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at,powerset):\n" }{MPLTEXT 1 0 74 "powerset(\{a,b,c\}); powerset(\{a,b
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20 "<*<\"<%I\"bG6\"I\"cGF&I\"aGF&<$F%F(<#F(<$F%F'<$F'F(<#F%<#F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "<&<\"<$I\"bG6\"I\"aGF&<#F'<#F%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "<$<\"<#I\"aG6\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT 206 27 "1.2. Rel\303\241ci\303\263k" }}
{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 57 
"1.2.1. Descartes-szorzat \303\251s rel\303\241ci\303\263." }}{PARA 0 
"" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eva
lb([x,y]=[y,x]);evalb([x,x,y]=[x,y,x]);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&f
alseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p:=[x,
y,z]; p[1]; p[2]; p[3];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%I\"xG6\"I
\"yGF$I\"zGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }}{PARA 11 ""
 1 "" {XPPMATH 20 "I\"yG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\""
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "descartesprod:=proc(L::l
ist(set)) local x,y,i,S,SS;\n" }{MPLTEXT 1 0 56 "if nops(L)=0 then ret
urn \{\} fi; S:=map(x->[x],L[1]);\n" }{MPLTEXT 1 0 38 "for i from 2 to
 nops(L) do SS:=\{\};\n" }{MPLTEXT 1 0 65 "  for x in S do for y in L[
i] do SS:=SS union \{[op(x),y]\} od;\n" }{MPLTEXT 1 0 23 "od; S:=SS od
; S; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 "descartesprod([]);
 descartesprod([\{1,2\}]);\n" }{MPLTEXT 1 0 65 "descartesprod([\{1,2\}
,\{a,b\}]);descartesprod([\{1,2\},\{\}]);\n" }{MPLTEXT 1 0 84 "descart
esprod([\{1,2\},\{a\},\{x,y\}]);descartesprod([\{1,2\},\{a,b,c\},\{x,y
\}]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"LG6\"-I%listG%*protec
tedG6#I$setGF)6'I\"xGF&I\"yGF&I\"iGF&I\"SGF&I#SSGF&F&F&C&@$/-I%nopsGF)
6#F%\"\"!O<\">F0-I$mapGF)6$f*6#F-F&6$I)operatorGF&I&arrowGF&F&7#F-F&F&
F&&F%6#\"\"\"?(F/\"\"#FGF5I%trueGF)C%>F1F:?&F-F0FJ?&F.&F%6#F/FJ>F1-I&u
nionGF)6$F1<#7$-I#opGF)F@F.>F0F1F0F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "%#%?G" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$7#\"\"\"7#\"\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "<&7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$F*F(" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"<&7%\"\"\"I\"aG6\"I\"xGF&7%F$F%I\"yGF&7%\"\"#F%F'7%F+F%F)" }}{PARA 11
 "" 1 "" {XPPMATH 20 "<.7%\"\"\"I\"aG6\"I\"xGF&7%F$F%I\"yGF&7%\"\"#F%F
'7%F+F%F)7%F$I\"cGF&F'7%F$F.F)7%F+F.F'7%F+F.F)7%F$I\"bGF&F'7%F$F3F)7%F
+F3F'7%F+F3F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "> "
 0 "" {MPLTEXT 1 0 64 "agent:=\{[D209,\"Peti\"],[KISZ1,\"Fleto\"],[Puf
3,\"Gyula\"]\};\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 212 "event:=\{[K
ISZ1,\"Balaton\",19930706],[Puf3,\"Nyugati\",19561108],[KISZ1,\"Motim
\",19961231],[D209,\"Paks\",20000103],[KISZ1,\"Fittelina\",19980320],[
D209,\"Gresham\",20010908],[KISZ1,\"Nomentana\",19951122]\};\n" }
{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 139 "descartesprod([agent,event]):se
lect(x->x[1][1]=x[2][1],%):map(x->[x[1][2],x[2][2],x[2][3],x[2][1]],%)
:active:=select(x->x[3]>19891023,%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20
 "<%7$I%D209G6\"Q%PetiF%7$I&KISZ1GF%Q&FletoF%7$I%Puf3GF%Q&GyulaF%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "<)7%I%Puf3G6\"Q(NyugatiF%\")36c>7%I&KISZ
1GF%Q(BalatonF%\")12$*>7%F)Q&MotimF%\")J7'*>7%I%D209GF%Q%PaksF%\").,+?
