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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:58"/><i><font color="Green">E42</font></i>: 
1 <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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:58"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E4:58"/><b>then </b><i><font color="Green">E43</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:58"/><i><font color="Green">E44</font></i>: 
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<a NAME="E1:58_1"/><i><font color="Green">E45</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:58"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<i><font color="Green">E46</font></i>:  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E4:58"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E2:58"><i><font color="Green">E41</font></i></a>, <a class="ref" href="integra1.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E3:58_1"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E4:58_1"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:58_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="integra1.html#E2:58_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:58"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:58"/><i><font color="Green">E46</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E2:58"><i><font color="Green">E41</font></i></a>, <a class="ref" href="integra1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E8:58"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" target="_self">upper_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <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">c<sub>2</sub></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">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</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">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:58"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D7" target="_self">Def7</a></i>;<br/>
 <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
 <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E7:58"><i><font color="Green">E46</font></i></a>, <a class="txt" href="integra1.html#E8:58"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E5:58"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E6:58"><i><font color="Green">E45</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/>
<b>hence </b><a NAME="E10:58"/>
 <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="integra1.html#T29" target="_self">Th29</a></i>;<br/>


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