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

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>

<b>assume </b><a NAME="E1:61"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/>

<a NAME="E2:61"/><b>then </b>
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">B</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> <b> such that </b><br/><a NAME="E4:61"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">rone</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">y</font> =  <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">B</font>;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="arytm_2.html#K8" target="_self">*'</a> <font color="Maroon">y</font> = 
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">rone</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a>, </i>
<br/>.= 
 <a href="arytm_3.html#K14">one</a> 
<b>by </b><i/>
;<br/>


</div>