7%F)Q*FittelinaF%\")?.)*>7%F0Q(GreshamF%\")34,?7%F)Q*NomentanaF%\")A6&
*>" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7&Q&Fleto6\"Q(BalatonF%\")12$*>I
&KISZ1GF%7&F$Q&MotimF%\")J7'*>F(7&F$Q*FittelinaF%\")?.)*>F(7&F$Q*Nomen
tanaF%\")A6&*>F(7&Q%PetiF%Q%PaksF%\").,+?I%D209GF%7&F3Q(GreshamF%\")34
,?F6" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 
{PARA 5 "" 0 "" {TEXT 207 25 "1.2.2. Megjegyz\303\251s." }}}{SECT 0 
{PARA 5 "" 0 "" {TEXT 207 43 "1.2.3. Bin\303\251r rel\303\241ci\303\26
3k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "bincartprod:=proc(X::set,Y::set) local x,y,Z; Z:=\{\}
;\n" }{MPLTEXT 1 0 67 "for x in X do for y in Y do Z:=Z union \{[x,y]
\}; od; od; Z; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$
setG%*protectedG'I\"YGF&F'6%I\"xGF&I\"yGF&I\"ZGF&F&F&C%>F.<\"?&F,F%I%t
rueGF(?&F-F*F3>F.-I&unionGF(6$F.<#7$F,F-F.F&F&F&" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 33 "bincartprod(\{1,2,3\},\{a,b\});\n" }}{PARA 11
 "" 1 "" {XPPMATH 20 "<(7$\"\"$I\"aG6\"7$F$I\"bGF&7$\"\"\"F(7$F*F%7$\"
\"#F(7$F-F%" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 57 "`type/ordpair`:=proc(x) type(x,list) and nops(x)=
2 end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 " type([a,b],ordpair);
 type([1,2,3],ordpair);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6
\"F%F%F%3-I%typeG%*protectedG6$F$I%listGF)/-I%nopsGF)F#\"\"#F%F%F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "binrel:=proc(R::set(ordpair),X::set,Y::set) local r;
\n" }{MPLTEXT 1 0 65 "if nargs<1 or nargs>3 then error \": needs 1..3 \+
arguments\" fi;\n" }{MPLTEXT 1 0 15 "for r in R do\n" }{MPLTEXT 1 0 
58 "  if nargs=2 and not (member(r[1],X) and member(r[2],X))\n" }
{MPLTEXT 1 0 24 "    then return false;\n" }{MPLTEXT 1 0 60 "  elif na
rgs=3 and not (member(r[1],X) and member(r[2],Y))\n" }{MPLTEXT 1 0 23 
"    then return false\n" }{MPLTEXT 1 0 7 "  fi;\n" }{MPLTEXT 1 0 14 "
od; true; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"-I$setG%*
protectedG6#I(ordpairGF&'I\"XGF&F('I\"YGF&F(6#I\"rGF&F&F&C%@$52%&nargs
G\"\"\"2\"\"$F6YQ7:~needs~1..3~argumentsF&?&F1F%I%trueGF)@&3/F6\"\"#43
-I'memberGF)6$&F16#F7F--FE6$&F16#FAF-OI&falseGF)3/F6F943FD-FE6$FKF/FMF
=F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "binrel(\{\});R:=
bincartprod(\{1,2,3\},\{a,b\});binrel(R);\n" }{MPLTEXT 1 0 86 "binrel(
R,\{0,1,2,3\},\{a,b,c\});binrel(R,\{a,b\},\{1,2,3\});binrel(R,\{1,2,3,
a,b\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"$I\"aG6\"7$F$I\"bGF&7$\"\"\"F(7$
F*F%7$\"\"#F(7$F-F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protecte
dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 ""
 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "
id:=proc(X) local x; map(x->[x,x],X); end;id(\{1,3,a\});\n" }}{PARA 11
 "" 1 "" {XPPMATH 20 "f*6#I\"XG6\"6#I\"xGF%F%F%-I$mapG%*protectedG6$f*
F&F%6$I)operatorGF%I&arrowGF%F%7$F'F'F%F%F%F$F%F%F%" }}{PARA 11 "" 1 "
" {XPPMATH 20 "<%7$\"\"\"F$7$\"\"$F&7$I\"aG6\"F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 131 "F:=\{\{1\},\{2\},\{1,2\}\}; FF:=bincartprod(
F,F);select(x->x[1] subset x[2],FF);select(x->x[1] subset x[2] and not
 x[1]=x[2],FF);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%<#\"\"\"<#\"\"#<$
F$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<+7$<$\"\"\"\"\"#F$7$<#F%<#F&7$F
(F$7$F)F(7$F)F)7$F)F$7$F$F(7$F$F)7$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "<'7$<$\"\"\"\"\"#F$7$<#F%F$7$<#F&F*7$F*F$7$F(F(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "<$7$<#\"\"\"<$F%\"\"#7$<#F'F&" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 53 "irem(13,5);X:=\{1,2,3,4,5,6\};XX:=bincartprod(X,X
):\n" }{MPLTEXT 1 0 37 "R:=select(x->irem(x[2],x[1])=0,XX);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<
(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$
\"\"\"\"\"#7$F$\"\"$7$F$\"\"%7$F$\"\"&7$F$\"\"'7$F%F%7$F'F-7$F)F)7$F%F
)7$F%F-7$F+F+7$F$F$7$F'F'7$F-F-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 
0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 44 "1.2.4. Rel\303
\241ci\303\263k gr\303\241fja." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 
"" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 85 "1.2.5. \303\211rt
elmez\303\251si tartom\303\241ny, \303\251rt\303\251kk\303\251szlet." 
}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "R:=\{[1,a],[1,b],[2,b],[3,d],[2,d],[4,e]\};\n" }{MPLTEXT 1 0 
56 "dmn:=proc(R::set(ordpair)) map(x->x[1],R);end; dmn(R);\n" }
{MPLTEXT 1 0 57 "rng:=proc(R::set(ordpair)) map(x->x[2],R); end; rng(R
);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$
\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&F&F&F&-I$ma
pGF)6$f*6#I\"xGF&F&6$I)operatorGF&I&arrowGF&F&&F16#\"\"\"F&F&F&F%F&F&F
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11
 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&F&
F&F&-I$mapGF)6$f*6#I\"xGF&F&6$I)operatorGF&I&arrowGF&F&&F16#\"\"#F&F&F
&F%F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"aGF$I\"dGF$I\"
eGF$" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 60 "1.2.
