<?xml version="1.0"?>
<div class="add">

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

<b>assume </b><a NAME="E1:54"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E2:54"/><i><font color="Green">E19</font></i>: 
(  <a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a>  )
 ;<br/>

<a NAME="E3:54"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a> </span>)</span></span><a href="measure5.html#K2">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a> </span>)</span></span><a href="measure5.html#K3">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a> </span>)</span></span><a href="measure5.html#K4">.[</a></span> )
 </span><b>by </b><i>, <a class="ref" href="measure6.html#T11" target="_self">Th11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>suppose </b></a><a NAME="E1:54_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a> </span>)</span></span><a href="measure5.html#K2">.[</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="measure5.html#D2">MEASURE5:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>suppose </b></a><a NAME="E1:54_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a> </span>)</span></span><a href="measure5.html#K3">.]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="measure5.html#D3">MEASURE5:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_3"><b>suppose </b></a><a NAME="E1:54_1_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a> </span>)</span></span><a href="measure5.html#K1">.]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_4"><b>suppose </b></a><a NAME="E1:54_1_4"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" target="_self">^^</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure6.html#K5" target="_self">^^</a> </span>)</span></span><a href="measure5.html#K4">.[</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="measure5.html#D4">MEASURE5:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
