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<b>let </b><font color="Maroon">A</font> be   <a href="measure5.html#NM1">Interval</a>;<br/>

<b>consider </b><font color="Maroon">s</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E2:8"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_1_1"/>
0 <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><a NAME="E2:8_1_2"/>
0 <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s</font></span><a href="measure5.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_2_1"/>
0 <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:8_1_2_1"/>
0 <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="measure5.html#T14">MEASURE5:14</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:8_1_2"/>
0 <a href="integra2.html#K1">*</a> <font color="Maroon">A</font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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