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<div class="add">

<b>set </b><font color="Maroon">S</font> =  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="integra1.html#M3" target="_self">Element</a> of  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> =  <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:27_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">D</font> being   <a href="integra1.html#M3" target="_self">Element</a> of  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font> holds <br/> ( <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] )
 <b>and </b><br/><a NAME="E3:27_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">D</font> being   <a href="integra1.html#M3" target="_self">Element</a> of  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S3">SEQ_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E4:27_1_1"/>
 <a href="subset_1.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:27_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be    <a href="integra1.html#M3" target="_self">Element</a> of  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:27_1_1_1"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:27_1_1_1"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:27_1_1_1"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:27_1_1"/>
for <font color="Olive">D</font> being   <a href="integra1.html#M3" target="_self">Element</a> of  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">A</font>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Olive">D</font>
 ;<br/>


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