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<div class="add">

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">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>assume </b><a NAME="E1:18"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> implies for <font color="Olive">T</font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>  st  <a href="integra2.html#K2">delta</a> <font color="Olive">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:18_1"/>
for <font color="Olive">T</font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>  st  <a href="integra2.html#K2">delta</a> <font color="Olive">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">T</font> be   <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:18_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:18_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K11">upper_integral</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E4:18_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E5:18_1_1"/>
 <a href="integra1.html#K11">upper_integral</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_1"/>
for <font color="Olive">T</font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>  st  <a href="integra2.html#K2">delta</a> <font color="Olive">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:18"/>
( ( for <font color="Olive">T</font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>  st  <a href="integra2.html#K2">delta</a> <font color="Olive">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 ) implies <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">T</font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>  st  <a href="integra2.html#K2">delta</a> <font color="Olive">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:18_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</font> <a href="integra1.html#R1">is_upper_integrable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="integra1.html#R2">is_lower_integrable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">T</font> being   <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:18_2"/><i><font color="Green">E50</font></i>: 
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>
<a NAME="E5:18_2"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:18_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:18_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:18_2"/><b>then </b>
 <a href="integra1.html#K11">upper_integral</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E9:18_2"/>
<font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
( <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> iff for <font color="Olive">T</font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>  st  <a href="integra2.html#K2">delta</a> <font color="Olive">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>


</div>
