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    <identity>
      <id>be8ed228-7b9f-4d88-9521-f6039e697cd5</id>
      <revision>39</revision>
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      <exponentialThreshold>5</exponentialThreshold>
      <fractions>decimal</fractions>
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      <margins left="39" right="39" top="39" bottom="39" />
      <header alignment="Center" color="#a9a9a9">&amp;[DATE] &amp;[TIME] - &amp;[FILENAME]</header>
      <footer alignment="Center" color="#a9a9a9">&amp;[PAGENUM] / &amp;[COUNT]</footer>
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  <region id="0" left="18" top="18" width="267" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p bold="true">Integrale mit SMath Bordmitteln</p>
    </text>
    <text lang="eng">
      <p bold="true">Integrals with native SMath functions</p>
    </text>
  </region>
  <region id="1" left="549" top="18" width="94" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Mathematik.sm">Mathematik</p>
    </text>
    <text lang="eng">
      <p href="Mathematik.sm">Mathematics</p>
    </text>
  </region>
  <region id="2" left="549" top="36" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
  </region>
  <region id="3" left="18" top="72" width="659" height="40" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Die Benutzung der Bordmittel ist nur zu empfehlen, wenn das Plugin Maxima nicht
zur Verfügung steht oder mit der gegebenen Aufgabe scheitert.</p>
    </text>
    <text lang="eng">
      <p>Usage of native integration functions is not recommended unless you do not have the
Maxima plugin or it fails with the given task.</p>
    </text>
  </region>
  <region id="4" left="18" top="126" width="699" height="328" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>SMath kann nur bestimmte Integrale (mit Integrationsgrenzen) und diese auch nur 
numerisch berechnen.

• Größen mit Maßeinheiten an den Integrationsgrenzen werden in Basiseinheiten 
    umgewandelt und dann als dimensionslos betrachtet.  

• Funktionen dürfen keine if()-Anweisung enthalten. Sollen stückweise stetige 
    Funktionen integriert werden, kann man sie durch arithmetische Ausdrücke mit 
    logischen Operatoren definieren. 

• Die Integrationsgenauigkeit wird global unter Hauptmenü&gt; Extras&gt; Einstellungen&gt; 
    Berechnung&gt; Integrale: Genauigkeit eingestellt. Der Wert gibt die Zahl der 
    Intervalle zwischen den Integrationsgrenzen an, die bei der numerischen 
    Berechnung verwendet wird. Im Zweifelsfall muss man den Wert erhöhen und prüfen, 
    in welchem Maße das Ergebnis von der Einstellung abhängt. Bei hinreichend glatten 
    Funktionen ist die Voreinstellung (100) gut geeignet, bei Funktionen mit 
    Unstetigkeiten können schnell höhere Werte erforderlich sein.

• Im Interesse der Genauigkeit ist es vorteilhaft, das Integrationsgebiet an 
    Unstetigkeitsstellen aufzuteilen.</p>
    </text>
    <text lang="eng">
      <p>SMath can only handle definite integrals (over a specified interval) and even that 
only numerically.

• Integration limits with units are transformed to base units and then treated as 
    dimensionless. 

• Integrands must not contain if() statements. If piecewise continuous functions 
    are to be integrated, then they have to be expressed using boolean operators.

• The number of intervals for numeric integration is set globally using Main menu&gt;
    Tools&gt; Options&gt; Calculation&gt; Integrals: Accuracy. If in doubt, increase the value 
    and check if the result depends on that setting. If the function is sufficiently
    smooth, then the default value (100) is appropriate. Functions with discontinuities
    may easily require much higher values.

