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<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> 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>assume </b><a NAME="E1:78"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<a NAME="E2:78"/>
not  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="integra1.html#D11" target="_self" title="INTEGRA1:def.11">Def11</a></i>;<br/>
<a NAME="E3:78"/><b>then </b><i><font color="Green" title="E60">A2</font></i>: 
not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E4:78"/>
not  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="integra1.html#D12" target="_self" title="INTEGRA1:def.12">Def12</a></i>;<br/>
<a NAME="E5:78"/><b>then </b><i><font color="Green" title="E61">A3</font></i>: 
not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E6:78"/>
for <font color="Olive" title="b1">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/> <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">b</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">D1</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:78_1"/><i><font color="Green" title="E62">A4</font></i>: 
( <font color="Maroon" title="c4">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">D1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:78_1"/><i><font color="Green" title="E63">A5</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:78_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="integra1.html#D11" target="_self" title="INTEGRA1:def.11">Def11</a></i>;<br/>
<a NAME="E5:78_1"/>
for <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:78_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">D2</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:78_1_1"/><i><font color="Green" title="E64">A6</font></i>: 
( <font color="Maroon" title="c6">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">D2</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:78_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:78_1_1"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="integra1.html#D12" target="_self" title="INTEGRA1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E5:78_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:78"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="integra1.html#T50" target="_self" title="INTEGRA1:th.50">Th50</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:78_1"/>
 <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="integra1.html#E5:78"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="integra1.html#E4:78_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="seq_4.html#T62" title="SEQ_4:th.62">SEQ_4:62</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:78"/>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="integra1.html#E3:78"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="seq_4.html#T60" title="SEQ_4:th.60">SEQ_4:60</a></i>;<br/>


</div>
