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<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> ;<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">D1</font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">D2</font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:78"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">D</font> being    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:78"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">D1</font> <a href="integra1.html#R4" target="_self">&lt;=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D2</font> <a href="integra1.html#R4" target="_self">&lt;=</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:78"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E5:78"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a>, <a class="txt" href="integra1.html#E6:78"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E6:78"/><i><font color="Green">E56</font></i>: 
 <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:78"/>
 <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a>, , <a class="txt" href="integra1.html#E5:78"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E8:78"/><b>then </b>
 <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:78"/>
 <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6" target="_self">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="integra1.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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