• For improved accuracy, the integration interval should be split at discontinuity 
    points.</p>
    </text>
  </region>
  <region id="5" left="18" top="468" width="715" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Unbestimmte Integrale oder symbolische Auswertung sind mit den eingebauten Funktionen 
in SMath nicht möglich. Alternativ kann das Plugin Maple Wrapper benutzt werden.</p>
    </text>
    <text lang="eng">
      <p>Indefinite integrals or symbolic evaluation are not possible within SMath. As an 
alternative you can use the plugin Maple Wrapper.</p>
    </text>
  </region>
  <region id="6" left="18" top="522" width="226" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Section math integration units.sm">Integrale mit Maßeinheiten</p>
    </text>
    <text lang="eng">
      <p href="Section math integration units.sm">Integrals with units</p>
    </text>
  </region>
  <region id="7" left="18" top="558" width="684" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand" style="string">surrender</e>
        <e type="function" preserve="true" args="1">MaximaControl</e>
      </input>
      <result action="numeric">
        <e type="operand" style="string">int(), diff(), sum() and lim() now are native SMath</e>
      </result>
    </math>
  </region>
  <region id="8" left="18" top="594" width="357" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p bold="true">Integration stückweise stetiger Funktionen</p>
    </text>
    <text lang="eng">
      <p bold="true">Integration of piecewise continuous functions</p>
    </text>
  </region>
  <region id="9" left="18" top="630" width="714" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Die Integrationsfunktion int() versagt, wenn Funktionen unter Verwendung von if() 
definiert werden. Es kommt dann die unsinnige Fehlermeldung, die Integrationsvariable 
sei nicht definiert.</p>
    </text>
    <text lang="eng">
      <p>The integration function int() fails if the integrand is defined using if() statements.
In this case, a senseless error message is thrown (integration variable not defined).</p>
    </text>
  </region>
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        <e type="bracket">(</e>
        <e type="operand">t</e>
        <e type="operand">1</e>
        <e type="operator" args="2">≤</e>
        <e type="bracket">(</e>
        <e type="operator" args="2">&amp;</e>
        <e type="operand">5</e>
        <e type="operand">t</e>
        <e type="operator" args="2">*</e>
        <e type="operand">0</e>
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        <e type="function" args="1">v</e>
        <e type="operand">t</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="4">int</e>
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        <e type="operand">#</e>
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  </region>
  <region id="13" left="18" top="810" width="655" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Der Ausweg besteht in der Verwendung von logischen Operatoren, mit denen man 
Ausdrücke definieren kann, die nur in einem bestimmten Bereich Eins, ansonsten 
aber Null sind:</p>
    </text>
    <text lang="eng">
      <p>As a workaround, you can define the function using boolean operators, which evaluate
to one if true and zero if false.</p>
    </text>
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  <region id="14" left="18" top="882" width="166" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
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        <e type="operand">t</e>
        <e type="operand">0</e>
        <e type="operator" args="2">≥</e>
        <e type="bracket">(</e>
        <e type="operator" args="2">*</e>
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        <e type="operand">1</e>
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        <e type="operand">x</e>
        <e type="function" args="1">w</e>
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  </region>
  <region id="16" left="432" top="882" width="160" height="66" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">t</e>
        <e type="function" args="1">w</e>
        <e type="operand">t</e>
        <e type="operand">2</e>
        <e type="operator" args="1">-</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="4">int</e>
      </input>
      <result action="numeric">
        <e type="operand">2.5333</e>
      </result>
    </math>
  </region>
  <region id="17" left="18" top="990" width="625" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Eine Konvergenzanalyse zeigt folgende Abhängigkeit des Ergebnisses von der 
eingestellten Genauigkeit:</p>
    </text>
    <text lang="eng">
      <p>A convergence study shows the dependence of the result from the accuracy setting.</p>
    </text>
  </region>
  <region id="18" left="45" top="1026" width="588" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p bold="true">(Hauptmenü&gt; Extras&gt; Einstellungen&gt; Berechnung&gt; Integrale: Genauigkeit)</p>
    </text>
    <text lang="eng">
      <p bold="true">Main menu&gt; Tools&gt; Options&gt; Calculation&gt; Integrals: Accuracy.</p>
    </text>
  </region>
  <region id="19" left="45" top="1053" width="377" height="40" color="#000000" bgColor="#ffffff">
    <picture>
      <raw format="png" encoding="base64"> </raw>
    </picture>
  </region>
  <region id="20" left="18" top="1116" width="107" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Genauigkeit </p>
    </text>
    <text lang="eng">
      <p>Accuracy</p>
    </text>
  </region>
  <region id="21" left="153" top="1116" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>exakt</p>
    </text>
    <text lang="eng">
      <p>exact</p>
    </text>
  </region>
  <region id="22" left="252" top="1116" width="37" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">100</e>
      </input>
    </math>
  </region>
  <region id="23" left="324" top="1116" width="45" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">1000</e>
