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{SECT 0 {EXCHG {PARA 216 "" 0 "" {TEXT 220 150 "In this Maple file, w
e compute the Casimir coefficients of the Lax matrix L associated to t
he Painlev\351 4 equation in relation with the spectral curve\n" }
{TEXT 214 114 "For convenience the irregular times and monodromies are
 denoted t_\{i,j\} at lambda=infinity and s_\{i,j\} at lambda=t" }}}
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{MPLTEXT 1 0 47 "DiaginftySheet1:=-t12*x-t11-t10/x+Unknown/x^2;\n" }
{MPLTEXT 1 0 48 "DiaginftySheet2:=-t22*x-t21-t20/x+Unknown2/x^2;\n" }
{MPLTEXT 1 0 45 "DiagtSheet1:=s10/(x-t)+u1+v1*x+Unknown3*x^2;\n" }
{MPLTEXT 1 0 45 "DiagtSheet2:=s20/(x-t)+u2+v2*x+Unknown4*x^2; " }}
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1 0 58 "series(DiaginftySheet1+DiaginftySheet2-P1(x),x=infinity);\n" }
{MPLTEXT 1 0 45 "series(DiagtSheet1+DiagtSheet2-P1(x),x=t,10);" }}
{PARA 11 "" 1 "" {XPPMATH 20 ",6*&,(I$t12G6\"!\"\"I$t22GF&F'I)Pinfty11
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "series(factor(series(SpectralCurve(
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1GF'F(F*I)Pinfty12GF'F(F(F-F(F(*&F)F(I$s10GF'F(F(*&F)F(I$s20GF'F(F(*&F
2F(F4F(F**&I$t20GF'F(F&F(F**&F)F(F<F(F(I)Pinfty02GF'F(*&,8*(F7F(I\"tGF
'F(F)F(F(*(F9F(FBF(F)F(F(*&F7F(F4F(F(*&I$s11GF'F(F)F(F(*&F9F(F4F(F(*&I
$s21GF'F(F)F(F(*&F2F(F<F(F**&F&F(I)Unknown2GF'F(F(*&F<F(F4F(F(*&F)F(FL
F(F*I%P012GF'F(F(F-F*F(*&,@*(F7F()FBF.F(F)F(F(*(F9F(FSF(F)F(F(*(F7F(FB
F(F4F(F(**F.F(FFF(FBF(F)F(F(*(F9F(FBF(F4F(F(**F.F(FIF(FBF(F)F(F(*&FOF(
FBF(F(*&F7F(F<F(F(*&FFF(F4F(F(*&F9F(F<F(F(*&FIF(F4F(F(*&F2F(FLF(F(*$)F
<F.F(F(*&F4F(FLF(F*I%P022GF'F(F(F,F*F(*&,F*(F7F()FB\"\"$F(F)F(F(*(F9F(
F`oF(F)F(F(*(F7F(FSF(F4F(F(**FaoF(FFF(FSF(F)F(F(*(F9F(FSF(F4F(F(**FaoF
(FIF(FSF(F)F(F(*&FOF(FSF(F(*(F7F(FBF(F<F(F(**F.F(FFF(FBF(F4F(F(*(F9F(F
BF(F<F(F(**F.F(FIF(FBF(F4F(F(*(F.F(F\\oF(FBF(F(*&F7F(FLF(F**&FFF(F<F(F
(*&F9F(FLF(F**&FIF(F<F(F(*(F.F(F<F(FLF(F*I%P032GF'F(F()F-FaoF*F(*&,L*(
F7F()FB\"\"%F(F)F(F(*(F9F(FgpF(F)F(F(*(F7F(F`oF(F4F(F(**FhpF(FFF(F`oF(
F)F(F(*(F9F(F`oF(F4F(F(**FhpF(FIF(F`oF(F)F(F(*&FOF(F`oF(F(*(F7F(FSF(F<
F(F(**FaoF(FFF(FSF(F4F(F(*(F9F(FSF(F<F(F(**FaoF(FIF(FSF(F4F(F(*(FaoF(F
\\oF(FSF(F(*(F7F(FBF(FLF(F***F.F(FFF(FBF(F<F(F(*(F9F(FBF(FLF(F***F.F(F
IF(FBF(F<F(F(*(FaoF(FbpF(FBF(F(*&FFF(FLF(F**&FIF(FLF(F**$)FLF.F(F(I%P0
42GF'F(F()F-FhpF*F(-I\"OG6$%*protectedGI(_syslibGF'6#*$)F-\"\"&F*F(" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Pinfty22:=t12*t22;\n" }
