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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">m</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">M</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">A</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font> )
 ;<br/>

<b>assume </b><a NAME="E2:40"/><i><font color="Green">E19</font></i>: 
( ( for <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">e</font> &amp; <font color="Olive">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">d</font> holds <br/><font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/>

<b>thus </b><a NAME="E3:40"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">M</font></span><a href="measure5.html#K4">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:40_1"/>
( <font color="Maroon">m</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">M</font> )
 
<b>by </b><i>, <a class="ref" href="measure6.html#T10" target="_self">Th10</a>, <a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a>, <a class="ref" href="supinf_1.html#T79">SUPINF_1:79</a></i>;<br/>
<a NAME="E4:40_1"/><b>then </b>
( <font color="Maroon">m</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#NR3" target="_self">&lt;</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="measure6.html#T10" target="_self">Th10</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>
<b>hence </b><a NAME="E5:40_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">M</font></span><a href="measure5.html#K4">.[</a></span>
 <b>by </b><i>, <a class="ref" href="measure6.html#T10" target="_self">Th10</a>, <a class="ref" href="measure5.html#D4">MEASURE5:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:40"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">M</font></span><a href="measure5.html#K4">.[</a></span>
 ;<br/>

<a NAME="E5:40"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E7:40"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">m</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#NR3" target="_self">&lt;</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i><a class="ref" href="measure6.html#T10" target="_self">Th10</a>, <a class="ref" href="measure5.html#D4">MEASURE5:def 4</a></i>;<br/>
<a NAME="E8:40"/>
 ex <font color="Olive">b</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2"/>
for <font color="Olive">b</font> being  <a href="supinf_1.html#NM3">R_eal</a>  holds <br/>(  not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> or  not <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:40_2"/><b>then </b>
for <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E3:40_2"/><b>then </b>
<font color="Maroon">x</font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:40_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">b</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E10:40"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>
<a NAME="E11:40"/><i><font color="Green">E34</font></i>: 
 ex <font color="Olive">a1</font>, <font color="Olive">x1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E12:40"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="txt" href="measure6.html#E10:40"><i><font color="Green">E33</font></i></a></i>;<br/>


</div>
