<?xml version="1.0"?>
<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="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">D</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:8"/>
 <a href="integra1.html#K3">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:8"/><b>then </b>
( <font color="Maroon">f</font> <a href="integra1.html#R1">is_upper_integrable_on</a> <font color="Maroon">A</font> &amp;  <a href="integra1.html#K11">upper_integral</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">f</font> <a href="integra1.html#R2">is_lower_integrable_on</a> <font color="Maroon">A</font> &amp;  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:8"/>
( <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp;  <a href="integra1.html#K13">integral</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a>, <a class="ref" href="integra1.html#D18">INTEGRA1:def 18</a></i>;<br/>


</div>
