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



<b>set </b><font color="Maroon">A</font> =  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font>;<br/>
<b>assume </b><a NAME="E1:34"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/>

<a NAME="E2:34"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:34_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:34_1"/>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:34_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:34"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">x</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:34"/><b>then </b><i><font color="Green">E21</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> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 
;<br/>
<b>thus </b><a NAME="E5:34"/>
<font color="Maroon">x</font> <a href="arytm_2.html#K8" target="_self">*'</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E4"><i><font color="Green">Lemma34</font></i></a>, </i>;<br/>


</div>
