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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="partfun1.html#NM1">PartFunc</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">D</font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font>;<br/>







<a NAME="E1:73"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:73"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:73"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</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#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E4:73"/><b>then </b><i><font color="Green">E53</font></i>: 
<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#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <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#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:73"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<a NAME="E1:73_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</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#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:73_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</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">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:73_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:73"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<b>hence </b><a NAME="E7:73"/>
<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#K15" target="_self">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div>