{MPLTEXT 1 0 27 "Pinfty12:=t11*t22+t12*t21;\n" }{MPLTEXT 1 0 52 "Pinft
y02:=-s10*t12-s20*t12-t10*t12+t10*t22+t11*t21;\n" }{MPLTEXT 1 0 15 "fa
ctor(residue(" }{MPLTEXT 1 0 46 "SpectralCurve(DiaginftySheet2)/x,x=in
finity));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 36 "CoherenceEquation:=s
10+s20+t10+t20;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6\"*&I$
t22GF$\"\"\"I$t12GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty12G6
\",&*&I$t11GF$\"\"\"I$t22GF$F(F(*&I$t12GF$F(I$t21GF$F(F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 ">I)Pinfty02G6\",,*&I$s10GF$\"\"\"I$t12GF$F(!\"\"*&
I$s20GF$F(F)F(F**&I$t10GF$F(F)F(F**&F.F(I$t22GF$F(F(*&I$t11GF$F(I$t21G
F$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I$t12G6\"\"\"\"I$t22GF%!\"
\"F&,*I$s10GF%F&I$s20GF%F&I$t10GF%F&I$t20GF%F&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I2CoherenceEquationG6\",*I$s10GF$\"\"\"I$s20GF$F'I$t10GF
$F'I$t20GF$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "factor(ser
ies(SpectralCurve(DiagtSheet1),x=t));\n" }{MPLTEXT 1 0 48 "factor(seri
es(SpectralCurve(DiagtSheet2),x=t));\n" }{MPLTEXT 1 0 54 "EQ1:=residue
((x-t)^3*SpectralCurve(DiagtSheet1),x=t);\n" }{MPLTEXT 1 0 13 "EQ2:=re
sidue(" }{MPLTEXT 1 0 5 "(x-t)" }{MPLTEXT 1 0 36 "^3*SpectralCurve(Dia
gtSheet2),x=t);\n" }{MPLTEXT 1 0 13 "EQ3:=residue(" }{MPLTEXT 1 0 5 "(
x-t)" }{MPLTEXT 1 0 36 "^2*SpectralCurve(DiagtSheet1),x=t);\n" }
{MPLTEXT 1 0 13 "EQ4:=residue(" }{MPLTEXT 1 0 5 "(x-t)" }{MPLTEXT 1 0 
36 "^2*SpectralCurve(DiagtSheet2),x=t);\n" }{MPLTEXT 1 0 13 "EQ5:=resi
due(" }{MPLTEXT 1 0 5 "(x-t)" }{MPLTEXT 1 0 34 "*SpectralCurve(DiagtSh
eet1),x=t);\n" }{MPLTEXT 1 0 13 "EQ6:=residue(" }{MPLTEXT 1 0 5 "(x-t)
" }{MPLTEXT 1 0 34 "*SpectralCurve(DiagtSheet2),x=t);\n" }{MPLTEXT 1 
0 13 "EQ7:=residue(" }{MPLTEXT 1 0 5 "(x-t)" }{MPLTEXT 1 0 36 "^0*Spec
tralCurve(DiagtSheet1),x=t);\n" }{MPLTEXT 1 0 13 "EQ8:=residue(" }
{MPLTEXT 1 0 5 "(x-t)" }{MPLTEXT 1 0 36 "^0*SpectralCurve(DiagtSheet2)
,x=t);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "P001 := s11+s21; \n" 
}{MPLTEXT 1 0 17 "P042 := s21*s11;\n" }{MPLTEXT 1 0 15 "simplify(EQ1);
\n" }{MPLTEXT 1 0 14 "simplify(EQ2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
+5,&I\"xG6\"\"\"\"I\"tGF%!\"\"I%P042GF%!\"%,(*&I$s10GF%F&I$s11GF%F&F(*
&F-F&I$s21GF%F&F(I%P032GF%F&!\"$,2*(F.F&)F'\"\"#F&I)Unknown3GF%F&F(*(F