6. Kiterjeszt\303\251s, lesz\305\261k\303\255t\303\251s." }}{PARA 0 ""
 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "R;sel
ect(x->(x[1]>1 and x[2]<>b),R);\n" }{MPLTEXT 1 0 71 "restrict:=proc(R:
:set(ordpair),X::set) select(x->(x[1] in X),R); end;\n" }{MPLTEXT 1 0 
27 "X:=\{2,3\};restrict(R,X);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$
\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$\"\"$I\"dG6\"7$\"\"#F%7$\"\"%I\"eG
F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"RG6\"-I$setG%*protectedG6#
I(ordpairGF&'I\"XGF&F(F&F&F&-I'selectGF)6$f*6#I\"xGF&F&6$I)operatorGF&
I&arrowGF&F&-I#inGF)6$&F36#\"\"\"F-F&F&6$F-9%F%F&F&F&" }}{PARA 11 "" 1
 "" {XPPMATH 20 "<$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$\"
\"#I\"bG6\"7$\"\"$I\"dGF&7$F$F)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 
0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 14 "1.2.7. Inverz."
 }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 59 "relinv:=proc(R::set(ordpair)) map(x->[x[2],x[1]],R); end;\n" }
{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 55 "R;dmn(R);rng(R);S:=relinv(R);dmn
(S);rng(S);relinv(S);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"
-I$setG%*protectedG6#I(ordpairGF&F&F&F&-I$mapGF)6$f*6#I\"xGF&F&6$I)ope
ratorGF&I&arrowGF&F&7$&F16#\"\"#&F16#\"\"\"F&F&F&F%F&F&F&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"d
GF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&\"\"\"\"\"#
\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"aGF$I\"dGF$I
\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$I\"bG6\"\"\"\"7$I\"aGF%F&7
$F$\"\"#7$I\"dGF%\"\"$7$F,F*7$I\"eGF%\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<&I\"bG6\"I\"aGF$I\"dGF$I\"eGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "
<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eG
F&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 
{PARA 5 "" 0 "" {TEXT 207 55 "1.2.8. Halmaz k\303\251pe \303\251s inve
rz k\303\251pe." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 62 "mapset:=proc(R::set(ordpair),A::set) rng(restr
ict(R,A)) end;\n" }{MPLTEXT 1 0 73 "invmapset:=proc(R::set(ordpair),A:
:set) rng(restrict(relinv(R),A)) end;\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 64 "R;S:=relinv(R);A:=\{1,4\};B:=\{b,e\}; mapset(R,A);map
set(R,B);\n" }{MPLTEXT 1 0 29 "invmapset(R,B);mapset(S,B);\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "f*6$'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&
'I\"AGF&F(F&F&F&-I$rngGF&6#-I)restrictGF&6$F%F-F&F&F&" }}{PARA 11 "" 1
 "" {XPPMATH 20 "f*6$'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&'I\"AGF
&F(F&F&F&-I$rngGF&6#-I)restrictGF&6$-I'relinvGF&6#F%F-F&F&F&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I
\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$I\"bG6
\"\"\"\"7$I\"aGF%F&7$F$\"\"#7$I\"dGF%\"\"$7$F,F*7$I\"eGF%\"\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<$I\"bG6\"I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"b
G6\"I\"aGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}{PARA 11 
"" 1 "" {XPPMATH 20 "<%\"\"\"\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "<%\"\"\"\"\"#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%
?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 32 "1.2.9. Kompoz\303\255ci
\303\263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 70 "relcomp:=proc(R::set(ordpair),S::set(ordpair)) local
 r,s,T; T:=\{\};\n" }{MPLTEXT 1 0 29 "for r in R do for s in S do\n" }
{MPLTEXT 1 0 53 "  if s[2]=r[1] then T:=T union \{[s[1],r[2]]\}; fi;\n
" }{MPLTEXT 1 0 17 "od; od; T; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 
1 0 45 "R;S:=\{[aa,1],[bb,3],[cc,5]\};relcomp(R,S);\n" }}{PARA 11 "" 1
 "" {XPPMATH 20 "f*6$'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&'I\"SGF
&F'6%I\"rGF&I\"sGF&I\"TGF&F&F&C%>F1<\"?&F/F%I%trueGF)?&F0F-F6@$/&F06#
\"\"#&F/6#\"\"\">F1-I&unionGF)6$F1<#7$&F0F>&F/F;F1F&F&F&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dG
F&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$I#aaG6\"\"
\"\"7$I#bbGF%\"\"$7$I#ccGF%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$
I#aaG6\"I\"bGF%7$F$I\"aGF%7$I#bbGF%I\"dGF%" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 37 "1.2.
