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

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

<b>thus </b><a NAME="E1:20"/>
<span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="measure5.html#K2" target="_self">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E23</font></i>: 
not <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="measure5.html#K2" target="_self">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="measure5.html#K2" target="_self">.[</a></span>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i>, </i>;<br/>
<a NAME="E4:20_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:20_1"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:20"/>
<span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="measure5.html#K4" target="_self">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E25</font></i>: 
not <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="measure5.html#K4" target="_self">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="measure5.html#K4" target="_self">.[</a></span>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i>, </i>;<br/>
<a NAME="E4:20_2"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:20_2"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:20"/><i><font color="Green">E28</font></i>: 
not <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="measure5.html#K3" target="_self">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="measure5.html#K3" target="_self">.]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i>, </i>;<br/>
<a NAME="E6:20"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:20"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>


</div>
