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

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

<a NAME="E1:22"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E2:22"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  implies ( <font color="Maroon">x</font> <a href="supinf_2.html#K2" target="_self">+</a> <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/>
<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">a</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<font color="Maroon">x</font> <a href="supinf_2.html#K2" target="_self">+</a> <a href="supinf_2.html#K1" target="_self">0.</a>  = 
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> 0
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>

;<br/>
<b>hence </b><a NAME="E4:22_1"/>
( <font color="Maroon">x</font> <a href="supinf_2.html#K2" target="_self">+</a> <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:22"/>
( <font color="Maroon">x</font> <a href="supinf_2.html#K2" target="_self">+</a> <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <a href="supinf_2.html#K1" target="_self">0.</a>  <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T1" target="_self">Th1</a>, </i>;<br/>


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