10. \303\201ll\303\255t\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "R;S:=\{[aa,1],[aa,2],[bb,3],
[cc,4],[dd,5]\};\n" }{MPLTEXT 1 0 30 "relcomp(R,S);rng(R);rng(%%);\n" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%
7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'
7$I#aaG6\"\"\"#7$I#ccGF%\"\"%7$I#ddGF%\"\"&7$F$\"\"\"7$I#bbGF%\"\"$" }
}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$I#aaG6\"I\"bGF%7$F$I\"aGF%7$I#bbGF%
I\"dGF%7$F$F+7$I#ccGF%I\"eGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG
6\"I\"aGF$I\"dGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I
\"aGF$I\"dGF$I\"eGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "T:=
\{[xx,aa],[xx,cc]\};relcomp(R,relcomp(S,T));relcomp(relcomp(R,S),T);\n
" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$7$I#xxG6\"I#aaGF%7$F$I#ccGF%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "<&7$I#xxG6\"I\"bGF%7$F$I\"aGF%7$F$I\"dGF
%7$F$I\"eGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&7$I#xxG6\"I\"bGF%7$F$I
\"aGF%7$F$I\"dGF%7$F$I\"eGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "relinv(relcomp(R,S));relcomp(relinv(S),relinv(R));\n" }}{PARA 11
 "" 1 "" {XPPMATH 20 "<'7$I\"bG6\"I#aaGF%7$I\"aGF%F&7$I\"dGF%I#bbGF%7$
F*F&7$I\"eGF%I#ccGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$I\"bG6\"I#aa
GF%7$I\"aGF%F&7$I\"dGF%I#bbGF%7$F*F&7$I\"eGF%I#ccGF%" }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 
207 37 "1.2.11. \303\201ll\303\255t\303\241s." }}{PARA 0 "" 0 "" {TEXT
 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "R;IX:=id(\{1,2,3
,4,5\});relcomp(R,IX);IY:=id(\{a,b,c,d,e\});relcomp(IY,R);\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I
\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$\"\"#F
$7$\"\"%F&7$\"\"&F(7$\"\"\"F*7$\"\"$F," }}{PARA 11 "" 1 "" {XPPMATH 20
 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"
eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$I\"aG6\"F$7$I\"bGF%F'7$I\"cG
F%F)7$I\"dGF%F+7$I\"eGF%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"
I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 
0 "" {TEXT 207 24 "1.2.12. Rendez\303\251s." }}{PARA 0 "" 0 "" {TEXT 
201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "istransitive:=pro
c(R::set(ordpair)) local r,s;\n" }{MPLTEXT 1 0 29 "for r in R do for s
 in R do\n" }{MPLTEXT 1 0 63 "  if r[2]=s[1] and not [r[1],s[2]] in R \+
then return false fi;\n" }{MPLTEXT 1 0 20 "od; od; true; end;\n" }
{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 93 "X:=\{1,2,3,4,5,6\};XX:=bincartpr
od(X,X):R:=select(x->irem(x[2],x[1])=0,XX);istransitive(R);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protectedG6#I(ordp
airGF&6$I\"rGF&I\"sGF&F&F&C$?&F-F%I%trueGF)?&F.F%F1@$3/&F-6#\"\"#&F.6#
\"\"\"4-I#inGF)6$7$&F-F:&F.F7F%OI&falseGF)F1F&F&F&" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "<(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<07$\"\"\"\"\"#7$F$\"\"$7$F$\"\"%7$F$\"\"&7$F$\"\"'7$F%F%
7$F'F-7$F)F)7$F%F)7$F%F-7$F+F+7$F$F$7$F'F'7$F-F-" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 54 "istrichotom:=proc(X::set,R::set(ordpair)) local x,y;\n" }
{MPLTEXT 1 0 41 "if not binrel(R,X) then return FAIL fi;\n" }{MPLTEXT 
1 0 29 "for x in X do for y in X do\n" }{MPLTEXT 1 0 72 "  if x<>y the
n if ([x,y] in R and [y,x] in R) or ((not [x,y] in R) and\n" }{MPLTEXT
 1 0 48 "    (not [y,x] in R)) then return false fi;fi;\n" }{MPLTEXT 
1 0 19 "od;od; true; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 19 "is
trichotom(X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG
%*protectedG'I\"RGF&-F'6#I(ordpairGF&6$I\"xGF&I\"yGF&F&F&C%@$4-I'binre
lGF&6$F*F%OI%FAILGF(?&F/F%I%trueGF(?&F0F%F:@$0F/F0@$53-I#inGF(6$7$F/F0
F*-FB6$7$F0F/F*45FAFEOI&falseGF(F:F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 
"isordering:=proc(X::set,R::set(ordpair)) local x;\n" }{MPLTEXT 1 0 
41 "if not binrel(R,X) then return FAIL fi;\n" }{MPLTEXT 1 0 44 "istra
nsitive(R) and istrichotom(X,R); end;\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 102 "X:=\{a,u,i,e,o\};R:=\{[a,u],[a,i],[a,e],[a,o],[u,i],
[u,e],[u,o],[i,e],[i,o],[e,o]\};isordering(X,R);\n" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&
6#I\"xGF&F&F&C$@$4-I'binrelGF&6$F*F%OI%FAILGF(3-I-istransitiveGF&6#F*-
I,istrichotomGF&6$F%F*F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'I\"iG6
\"I\"aGF$I\"uGF$I\"oGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<,7$I
\"aG6\"I\"uGF%7$F$I\"iGF%7$F$I\"eGF%7$F$I\"oGF%7$F&F(7$F&F*7$F&F,7$F(F
*7$F(F,7$F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 ""
 0 "" {TEXT 207 21 "1.2.13. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 22 "1.2.