      </input>
    </math>
  </region>
  <region id="24" left="396" top="1116" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">10000</e>
      </input>
    </math>
  </region>
  <region id="25" left="27" top="1143" width="86" height="66" color="#000000" bgColor="#ffffff" fontSize="10">
    <math optimize="2">
      <input>
        <e type="operand">t</e>
        <e type="function" args="1">w</e>
        <e type="operand">t</e>
        <e type="operand">2</e>
        <e type="operator" args="1">-</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="4">int</e>
      </input>
    </math>
  </region>
  <region id="26" left="153" top="1170" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">2.5000</e>
      </input>
    </math>
  </region>
  <region id="27" left="234" top="1170" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">2.5333</e>
      </input>
    </math>
  </region>
  <region id="28" left="324" top="1170" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">2.5033</e>
      </input>
    </math>
  </region>
  <region id="29" left="396" top="1170" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">2.5003</e>
      </input>
    </math>
  </region>
  <region id="30" left="18" top="1215" width="592" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Aufgrund der Unstetigkeit bei t=1 kommt es bei der Standardeinstellung 
Genauigkeit 100) zu einem Fehler von über einem Prozent.</p>
    </text>
    <text lang="eng">
      <p>Due to the discontinuity at point t=1, the error is greater than 1% with the standard
setting for accuracy (100 intervals)</p>
    </text>
  </region>
  <region id="31" left="18" top="1269" width="284" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p bold="true">Integration mit variablen Grenzen</p>
    </text>
    <text lang="eng">
      <p bold="true">Integration with variable limits</p>
    </text>
  </region>
  <region id="32" left="18" top="1305" width="690" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Man kann Funktionen definieren, die Integrationen mit variablen Grenzen enthalten: </p>
    </text>
    <text lang="eng">
      <p>It is possible to define functions involving integrals with variable limits:</p>
    </text>
  </region>
  <region id="33" left="27" top="1341" width="109" height="66" color="#000000" bgColor="#ffffff" fontSize="10">
    <math optimize="2" fractionType="auto">
      <input>
        <e type="operand">x</e>
        <e type="function" args="1">f</e>
        <e type="operand">y</e>
        <e type="operand">y</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">x</e>
        <e type="function" preserve="true" args="4">int</e>
        <e type="operator" args="2">:</e>
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    </math>
  </region>
  <region id="34" left="171" top="1350" width="514" height="40" border="true" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Achtung: Benennen Sie die Grenze und die Integrationsvariable
unterschiedlich!</p>
    </text>
    <text lang="eng">
      <p>Take care to use different variable for limit(here x) and 
integration (here x)!</p>
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  <region id="35" left="18" top="1404" width="128" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">M</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="2">matrix</e>
        <e type="operator" args="2">:</e>
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    </math>
  </region>
  <region id="36" left="171" top="1404" width="188" height="46" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">X</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">0.5</e>
        <e type="operator" args="2">+</e>
        <e type="function" preserve="true" args="3">range</e>
        <e type="operand">M</e>
        <e type="operand">M</e>
        <e type="operand">X</e>
        <e type="operand">X</e>
        <e type="function" args="1">f</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="4">mat</e>
        <e type="function" preserve="true" args="2">stack</e>
        <e type="operator" args="2">:</e>
        <e type="function" preserve="true" args="3">for</e>
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    </math>
  </region>
  <region id="37" left="378" top="1404" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <math error="9">
      <input>
        <e type="operand">M</e>
      </input>
      <result action="numeric">
        <e type="operand">#</e>
      </result>
    </math>
  </region>
  <region id="38" left="18" top="1521" width="693" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Ausdrücke mit variablen Grenzen lassen sich allerdings nicht ohne weiteres plotten.</p>
    </text>
    <text lang="eng">
      <p>Expressions with variable integration limits may fail to be plotted.</p>
    </text>
  </region>
  <region id="39" left="18" top="1548" width="135" height="107" color="#000000" bgColor="#ffffff" fontSize="10">
    <plot error="2" type="2d" render="lines" scale_x="3.68470883775986" scale_y="3.68470883775986" scale_z="3.68470883775986" rotate_x="0" rotate_y="0" rotate_z="0" transpose_x="-4" transpose_y="-23" transpose_z="0">
      <input>
        <e type="operand">y</e>
        <e type="operand">y</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">x</e>
        <e type="function" preserve="true" args="4">int</e>
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    </plot>
  </region>
  <region id="40" left="162" top="1548" width="135" height="107" color="#000000" bgColor="#ffffff" fontSize="10">
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        <e type="operand">y</e>
        <e type="operand">y</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">x</e>
        <e type="function" preserve="true" args="4">int</e>
        <e type="function" preserve="true" args="1">eval</e>
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  </region>
  <region id="41" left="306" top="1548" width="135" height="107" color="#000000" bgColor="#ffffff" fontSize="10">
