077400000004E080A4E7DA8F47786A8F47701
00000000003000040000002D0105001C000000FB0210FF00000000000090010000000207000000
53796D626F6C00774000000088070AD67DA8F47786A8F4770100000000003000040000002D0106
001C000000FB0210FF0000000000009001000000020700000054696D6573204E657720526F6D61
6E007DA8F47786A8F4770100000000003000040000002D010700040000002D010300040000002D
010500040000002D0106000500000009020000FF0007000000210501002D009801880004000000
2D010700040000002D010300040000002D01040007000000210501006B0027015A010700000021
0501006E00200259011C000000FB0210FF000000000000900100000001070000005346204D6174
68204578740053080A0B7DA8F47786A8F4770100000000003000040000002D0108001C000000FB
0210FF000000000000900100000002070000005346204D617468204578740058080A3F7DA8F477
86A8F4770100000000003000040000002D0109000700000021050100C50047013C010700000021
050100C50047015E0108000000FA0200000000000000000000040000002D010A0004000000F001
000007000000FC020000FFFFFF000000040000002D01000004000000F00101001C000000FB0210
00070000000000BC02000000000102022253797374656D00774000000066080A187DA8F47786A8
F4770100000000003000040000002D010100040000002701FFFF04000000F001020004000000F0
01030004000000F001040004000000F001050004000000F001060004000000F001070004000000
F001080004000000F0010900040000002701FFFF030000000000
}\plain\f4\fs22\cf1\protect
\par
\par \pard\ri4\plain\f3\fs22\cf0 07.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}REtoPol(k*(k+1)^2,-k*(k+1),k^2*(k+1),k)
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect "Teil 1"
\par {\pict\wmetafile8\picw564\pich827\picwgoal319\pichgoal468
0100090000031701000005001C0000000000050000000B0200000000050000000C023B03340203
0000001E00030000001E00050000000C02D5014001050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02
0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000
00417269616C00F47740000000F1070A1E7DA8F47786A8F4770100000000003000040000002D01
020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200
00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65
73204E657720526F6D616E007DA8F47786A8F4770100000000003000040000002D0103000B0000
0026060F000C004D6174685479706500007B000500000009020000FF0007000000210501003100
3B01640008000000FA0200000000000000000000040000002D01040004000000F0010000070000
00FC020000FFFFFF000000040000002D01000004000000F00101001C000000FB02100007000000
0000BC02000000000102022253797374656D00774000000063080AD47DA8F47786A8F477010000
0000003000040000002D010100040000002701FFFF04000000F001020004000000F00103000400
00002701FFFF030000000000
}\plain\f4\fs22\cf1\protect
\par
\par \pard\ri4\plain\f3\fs22\cf0 08.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}REtoPol(k+1,-1,1,k)
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect "Teil 1"
\par {\pict\wmetafile8\picw4376\pich827\picwgoal2480\pichgoal468
010009000003A301000005001C0000000000050000000B0200000000050000000C023B03181103
0000001E00030000001E00050000000C02D501B109050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02
0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000
00417269616C00F4774000000058080A427DA8F47786A8F4770100000000003000040000002D01
020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200
00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65
73204E657720526F6D616E007DA8F47786A8F4770100000000003000040000002D0103000B0000
0026060F000C004D6174685479706500007B000500000009020000FF0007000000210501002200
3B01640007000000210501006B003B01C600070000002105010065003B013E0107000000210501
0069003B01A90107000000210501006E003B01EC01070000002105010065003B01640207000000
2105010020003B01CF02070000002105010050003B010B0307000000210501006F003B01900307
000000210501006C003B010804070000002105010079003B014B0407000000210501006E003B01
C30407000000210501006F003B013B0507000000210501006D003B01B30507000000210501006C
003B016B060700000021050100F6003B01AE06070000002105010073003B012607070000002105
010075003B01830707000000210501006E003B01FB07070000002105010067003B017308070000
002105010022003B01EB0808000000FA0200000000000000000000040000002D01040004000000
F001000007000000FC020000FFFFFF000000040000002D01000004000000F00101001C000000FB
021000070000000000BC02000000000102022253797374656D00774000000063080AD57DA8F477
86A8F4770100000000003000040000002D010100040000002701FFFF04000000F0010200040000
00F0010300040000002701FFFF030000000000
}\plain\f4\fs22\cf1\protect
\par
\par \pard\ri4\plain\f3\fs22\cf0 09.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}a:=1/(k+1)
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect {\pict\wmetafile8\picw1318\pich1141\picwgoal747\pichgoal646
0100090000033C0200000B001C0000000000050000000B0200000000050000000C027504260503
0000001E00030000001E00050000000C028702EB02050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02
0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000
00417269616C00F4774000000053080A0F7DA8F47786A8F4770100000000003000040000002D01
020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200
00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65
73204E657720526F6D616E007DA8F47786A8F4770100000000003000040000002D0103000B0000
0026060F000C004D617468547970650000C4001C000000FB0210FF000000000000900101000001
0700000054696D6573204E657720526F6D616E007DA8F47786A8F4770100000000003000040000
002D010400040000002D0103001C000000FB0210FF000000000000900100000002070000005379
6D626F6C00774000000088070AD77DA8F47786A8F4770100000000003000040000002D0105001C
