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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>and </b><br/><a NAME="E4:19"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:19_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra5.html#D4">INTEGRA5:def 4</a></i>;<br/><b>thus </b><a NAME="E3:19_1"/>
<font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:19_1"/>
<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <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"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E5:19_1"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="integra1.html#T6">INTEGRA1:6</a></i>;<br/><a NAME="E6:19_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E7:19_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> <a href="integra1.html#R3">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp;  <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <span class="p2"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="integra4.html#T6">INTEGRA4:6</a></i>;<br/><b>hence </b><a NAME="E8:19_1"/>
<font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i><a class="ref" href="integra5.html#D2">INTEGRA5:def 2</a></i>;<br/><a NAME="E9:19_1"/>
( <font color="Maroon">c</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">b</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E10:19_1"/><b>then </b>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span>
 <b>by </b><i><a class="ref" href="integra5.html#T19">INTEGRA5:19</a>, </i>;<br/><a NAME="E11:19_1"/><b>then </b>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="integra4.html#T6">INTEGRA4:6</a></i>;<br/><b>hence </b><a NAME="E12:19_1"/>
 <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19"/>
( <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="integra5.html#K3">']</a></span> &amp; <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3">['</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="integra5.html#K3">']</a></span> &amp;  <a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra5.html#K4">integral</a> <font color="Maroon">f</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, , , , , </i>;<br/>


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