0F&F5F&F7F&F(*(F.F&F'F&I#v1GF%F&F(*(F0F&F'F&F:F&F(*&F-F&I$s20GF%F&F(*&
F.F&I#u1GF%F&F(*&F0F&F?F&F(I%P022GF%F&!\"#,@*(F-F&F5F&F7F&F&*(F=F&F5F&
F7F&F(*(F-F&F'F&I$t12GF%F&F&*(F-F&F'F&I$t22GF%F&F&*(F-F&F'F&F:F&F&**F6
F&F.F&F'F&F7F&F(*(F=F&F'F&F:F&F(**F6F&F0F&F'F&F7F&F(*&F-F&I$t11GF%F&F&
*&F-F&I$t21GF%F&F&*&F-F&F?F&F&*&F.F&F:F&F(*&F=F&F?F&F(*&F0F&F:F&F(I%P0
12GF%F&F(,\\o*&)F'\"\"%F&)F7F6F&F&*()F'\"\"$F&FGF&F7F&F&*(FgnF&FIF&F7F
&F&**F6F&FgnF&F:F&F7F&F&*(F5F&FOF&F7F&F&*(F5F&FGF&FIF&F&*(F5F&FGF&F:F&
F&*(F5F&FQF&F7F&F&*(F5F&FIF&F:F&F&**F6F&F5F&F?F&F7F&F&*&F5F&)F:F6F&F&*
*F6F&F-F&F'F&F7F&F&**F6F&F=F&F'F&F7F&F(*(F'F&FOF&FIF&F&*(F'F&FOF&F:F&F
&*(F'F&FGF&FQF&F&*(F'F&FGF&F?F&F&*(F'F&FQF&F:F&F&*(F'F&FIF&F?F&F&**F6F
&F'F&F?F&F:F&F&*&F-F&FIF&F&*&F-F&F:F&F&*&F.F&F7F&F(*&F=F&FGF&F(*&F=F&F
:F&F(*&F0F&F7F&F(*&I$t10GF%F&FGF&F(*&FcpF&FIF&F&*&FOF&FQF&F&*&FOF&F?F&
F&*&FQF&F?F&F&*$)F?F6F&F&\"\"!,J*(FZF&FgnF&FenF&F&**FhnF&F5F&FGF&F7F&F
&**FhnF&F5F&FIF&F7F&F&**\"\"'F&F5F&F:F&F7F&F&**F6F&F'F&FOF&F7F&F&**F6F
&F'F&FGF&FIF&F&**F6F&F'F&FGF&F:F&F&**F6F&F'F&FQF&F7F&F&**F6F&F'F&FIF&F
:F&F&**FZF&F'F&F?F&F7F&F&*(F6F&F'F&FboF&F&*&F-F&F7F&F&*&F=F&F7F&F(*&FO
F&FIF&F&*&FOF&F:F&F&*&FGF&FQF&F&*&FGF&F?F&F&*&FQF&F:F&F&*&FIF&F?F&F&*(
F6F&F?F&F:F&F&F&,8*(F`qF&F5F&FenF&F&**FhnF&F'F&FGF&F7F&F&**FhnF&F'F&FI
F&F7F&F&**F`qF&F'F&F:F&F7F&F&*&FOF&F7F&F&*&FGF&FIF&F&*&FGF&F:F&F&*&FQF
&F7F&F&*&FIF&F:F&F&*(F6F&F?F&F7F&F&*$FboF&F&F6*&F7F&,**(FZF&F'F&F7F&F&
FGF&FIF&*&F6F&F:F&F&F&Fhn*$FenF&FZ" }}{PARA 11 "" 1 "" {XPPMATH 20 "+5
,&I\"xG6\"\"\"\"I\"tGF%!\"\"I%P042GF%!\"%,(*&I$s11GF%F&I$s20GF%F&F(*&F
.F&I$s21GF%F&F(I%P032GF%F&!\"$,2*(F-F&)F'\"\"#F&I)Unknown4GF%F&F(*(F0F
&F5F&F7F&F(*(F-F&F'F&I#v2GF%F&F(*(F0F&F'F&F:F&F(*&I$s10GF%F&F.F&F(*&F-
F&I#u2GF%F&F(*&F0F&F?F&F(I%P022GF%F&!\"#,@*(F=F&F5F&F7F&F(*(F.F&F5F&F7
F&F&*(F=F&F'F&F:F&F(**F6F&F-F&F'F&F7F&F(*(F.F&F'F&I$t12GF%F&F&*(F.F&F'
F&I$t22GF%F&F&*(F.F&F'F&F:F&F&**F6F&F0F&F'F&F7F&F(*&F=F&F?F&F(*&F-F&F:
F&F(*&F.F&I$t11GF%F&F&*&F.F&I$t21GF%F&F&*&F.F&F?F&F&*&F0F&F:F&F(I%P012
GF%F&F(,\\o*&)F'\"\"%F&)F7F6F&F&*()F'\"\"$F&FIF&F7F&F&*(FgnF&FKF&F7F&F
&**F6F&FgnF&F:F&F7F&F&*(F5F&FQF&F7F&F&*(F5F&FIF&FKF&F&*(F5F&FIF&F:F&F&
*(F5F&FSF&F7F&F&*(F5F&FKF&F:F&F&**F6F&F5F&F?F&F7F&F&*&F5F&)F:F6F&F&**F
6F&F=F&F'F&F7F&F(**F6F&F.F&F'F&F7F&F&*(F'F&FQF&FKF&F&*(F'F&FQF&F:F&F&*
(F'F&FIF&FSF&F&*(F'F&FIF&F?F&F&*(F'F&FSF&F:F&F&*(F'F&FKF&F?F&F&**F6F&F
'F&F?F&F:F&F&*&F=F&FIF&F(*&F=F&F:F&F(*&F-F&F7F&F(*&F.F&FKF&F&*&F.F&F:F