    <plot error="2" type="2d" render="lines" scale_x="3.68470883775986" scale_y="3.68470883775986" scale_z="3.68470883775986" rotate_x="0" rotate_y="0" rotate_z="0" transpose_x="-4" transpose_y="-23" transpose_z="0">
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        <e type="operand">x</e>
        <e type="function" args="1">f</e>
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  </region>
  <region id="42" left="18" top="1719" width="654" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Beim Standard-Diagrammbereich gelingt der Plot erst, wenn man den Ausdruck als
Funktion definiert und bei Definition oder Aufruf eval() anwendet</p>
    </text>
    <text lang="eng">
      <p>Using the built-in 2D plot region, the plot succeeds only if you define the 
expression as function and use eval() either in the definition or in the call.</p>
    </text>
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        <e type="operand">x</e>
        <e type="function" args="1">f.e</e>
        <e type="operand">y</e>
        <e type="operand">y</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
        <e type="operand">x</e>
        <e type="function" preserve="true" args="4">int</e>
        <e type="function" preserve="true" args="1">eval</e>
        <e type="operator" args="2">:</e>
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    <plot type="2d" render="lines" scale_x="3.68470883775986" scale_y="3.68470883775986" scale_z="3.68470883775986" rotate_x="0" rotate_y="0" rotate_z="0" transpose_x="-4" transpose_y="-23" transpose_z="0">
      <input>
        <e type="operand">x</e>
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  <region id="46" left="18" top="1899" width="729" height="88" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Beim Diagrammbereich aus dem Plugin X-Y Plot:
- muss entweder auch eine Funktion verwendet werden (ohne dass eval() erforderlich ist) 
- oder die Integrationsvariable, die im Ausdruck verwendet wird, muss mit einem Wert 
    belegt werden, damit der Plotbereich nicht annimmt, das der Ausdruck von zwei 
    Variablen abhängig ist.</p>
    </text>
    <text lang="eng">
      <p>Using the X-Y Plot-plugin, you can 
- either use a function (without need to use eval())
- or pre-define the dummy integration variable with whatever value, just to make sure
    that the plot region does not complain about a function of two variables.</p>
    </text>
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      <title2d title="Title" titlefont="Arial, 12pt" titlefontcolor="Black" />
      <xylabel labelfont="Arial, 10pt" labelfontcolor="Black" tickfont="Courier New, 10pt" tickfontcolor="Black" xlabel="x" ylabel="y" y2label="y2" />
      <legend isbordervisible="true" islegendvisible="false" legendbackcolor="White" legendbordercolor="Black" legendfont="Arial, 8pt" legendposition="NorthEast" textcolor="Black" />
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      <input>
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        <e type="operand">y</e>
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
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        <trace seriesname="" isy2data="false" isvisible="true" plotmethod="Lines" linecolor="Blue" linethickness="1" linepattern="Solid" symbolsize="8" symboltype="None" symbolborderthickness="1" symbolbordercolor="Black" symbolfillcolor="White" />
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        <e type="operand">x</e>
        <e type="function" args="1">f</e>
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        <trace seriesname="" isy2data="false" isvisible="true" plotmethod="Lines" linecolor="Blue" linethickness="1" linepattern="Solid" symbolsize="8" symboltype="None" symbolborderthickness="1" symbolbordercolor="Black" symbolfillcolor="White" />
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  <region id="51" top="2196" color="#000000" bgColor="#ffffff">
    <area collapsed="false">
      <title lang="ger">
        <p>Falsche Ergebnisse bei bestimmten Anzeigeeinstellungen (Bruchdarstellung, Optimierung)</p>
      </title>
      <title lang="eng">
        <p>Wrong results for certain display settings (fraction, optimization)</p>
      </title>
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          <p>auto, numeric</p>
        </description>
        <input>
          <e type="operand">x</e>
          <e type="operand">x</e>
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          <e type="operand">x</e>
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        </description>
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          <e type="operand">x</e>
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        </description>
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          <e type="operand">x</e>
          <e type="operand">1</e>
          <e type="operator" args="1">-</e>
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        <description active="true" position="Top" lang="ger">
          <p>decimal, symbolic</p>
        </description>
        <input>
          <e type="operand">x</e>
          <e type="operand">x</e>
          <e type="operand">1</e>
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    <region id="70" top="2754" color="#000000" bgColor="#ffffff">
      <area terminator="true" />
    </region>
  </region>
  <region id="71" left="18" top="2871" width="94" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Mathematik.sm">Mathematik</p>
    </text>
    <text lang="eng">
      <p href="Mathematik.sm">Mathematics</p>
    </text>
  </region>
  <region id="72" left="18" top="2889" width="61" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
  </region>
  <region id="73" left="18" top="2925" width="450" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p italic="true">$Author: mkraska $
$Date: 2013-09-20 23:03:21 +0200 (Fr, 20. Sep 2013) $ </p>
    </text>
  </region>
</regions>