000000FB0210FF0000000000009001000000020700000053796D626F6C0077400000004E080A4F
7DA8F47786A8F4770100000000003000040000002D0106001C000000FB0210FF00000000000090
01000000020700000054696D6573204E657720526F6D616E007DA8F47786A8F477010000000000
3000040000002D010700040000002D0103000500000009020000FF000700000021050100310023
013901040000002D01040007000000210501006B0020028100040000002D010500040000002D01
060007000000210501002B0020023201040000002D010700040000002D01030007000000210501
0031002002F2011C000000FB0210FF000000000000900100000001070000005346204D61746820
45787400E1070AAE7DA8F47786A8F4770100000000003000040000002D0108001C000000FB0210
FF000000000000900100000002070000005346204D6174682045787400ED070AE47DA8F47786A8
F4770100000000003000040000002D0109000700000021050100C5004001640007000000210501
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000000040000002D01000004000000F00101001C000000FB021000070000000000BC0200000000
0102022253797374656D00774000000066080A197DA8F47786A8F4770100000000003000040000
002D010100040000002701FFFF04000000F001020004000000F001030004000000F00104000400
0000F001050004000000F001060004000000F001070004000000F001080004000000F001090004
0000002701FFFF030000000000
}\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect
\par \pard\ri4\plain\f3\fs22\cf0 10.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}rat:=normal(expand(subs(a,k=k+1)/a))
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect {\pict\wmetafile8\picw1318\pich1154\picwgoal747\pichgoal654
010009000003760200000B001C0000000000050000000B0200000000050000000C028204260503
0000001E00030000001E00050000000C028E02EB02050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
0004000000F001000008000000FA0200000000000000000000040000002D01000007000000FC02
0000000000000000040000002D0101001C000000FB0210FF000000000000900100000001070000
00417269616C00F47740000000F1070A227DA8F47786A8F4770100000000003000040000002D01
020005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200
00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65
73204E657720526F6D616E007DA8F47786A8F4770100000000003000040000002D0103000B0000
0026060F000C004D617468547970650000C4001C000000FB0210FF000000000000900101000001
0700000054696D6573204E657720526F6D616E007DA8F47786A8F4770100000000003000040000
002D010400040000002D0103001C000000FB0210FF000000000000900100000002070000005379
6D626F6C0077400000004E080A507DA8F47786A8F4770100000000003000040000002D0105001C
000000FB0210FF0000000000009001000000020700000053796D626F6C00774000000088070AD8
7DA8F47786A8F4770100000000003000040000002D0106001C000000FB0210FF00000000000090
01000000020700000054696D6573204E657720526F6D616E007DA8F47786A8F477010000000000
3000040000002D010700040000002D010300040000002D010400040000002D010300040000002D
010500040000002D010600040000002D010700040000002D0103000500000009020000FF000400
00002D01040007000000210501006B0027018100040000002D010500040000002D010600070000
00210501002B0027013201040000002D010700040000002D010300070000002105010031002701
F201040000002D01040007000000210501006B0027028100040000002D010500040000002D0106
0007000000210501002B0027023201040000002D010700040000002D0103000700000021050100
32002702F2011C000000FB0210FF000000000000900100000001070000005346204D6174682045
78740058080A447DA8F47786A8F4770100000000003000040000002D0108001C000000FB0210FF
000000000000900100000002070000005346204D617468204578740053080A107DA8F47786A8F4
770100000000003000040000002D0109000700000021050100C500470164000700000021050100
C5004701F4000700000021050100C500470184010700000021050100C5004701F70108000000FA
0200000000000000000000040000002D010A0004000000F001000007000000FC020000FFFFFF00
0000040000002D01000004000000F00101001C000000FB021000070000000000BC020000000001
02022253797374656D00774000000066080A1A7DA8F47786A8F477010000000000300004000000
2D010100040000002701FFFF04000000F001020004000000F001030004000000F0010400040000
00F001050004000000F001060004000000F001070004000000F001080004000000F00109000400
00002701FFFF030000000000
}\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect
\par \pard\ri4\plain\f3\fs22\cf0 11.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}u:=numer(rat):
\par \pard\li600\ri1\fi-300\plain\f4\fs22\cf2 v:=denom(rat):
\par \pard\ri4\plain\f5\fs22\cf0 12.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}h:=1
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect {\pict\wmetafile8\picw564\pich827\picwgoal319\pichgoal468
0100090000031701000005001C0000000000050000000B0200000000050000000C023B03340203
0000001E00030000001E00050000000C02D5014001050000000B020000000014000000FF060000
0600F602000024000100020000800080FF7FFF7F02000080FF7F0080FF7F0200040000002D0100
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\par \pard\ri4\plain\f5\fs22\cf0 14.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}GradSchranke(coeff(lhs(RE),g[k+1]),coeff(lhs(RE),g[k]),rhs(RE),k)
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect "Teil 2b"
\par {\pict\wmetafile8\picw564\pich827\picwgoal319\pichgoal468
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\par
\par \pard\ri4\plain\f5\fs22\cf0 15.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}REtoPol(coeff(lhs(RE),g[k+1]),coeff(lhs(RE),g[k]),rhs(RE),k)
\par \pard\li50\ri6\plain\f4\fs22\cf1\protect
\par \pard\li50\ri2\plain\f4\fs22\cf1\protect "Teil 2b"
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\par
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs22\cf2 {\pntext\f1\'b7\tab}
\par }