&F&*&F0F&F7F&F(*&I$t10GF%F&FIF&F(*&FcpF&FKF&F&*&FQF&FSF&F&*&FQF&F?F&F&
*&FSF&F?F&F&*$)F?F6F&F&\"\"!,J*(FZF&FgnF&FenF&F&**FhnF&F5F&FIF&F7F&F&*
*FhnF&F5F&FKF&F7F&F&**\"\"'F&F5F&F:F&F7F&F&**F6F&F'F&FQF&F7F&F&**F6F&F
'F&FIF&FKF&F&**F6F&F'F&FIF&F:F&F&**F6F&F'F&FSF&F7F&F&**F6F&F'F&FKF&F:F
&F&**FZF&F'F&F?F&F7F&F&*(F6F&F'F&FboF&F&*&F=F&F7F&F(*&F.F&F7F&F&*&FQF&
FKF&F&*&FQF&F:F&F&*&FIF&FSF&F&*&FIF&F?F&F&*&FSF&F:F&F&*&FKF&F?F&F&*(F6
F&F?F&F:F&F&F&,8*(F`qF&F5F&FenF&F&**FhnF&F'F&FIF&F7F&F&**FhnF&F'F&FKF&
F7F&F&**F`qF&F'F&F:F&F7F&F&*&FQF&F7F&F&*&FIF&FKF&F&*&FIF&F:F&F&*&FSF&F
7F&F&*&FKF&F:F&F&*(F6F&F?F&F7F&F&*$FboF&F&F6*&F7F&,**(FZF&F'F&F7F&F&FI
F&FKF&*&F6F&F:F&F&F&Fhn*$FenF&FZ" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$E
Q1G6\"I%P042GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$EQ2G6\"I%P042GF$" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">I$EQ3G6\",(*&I$s10GF$\"\"\"I$s11GF$F(
!\"\"*&F'F(I$s21GF$F(F*I%P032GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
$EQ4G6\",(*&I$s11GF$\"\"\"I$s20GF$F(!\"\"*&F)F(I$s21GF$F(F*I%P032GF$F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$EQ5G6\",2*(I$s11GF$\"\"\")I\"tGF$
\"\"#F(I)Unknown3GF$F(!\"\"*(I$s21GF$F(F)F(F,F(F-*(F'F(F*F(I#v1GF$F(F-
*(F/F(F*F(F1F(F-*&I$s10GF$F(I$s20GF$F(F-*&F'F(I#u1GF$F(F-*&F/F(F7F(F-I
%P022GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$EQ6G6\",2*(I$s11GF$\"\"
\")I\"tGF$\"\"#F(I)Unknown4GF$F(!\"\"*(I$s21GF$F(F)F(F,F(F-*(F'F(F*F(I
#v2GF$F(F-*(F/F(F*F(F1F(F-*&I$s10GF$F(I$s20GF$F(F-*&F'F(I#u2GF$F(F-*&F
/F(F7F(F-I%P022GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$EQ7G6\",@*(I$
s10GF$\"\"\")I\"tGF$\"\"#F(I)Unknown3GF$F(F(*(I$s20GF$F(F)F(F,F(!\"\"*
(F'F(F*F(I$t12GF$F(F(*(F'F(F*F(I$t22GF$F(F(*(F'F(F*F(I#v1GF$F(F(**F+F(
I$s11GF$F(F*F(F,F(F/*(F.F(F*F(F5F(F/**F+F(I$s21GF$F(F*F(F,F(F/*&F'F(I$
t11GF$F(F(*&F'F(I$t21GF$F(F(*&F'F(I#u1GF$F(F(*&F7F(F5F(F/*&F.F(F@F(F/*
&F:F(F5F(F/I%P012GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$EQ8G6\",@*(
I$s10GF$\"\"\")I\"tGF$\"\"#F(I)Unknown4GF$F(!\"\"*(I$s20GF$F(F)F(F,F(F
(*(F'F(F*F(I#v2GF$F(F-**F+F(I$s11GF$F(F*F(F,F(F-*(F/F(F*F(I$t12GF$F(F(
*(F/F(F*F(I$t22GF$F(F(*(F/F(F*F(F1F(F(**F+F(I$s21GF$F(F*F(F,F(F-*&F'F(
I#u2GF$F(F-*&F3F(F1F(F-*&F/F(I$t11GF$F(F(*&F/F(I$t21GF$F(F(*&F/F(F<F(F
(*&F:F(F1F(F-I%P012GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P001G6\",
&I$s11GF$\"\"\"I$s21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P042G6\"