14. Intervallumok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 63 "int_o_o:=proc(X::set,R::set(ordpair),x,y) l
ocal S,z; S:=\{\};\n" }{MPLTEXT 1 0 16 "for z in X do \n" }{MPLTEXT 1 
0 59 "  if [x,z] in R and [z,y] in R then S:=S union \{z\}; fi;\n" }
{MPLTEXT 1 0 13 "od; S; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 63 
"int_o_c:=proc(X::set,R::set(ordpair),x,y) local S,z; S:=\{\};\n" }
{MPLTEXT 1 0 16 "for z in X do \n" }{MPLTEXT 1 0 68 "  if [x,z] in R a
nd ([z,y] in R or z=y) then S:=S union \{z\}; fi;\n" }{MPLTEXT 1 0 13 
"od; S; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 61 "int_i_o:=proc(X
::set,R::set(ordpair),x) local S,y; S:=\{\};\n" }{MPLTEXT 1 0 16 "for \+
y in X do \n" }{MPLTEXT 1 0 44 "  if [y,x] in R then S:=S union \{y\};
 fi;\n" }{MPLTEXT 1 0 13 "od; S; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "f*6&'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&I\"xGF&I\"
yGF&6$I\"SGF&I\"zGF&F&F&C%>F1<\"?&F2F%I%trueGF(@$3-I#inGF(6$7$F.F2F*-F
;6$7$F2F/F*>F1-I&unionGF(6$F1<#F2F1F&F&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6&'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&I
\"xGF&I\"yGF&6$I\"SGF&I\"zGF&F&F&C%>F1<\"?&F2F%I%trueGF(@$3-I#inGF(6$7
$F.F2F*5-F;6$7$F2F/F*/F2F/>F1-I&unionGF(6$F1<#F2F1F&F&F&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordp
airGF&I\"xGF&6$I\"SGF&I\"yGF&F&F&C%>F0<\"?&F1F%I%trueGF(@$-I#inGF(6$7$
F1F.F*>F0-I&unionGF(6$F0<#F1F0F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "X;R;int_o_o(X,R,u,o);int_o_c(X,R,a,o);int_i_o(X,R,o);
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'I\"iG6\"I\"aGF$I\"uGF$I\"oGF$I\"
eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<,7$I\"aG6\"I\"uGF%7$F$I\"iGF%7$
F$I\"eGF%7$F$I\"oGF%7$F&F(7$F&F*7$F&F,7$F(F*7$F(F,7$F*F," }}{PARA 11 "
" 1 "" {XPPMATH 20 "<$I\"iG6\"I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"<&I\"iG6\"I\"uGF$I\"oGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I
\"iG6\"I\"aGF$I\"uGF$I\"eGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "$ 2..6;$ 1..9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\"$\"\"
%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+\"\"\"\"\"#\"\"$\"\"%\"
\"&\"\"'\"\"(\"\")\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#
%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 44 "1.2.15. Legkisebb \303
\251s legnagyobb elem." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "minimal:=proc(X::set,R::set(ordpair
)) local x,y,f;\n" }{MPLTEXT 1 0 25 "for x in X do f:=true; \n" }
{MPLTEXT 1 0 74 "  for y in X do if x<>y and not [x,y] in R then f:=fa
lse; break; fi; od;\n" }{MPLTEXT 1 0 26 "  if f then return x fi;\n" }
{MPLTEXT 1 0 16 "od; NULL; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 
52 "maximal:=proc(X::set,R::set(ordpair)) local x,y,f;\n" }{MPLTEXT 1 
0 25 "for x in X do f:=true; \n" }{MPLTEXT 1 0 74 "  for y in X do if \+
x<>y and not [y,x] in R then f:=false; break; fi; od;\n" }{MPLTEXT 1 
0 26 "  if f then return x fi;\n" }{MPLTEXT 1 0 16 "od; NULL; end;\n" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF
&-F'6#I(ordpairGF&6%I\"xGF&I\"yGF&I\"fGF&F&F&C$?&F/F%I%trueGF(C%>F1F4?
&F0F%F4@$30F/F04-I#inGF(6$7$F/F0F*C$>F1I&falseGF([@$F1OF/I%NULLGF(F&F&
F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I
\"RGF&-F'6#I(ordpairGF&6%I\"xGF&I\"yGF&I\"fGF&F&F&C$?&F/F%I%trueGF(C%>
F1F4?&F0F%F4@$30F/F04-I#inGF(6$7$F0F/F*C$>F1I&falseGF([@$F1OF/I%NULLGF
(F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "X;R;minimal(X,R)
;maximal(X,R);minimal(\{\},R);maximal(\{\},R);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<'I\"iG6\"I\"aGF$I\"uGF$I\"oGF$I\"eGF$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "<,7$I\"aG6\"I\"uGF%7$F$I\"iGF%7$F$I\"eGF%7$F$I\"oGF%7$F
&F(7$F&F*7$F&F,7$F(F*7$F(F,7$F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"
aG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"oG6\"" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 24 
"1.2.16. Korl\303\241tok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "islowerbound:=proc(X::set,R::set(or
dpair),Y::set,x) local y;\n" }{MPLTEXT 1 0 68 "for y in Y do if x<>y a
nd not [x,y] in R then return false fi; od;\n" }{MPLTEXT 1 0 12 "true;
 end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 62 "isupperbound:=proc(X::
