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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:13_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  <a href="subset_1.html#K8">/\</a> <a href="arytm_3.html#K7">RAT+</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  <a href="subset_1.html#K8">/\</a> <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:13_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/>

<a NAME="E3:13_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:13_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E2:13_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E5:13_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:13_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:13_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T99">ARYTM_3:99</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:13_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma11</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:13_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  <a href="subset_1.html#K8">/\</a> <a href="arytm_3.html#K7">RAT+</a>  )
 <b>by </b><i><a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma12</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13"/>
<a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  <a href="subset_1.html#K8">/\</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div>