*&I$s21GF$\"\"\"I$s11GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&I$s21G6
\"\"\"\"I$s11GF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&I$s21G6\"\"\"\"I
$s11GF$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(EQ3,P032
):\n" }{MPLTEXT 1 0 25 "P032 := s21*s10+s20*s11;\n" }{MPLTEXT 1 0 15 "
simplify(EQ3);\n" }{MPLTEXT 1 0 15 "simplify(EQ4);\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I%P032G6\",&*&I$s10GF$\"\"\"I$s21GF$F(F(*&I$s11GF$F(I
$s20GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&I$s11G6\"\"\"\",&I$s1
0GF%F&I$s20GF%!\"\"F&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&I$s21G6\"\"
\"\",&I$s10GF$F%I$s20GF$!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "s11:=0:\n" }{MPLTEXT 1 0 8 "s21:=0:\n" }{MPLTEXT 1 0 16 "solve(
EQ5,P022);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 15 "P022:=s10*s20;\n" }
{MPLTEXT 1 0 14 "simplify(EQ5);" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "*&I$s10G6\"\"\"\"I$s20GF$F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I%P022G6\"*&I$s10GF$\"\"\"I$s20GF$F'" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 221 28 "Summary o
f the coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "P1:=
unapply(P1(lambda),lambda);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "P2
:=unapply(P2(lambda),lambda);\n" }{MPLTEXT 1 0 125 "tdP2:=unapply(s10*
s20/(lambda-t)^2-t12*s10-t12*s20-t12*t10+t10*t22+t11*t21+(t11*t22+t12*
t21)*lambda+t12*t22*lambda^2,lambda);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "
" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF
$F$,**&,&I$s10GF$\"\"\"I$s20GF$F/F/,&F'F/I\"tGF$!\"\"F3F/I$t11GF$F3I$t
21GF$F3*&,&I$t12GF$F3I$t22GF$F3F/F'F/F/F$F$F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I#P2G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,4*(
I$s10GF$\"\"\"I$s20GF$F.),&F'F.I\"tGF$!\"\"\"\"#F3F.*&I%P012GF$F.F1F3F
.*&I$t12GF$F.F-F.F3*&F8F.F/F.F3*&F8F.I$t10GF$F.F3*&F;F.I$t22GF$F.F.*&I
$t11GF$F.I$t21GF$F.F.*&,&*&F?F.F=F.F.*&F8F.F@F.F.F.F'F.F.*(F=F.F8F.)F'
F4F.F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6#I'lambdaG
F$F$6$I)operatorGF$I&arrowGF$F$,2*(I$s10GF$\"\"\"I$s20GF$F.),&F'F.I\"t
GF$!\"\"\"\"#F3F.*&I$t12GF$F.F-F.F3*&F6F.F/F.F3*&F6F.I$t10GF$F.F3*&F9F
.I$t22GF$F.F.*&I$t11GF$F.I$t21GF$F.F.*&,&*&F=F.F;F.F.*&F6F.F>F.F.F.F'F
.F.*(F;F.F6F.)F'F4F.F.F$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}
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