set,R::set(ordpair),Y::set,x) local y;\n" }{MPLTEXT 1 0 68 "for y in Y
 do if x<>y and not [y,x] in R then return false fi; od;\n" }{MPLTEXT 
1 0 12 "true; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6\"I$s
etG%*protectedG'I\"RGF&-F'6#I(ordpairGF&'I\"YGF&F'I\"xGF&6#I\"yGF&F&F&
C$?&F2F/I%trueGF(@$30F0F24-I#inGF(6$7$F0F2F*OI&falseGF(F5F&F&F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6\"I$setG%*protectedG'I\"RGF&-
F'6#I(ordpairGF&'I\"YGF&F'I\"xGF&6#I\"yGF&F&F&C$?&F2F/I%trueGF(@$30F0F
24-I#inGF(6$7$F2F0F*OI&falseGF(F5F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "X;R;islowerbound(X,R,\{u,i,e\},a);isupperbound(X,R,\{
a,u,i\},i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'I\"iG6\"I\"aGF$I\"uGF$I
\"oGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<,7$I\"aG6\"I\"uGF%7$F$
I\"iGF%7$F$I\"eGF%7$F$I\"oGF%7$F&F(7$F&F*7$F&F,7$F(F*7$F(F,7$F*F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 71 "lowerbounds:=proc(X::set,R::set(ordpair),Y::set) local S,x; S
:=\{\}; \n" }{MPLTEXT 1 0 15 "for x in X do\n" }{MPLTEXT 1 0 55 "  if \+
islowerbound(X,R,Y,x) then S:=S  union \{x\} fi;\n" }{MPLTEXT 1 0 13 "
od; S; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*pr
otectedG'I\"RGF&-F'6#I(ordpairGF&'I\"YGF&F'6$I\"SGF&I\"xGF&F&F&C%>F1<
\"?&F2F%I%trueGF(@$-I-islowerboundGF&6&F%F*F/F2>F1-I&unionGF(6$F1<#F2F
1F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "upperbounds:=pro
c(X::set,R::set(ordpair),Y::set) local S,x; S:=\{\};\n" }{MPLTEXT 1 0 
15 "for x in X do\n" }{MPLTEXT 1 0 55 "  if isupperbound(X,R,Y,x) then
 S:=S  union \{x\} fi;\n" }{MPLTEXT 1 0 13 "od; S; end;\n" }}{PARA 11 
"" 1 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ord
pairGF&'I\"YGF&F'6$I\"SGF&I\"xGF&F&F&C%>F1<\"?&F2F%I%trueGF(@$-I-isupp
erboundGF&6&F%F*F/F2>F1-I&unionGF(6$F1<#F2F1F&F&F&" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 55 "X;R;lowerbounds(X,R,\{i,e\}); upperbounds(X
,R,\{i,e\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'I\"iG6\"I\"aGF$I\"uGF$
I\"oGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<,7$I\"aG6\"I\"uGF%7$F
$I\"iGF%7$F$I\"eGF%7$F$I\"oGF%7$F&F(7$F&F*7$F&F,7$F(F*7$F(F,7$F*F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"iG6\"I\"aGF$I\"uGF$" }}{PARA 11 "" 
1 "" {XPPMATH 20 "<$I\"oG6\"I\"eGF$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "inf:=proc(X::set,R::set(ordpair),Y::set)\n" }{MPLTEXT
 1 0 37 "maximal(lowerbounds(X,R,Y),R); end;\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 42 "sup:=proc(X::set,R::set(ordpair),Y::set)\n" }{MPLTEXT
 1 0 37 "minimal(upperbounds(X,R,Y),R); end;\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 36 "inf(X,R,\{i,e\});sup(X,R,\{i,e\});\n" }}{PARA 11 "" 1
 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpair
GF&'I\"YGF&F'F&F&F&-I(maximalGF&6$-I,lowerboundsGF&6%F%F*F/F*F&F&F&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*protectedG'I\"RGF&
-F'6#I(ordpairGF&'I\"YGF&F'F&F&F&-I(minimalGF&6$-I,upperboundsGF&6%F%F
*F/F*F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"iG6\"" }}{PARA 11 "" 1
 "" {XPPMATH 20 "I\"eG6\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "
%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 34 "1.2.17. J\303\263lren
dez\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 60 "iswellordering:=proc(X::set,R::set(ordpair)) loc
al f,Y,P; \n" }{MPLTEXT 1 0 48 "f:=isordering(X,R); if not f then retu
rn f fi;\n" }{MPLTEXT 1 0 17 "P:=powerset(X);\n" }{MPLTEXT 1 0 15 "for
 Y in P do\n" }{MPLTEXT 1 0 69 "  if Y<>\{\} then f:=minimal(Y,R); if \+
f=NULL then return f; fi; fi;\n" }{MPLTEXT 1 0 16 "od; true; end;\n" }
}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&
-F'6#I(ordpairGF&6%I\"fGF&I\"YGF&I\"PGF&F&F&C'>F/-I+isorderingGF&6$F%F
*@$4F/OF/>F1-_I)combinatG6$F(I(_syslibGF&I)powersetGF&6#F%?&F0F1I%true
GF(@$0F0<\"C$>F/-I(minimalGF&6$F0F*@$/F/I%NULLGF(F9FCF&F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "X;R;iswellordering(X,R);\n" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "<'I\"iG6\"I\"aGF$I\"uGF$I\"oGF$I\"eGF$
" }}{PARA 11 "" 1 "" {XPPMATH 20 "<,7$I\"aG6\"I\"uGF%7$F$I\"iGF%7$F$I
\"eGF%7$F$I\"oGF%7$F&F(7$F&F*7$F&F,7$F(F*7$F(F,7$F*F," }}{PARA 11 "" 1
 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.
18. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "
" 0 "" {TEXT 206 29 "1.3. F\303\274ggv\303\251nyek" }}{PARA 0 "" 0 "" 
{TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 30 "1.3.1. F\303\27
4ggv\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 62 "isfunction:=proc(f::set(ordpair),X::set,Y::set
) local x,y,S;\n" }{MPLTEXT 1 0 43 "if not binrel(args) then return fa
lse fi;\n" }{MPLTEXT 1 0 29 "for x in dmn(f) do S:=\{\};\n" }{MPLTEXT 
1 0 22 "  for y in rng(f) do\n" }{MPLTEXT 1 0 45 "    if [x,y] in f th
en S:=S union \{y\} fi;\n" }{MPLTEXT 1 0 42 "  od; if nops(S)>1 then r
eturn false fi;\n" }{MPLTEXT 1 0 16 "od; true; end;\n" }}{PARA 11 "" 1
 "" {XPPMATH 20 "f*6%'I\"fG6\"-I$setG%*protectedG6#I(ordpairGF&'I\"XGF
&F('I\"YGF&F(6%I\"xGF&I\"yGF&I\"SGF&F&F&C%@$4-I'binrelGF&6#%%argsGOI&f
alseGF)?&F1-I$dmnGF&6#F%I%trueGF)C%>F3<\"?&F2-I$rngGF&F@FA@$-I#inGF)6$
7$F1F2F%>F3-I&unionGF)6$F3<#F2@$2\"\"\"-I%nopsGF)6#F3F;FAF&F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f:=id(\{a,b\});isfunction(f)
;isfunction(f,\{a,b,1\});\n" }{MPLTEXT 1 0 71 "isfunction(f,\{a,b\},\{
1,2\});f:=\{[a,1],[b,2],[a,2]\};isfunction(f);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<$7$I\"aG6\"F$7$I\"bGF%F'" }}{PARA 11 "" 1 "" {XPPMATH 20
 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protec
tedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11
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1 0 37 "X:=\{1,2,3\}; XX:=bincartprod(X,X):\n" }{MPLTEXT 1 0 61 "R:=se
lect(x->x[1]<x[2],XX);Y:=X;S:=select(x->x[1]>x[2],XX);\n" }{MPLTEXT 1 
0 2 "\n" }{MPLTEXT 1 0 35 "f:=x->x; isincreasing(f,X,R,Y,S);\n" }
{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 37 "f:=x->4-x; isincreasing(f,Y,S,X,
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45-FN6$7$-F%6#F6-F%6#F7F4/FVFXOI&falseGF(FIF&F&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$
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\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$\"\"#\"\"\"7$\"\"$F%7$F'
F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arr
owGF%F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arrowG
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{SECT 0 {PARA 5 "" 0 "" {TEXT 207 23 "1.3.4. Csal\303\241dok." }}
{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "issetfamily:=proc(Iset::set,f::procedure) local i;\n" }{MPLTEXT 
1 0 62 "for i in Iset do if not type(f(i),set) then return false fi;\n
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1 0 39 "f:='f';f(1):=\{a,b\};f(2):=\{b,c,d\};\n" }{MPLTEXT 1 0 50 "iss
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{XPPMATH 20 "f*6$'I%IsetG6\"I$setG%*protectedG'I\"fGF&I*procedureGF(6#
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{XPPMATH 20 "<$I\"aG6\"I\"bGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"b
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{PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" 
{TEXT 207 34 "1.3.5. De Morgan-szab\303\241lyok." }}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "
" "%#%?G" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 25 "1.3.6. Megjegyz\303
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}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "`intersect`();" }}{PARA 8 "
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{PARA 5 "" 0 "" {TEXT 207 66 "1.3.8. T\303\266bbv\303\241ltoz\303\263s
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot3d(sin(x*y),x=-Pi..Pi,y=-Pi..Pi
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 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 
207 29 "1.3.10. P\303\251ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "X:=\{1,2,3\};P:=combinat
[powerset](X);isbinop(P,(x,y)->x union y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<*<\"
<#\"\"\"<#\"\"#<$F%F'<$F'\"\"$<#F*<$F%F*<%F%F'F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2
 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 87 "1.3.11. P\303
\251ld\303\241k: m\305\261velet megad\303\241sa t\303\241bl\303\241zat
tal." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "true and true; true and false; false and true; false \+
and false;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 103 "T:=table();T[tru
e,true]:=true;T[true,false]:=false;T[false,true]:=false;T[false,false]
:=false;print(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20
 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"I&falseG%*
protectedGE\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1
 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&TABLEG%*prot
ectedG6#7&/6$I%trueGF$I&falseGF$F*/6$F*F)F*/6$F)F)F)/6$F*F*F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "X:=\{true,false\};XX:=bincar
tprod(X,X);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 63 "map(x->cat(\"\",
x[1],\" and \",x[2],\"=\",x[1] and x[2]),XX);\n" }{MPLTEXT 1 0 61 "map
(x->cat(\"\",x[1],\" or \",x[2],\"=\",x[1] or x[2]),XX);\n" }{MPLTEXT 
1 0 71 "map(x->cat(\"\",x[1],\" implies \",x[2],\"=\",x[1] implies x[2
]),XX);\n" }{MPLTEXT 1 0 66 "map(x->cat(\"\",x[1],\" iff \",x[2],\"=\"
,evalb(x[1]=x[2])),XX);\n" }{MPLTEXT 1 0 63 "map(x->cat(\"\",x[1],\" x
or \",x[2],\"=\",x[1] xor x[2]),XX);\n" }}{PARA 11 "" 1 "" {XPPMATH 20
 "<$I&falseG%*protectedGI%trueGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&7
$I&falseG%*protectedGI%trueGF%7$F&F$7$F&F&7$F$F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<&Q5false~and~true=false6\"Q5true~and~false=falseF$Q6fals
