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<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="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="integra5.html#T9">INTEGRA5:9</a>, <a class="ref" href="integra5.html#D2">INTEGRA5:def 2</a></i>;<br/>
<a NAME="E3:10"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E4:10"/><b>then </b>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E5:10"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E6:10"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T65">RFUNCT_1:65</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="integra5.html#K1">||</a> <font color="Maroon">A</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="integra2.html#T31">INTEGRA2:31</a></i>;<br/>
<b>hence </b><a NAME="E8:10"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="integra5.html#R1">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="integra5.html#D2">INTEGRA5:def 2</a></i>;<br/>

<a NAME="E9:10"/>
 <a href="integra5.html#K2">integral</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="integra5.html#K1">||</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T65">RFUNCT_1:65</a></i>;<br/>
<b>hence </b><a NAME="E10:10"/>
 <a href="integra5.html#K2">integral</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K2">integral</a> <font color="Maroon">f</font>,<font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra2.html#T31">INTEGRA2:31</a></i>;<br/>


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