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

<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>assume </b><a NAME="E1:79"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:79"/>
not  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" target="_self">upper_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="integra1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E3:79"/><b>then </b><i><font color="Green">E61</font></i>: 
not  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" target="_self">upper_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E4:79"/>
not  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" target="_self">lower_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="integra1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E5:79"/><b>then </b><i><font color="Green">E62</font></i>: 
not  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" target="_self">lower_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E6:79"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" target="_self">upper_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/> <a href="integra1.html#K12" target="_self">lower_integral</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" target="_self">upper_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="integra1.html#M3" target="_self">Element</a> of  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:79_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" target="_self">upper_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" target="_self">upper_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:79_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:79_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E5:79_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" target="_self">lower_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:79_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" target="_self">lower_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="integra1.html#M3" target="_self">Element</a> of  <a href="integra1.html#K8" target="_self">divs</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:79_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" target="_self">lower_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" target="_self">lower_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:79_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:79_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="integra1.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E5:79_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <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>
 <b>by </b><i><a class="txt" href="integra1.html#E1:79"><i><font color="Green">E60</font></i></a>, <a class="ref" href="integra1.html#T50" target="_self">Th50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:79_1"/><b>then </b>
 <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"><a href="integra1.html#K10" target="_self">lower_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:79"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra1.html#E4:79_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<b>hence </b><a NAME="E7:79_1"/>
 <a href="integra1.html#K12" target="_self">lower_integral</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:79"/><b>then </b>
 <a href="integra1.html#K12" target="_self">lower_integral</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" target="_self">upper_sum_set</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:79"><i><font color="Green">E61</font></i></a>, <a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/>
<b>hence </b><a NAME="E8:79"/>
 <a href="integra1.html#K11" target="_self">upper_integral</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K12" target="_self">lower_integral</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