e~and~false=falseF$Q3true~and~true=trueF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<&Q2true~or~true=true6\"Q3true~or~false=trueF$Q5false~or~
false=falseF$Q3false~or~true=trueF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<
&Q8false~implies~true=true6\"Q7true~implies~true=trueF$Q9true~implies~
false=falseF$Q9false~implies~false=trueF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<&Q5false~iff~true=false6\"Q5true~iff~false=falseF$Q3true
~iff~true=trueF$Q5false~iff~false=trueF$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "<&Q4true~xor~false=true6\"Q6false~xor~false=falseF$Q4false~xor~tru
e=trueF$Q4true~xor~true=falseF$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 
0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 53 "1.3.12. M\305\2
61veletek f\303\274ggv\303\251nyekkel." }}{PARA 0 "" 0 "" {TEXT 201 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "f:=x->x^2; g:=x->x^3; (
f*g)(2);(f*g)(3);(f/g)(2);(f/g)(0);(g/f)(0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arrowGF%F%*$)F$\"\"#\"\"\"
F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I
&arrowGF%F%*$)F$\"\"$\"\"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#
K" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$V#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "#\"\"\"\"\"#" }}{PARA 8 "" 1 "" {TEXT 209 42 "Error, numeric excep
tion: division by zero" }}{PARA 8 "" 1 "" {TEXT 209 42 "Error, numeric
 exception: division by zero" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "
" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "1.3.13. P\303\251l
d\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 78 "f:=[true,true,false,false];g:=[true,false,true,fal
se];zip((x,y)->x and y,f,g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&I%true
G%*protectedGF#I&falseGF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&I%trueG
%*protectedGI&falseGF$F#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&I%trueG%
*protectedGI&falseGF$F%F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "
%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "1.3.14. P\303\251ld\3
03\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "f:=x->x^2; g:=x->x^3;\n" }{MPLTEXT 1 0 20 "(f+g)(2);(
f+g)(3);\n" }{MPLTEXT 1 0 20 "(f*g)(2);(f*g)(3);\n" }{MPLTEXT 1 0 29 "
(f/g)(2);(f/g)(0);(g/f)(0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"
xG6\"F%6$I)operatorGF%I&arrowGF%F%*$)F$\"\"#\"\"\"F%F%F%" }}{PARA 11 "
" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arrowGF%F%*$)F$\"\"
$\"\"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#7" }}{PARA 11 "" 1 "
" {XPPMATH 20 "\"#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#K" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"$V#" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"\"\"
\"#" }}{PARA 8 "" 1 "" {TEXT 209 42 "Error, numeric exception: divisio
n by zero" }}{PARA 8 "" 1 "" {TEXT 209 42 "Error, numeric exception: d
ivision by zero" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}
}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 61 "1.3.15. M\305\261velettart\303
\263 lek\303\251pez\303\251sek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "ishom:=proc(phi::procedure,X
::set,f::procedure,Y::set,g::procedure)\n" }{MPLTEXT 1 0 12 "local x,y
;\n" }{MPLTEXT 1 0 52 "if not isarrowfromto(phi,X,Y) then return FAIL \+
fi;\n" }{MPLTEXT 1 0 42 "if not isbinop(X,f) then return FAIL fi;\n" }
{MPLTEXT 1 0 42 "if not isbinop(Y,g) then return FAIL fi;\n" }{MPLTEXT
 1 0 29 "for x in X do for y in X do\n" }{MPLTEXT 1 0 58 "  if phi(f(x
,y))<>g(phi(x),phi(y)) then return false fi;\n" }{MPLTEXT 1 0 19 "od; \+
od; true end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 70 "X:=\{true,fals
e\};Y:=X;ishom(x->x,X,(x,y)->x and y,Y,(x,y)->x or y);\n" }{MPLTEXT 1 
0 2 "\n" }{MPLTEXT 1 0 52 "ishom(x-> not x,X,(x,y)->x and y,Y,(x,y)->x
 or y);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6''I$phiG6\"I*procedureG%
*protectedG'I\"XGF&I$setGF('I\"fGF&F''I\"YGF&F+'I\"gGF&F'6$I\"xGF&I\"y
GF&F&F&C'@$4-I.isarrowfromtoGF&6%F%F*F/OI%FAILGF(@$4-I(isbinopGF&6$F*F
-F;@$4-F@6$F/F1F;?&F3F*I%trueGF(?&F4F*FG@$0-F%6#-F-F2-F16$-F%6#F3-F%6#
F4OI&falseGF(FGF&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I&falseG%*pro
tectedGI%trueGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I&falseG%*protecte
dGI%trueGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 
21 "1.3.16. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a:='a':x:='x':y:='y':a^(x+y);expand
(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ")I\"aG6\",&I\"xGF$\"\"\"I\"yGF$F
'" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&)I\"aG6\"I\"xGF%\"\"\")F$I\"yGF%F
'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT 205 16 "2. Sz\303\241mok" }}{EXCHG {PARA 0 "" 0 
"" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 34 "3
. Hat\303\241r\303\251rt\303\251k" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2
 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 51 "4. Differenci
\303\241lsz\303\241m\303\255t\303\241s" }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 47 "5. I
ntegr\303\241lsz\303\241m\303\255t\303\241s" }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 2 0 "" "%#%?G" }}}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {XPPEDIT 19 1 "" "%#%?G" }}}}
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