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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="integra1.html#M2" title="INTEGRA1:mode.2">Division</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c5">D</font> be    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Maroon" title="c4">S</font>;<br/>







<b>assume </b><a NAME="E1:84"/><i><font color="Green" title="E65">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">g</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">F</font> =  <a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font>;<br/>
<b>set </b><font color="Maroon" title="c7">G</font> =  <a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font>;<br/>
<b>set </b><font color="Maroon" title="c8">H</font> =  <a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c5">D</font>;<br/>
<a NAME="E2:84"/><i><font color="Green" title="E66">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def7</a></i>;<br/>
<a NAME="E3:84"/><i><font color="Green" title="E67">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def7</a></i>;<br/>
<a NAME="E4:84"/><i><font color="Green" title="E68">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def7</a></i>;<br/>
<a NAME="E5:84"/><i><font color="Green" title="E69">A5</font></i>: 
 <a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E2:84"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E6:84"/><i><font color="Green" title="E70">A6</font></i>: 
 <a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E3:84"><i><font color="Green" title="E67">A3</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E7:84"/><i><font color="Green" title="E71">A7</font></i>: 
 <a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c5">D</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E4:84"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E8:84"/><i><font color="Green" title="E72">A8</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font></span>)</span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E5:84"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="integra1.html#E6:84"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="finseq_2.html#T140" title="FINSEQ_2:th.140">FINSEQ_2:140</a></i>;<br/>
<a NAME="E9:84"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:84_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font></span>)</span>
 ;<br/>

<a NAME="E2:84_1"/><b>then </b>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:84_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:84"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="integra1.html#T55" target="_self" title="INTEGRA1:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E4:84_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra1.html#E5:84"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="integra1.html#E6:84"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="rvsum_1.html#T27" title="RVSUM_1:th.27">RVSUM_1:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:84"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E7:84"><i><font color="Green" title="E71">A7</font></i></a>, <a class="txt" href="integra1.html#E8:84"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="rvsum_1.html#T112" title="RVSUM_1:th.112">RVSUM_1:112</a></i>;<br/>
<b>hence </b><a NAME="E11:84"/>
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c5">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E5:84"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="integra1.html#E6:84"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="rvsum_1.html#T119" title="RVSUM_1:th.119">RVSUM_1:119</a></i>;<br/>


</div>
