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<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1" target="_self">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">D1</font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">D2</font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">D1</font> <a href="integra1.html#R4" target="_self">&lt;=</a> <font color="Maroon">D2</font>
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>and </b><br/><a NAME="E3:70"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:70"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:70_1"/>
<font color="Maroon">i</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="binarith.html#T5">BINARITH:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:70"/><i><font color="Green">E56</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<a NAME="E1:70_2"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:70_2"/>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T3" target="_self">Th3</a>, <a class="ref" href="integra1.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E3:70_2"/>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E5:70"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:70"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:70"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:70"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K18" target="_self">PartSums</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:70"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K18" target="_self">PartSums</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#T3" target="_self">Th3</a>, <a class="ref" href="integra1.html#T4" target="_self">Th4</a>, <a class="txt" href="integra1.html#E2:70"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E3:70"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra1.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E10:70"/>
<span class="p1">(<span class="default"><a href="integra1.html#K18" target="_self">PartSums</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K18" target="_self">PartSums</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E4:70"><i><font color="Green">E55</font></i></a>, </i>;<br/>


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