<?xml version="1.0"?>
<div class="add">

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







<b>assume </b><a NAME="E1:52"/><i><font color="Green">E49</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="E2:52"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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">D</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<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><a NAME="E1:52_1"/><i><font color="Green">E54</font></i>: 
<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">D</font></span>)</span>
 ;<br/>

<a NAME="E2:52_1"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E3:52_1"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:52_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:52_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_1_1"/><b>then </b>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:52_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E5:52_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:52_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:52_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="seq_4.html#T65">SEQ_4:65</a></i>;<br/>
<a NAME="E6:52_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K3" target="_self">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:52_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:52"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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">D</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<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><a NAME="E1:52_2"/><i><font color="Green">E61</font></i>: 
<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">D</font></span>)</span>
 ;<br/>

<a NAME="E2:52_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:52_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:52"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rvsum_1.html#K15">Sum</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">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:52_3"/><i><font color="Green">E65</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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">D</font></span>)</span> holds <br/><font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:52_3"/><i><font color="Green">E66</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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">D</font></span>)</span> holds <br/><font color="Maroon">q</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E5:52_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:52_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E7:52_3"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></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="E8:52_3"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1"><b>proof </b></a><div class="add">

<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><a NAME="E1:52_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 ;<br/>

<a NAME="E2:52_3_1"/><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> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:52_3_1"/><b>then </b>
<font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:52_3_1"/>
<font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:52_3"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_2"><b>now </b></a><div class="add"><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><a NAME="E1:52_3_2"/><i><font color="Green">E70</font></i>: 
<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">D</font></span>)</span>
 ;<br/><a NAME="E2:52_3_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">q</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:52_3_2"/>
<font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:52_3_2"/>
<font color="Maroon">q</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:52_3"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rvsum_1.html#K15">Sum</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">D</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:52"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rvsum_1.html#K15">Sum</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">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>;<br/>
<a NAME="E6:52"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3" target="_self">vol</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:52"/>
 <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
