<?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> , <font color="Maroon">g</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="E1:85"/><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> &amp; <font color="Maroon">g</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> )
 ;<br/>

<b>set </b><font color="Maroon">F</font> =  <a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font>;<br/>
<a NAME="E2:85"/><i><font color="Green">E53</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">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="E3:85"/><i><font color="Green">E54</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">g</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="E4:85"/><i><font color="Green">E55</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<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="E5:85"/><i><font color="Green">E56</font></i>: 
 <a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> is    <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/>
<a NAME="E6:85"/><i><font color="Green">E57</font></i>: 
 <a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font> is    <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/>
<a NAME="E7:85"/><i><font color="Green">E58</font></i>: 
 <a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font> is    <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/>
<a NAME="E8:85"/><i><font color="Green">E59</font></i>: 
<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> <a href="rvsum_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span> is    <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#T140">FINSEQ_2:140</a></i>;<br/>
<a NAME="E9:85"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</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="integra1.html#K14" target="_self">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</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">D</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:85_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">j</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="E2:85_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</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">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</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">g</font>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E3:85_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</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">D</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="rvsum_1.html#T27">RVSUM_1:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:85"/><b>then </b>
 <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"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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">D</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/>
<a NAME="E11:85"/><b>then </b>
 <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"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</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> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>;<br/>
<a NAME="E12:85"/><b>then </b>
 <a href="integra1.html#K6" target="_self">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</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> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</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> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E13:85"/><b>then </b>
 <a href="integra1.html#K6" target="_self">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</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> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E14:85"/>
 <a href="integra1.html#K6" target="_self">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span>
 ;<br/>


</div>
