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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_1_1_1"/>
 <a href="arytm_3.html#K14">one</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  </span>}</span> 
 ;<br/><a NAME="E2:2_1_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#R1">=</a>  <a href="arytm_3.html#K14">one</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  )
 ;<br/><b>hence </b><a NAME="E3:2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E2:2_1_1"/><b>then </b><i><font color="Green">E1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
;<br/>
<a NAME="E3:2_1_1"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/>
<a NAME="E4:2_1_1"/><b>then </b>
 <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/>
<a NAME="E5:2_1_1"/><b>then </b><i><font color="Green">E2</font></i>: 
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  </span>}</span> 
 
;<br/>
<a NAME="E6:2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></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="E1:2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  </span>}</span> 
 ;<br/>

<a NAME="E2:2_1_1_2"/><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#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  )
 
;<br/>
<b>hence </b><a NAME="E3:2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  </span>}</span>  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#E5:2_1_1"><i><font color="Green">E2</font></i></a></i>;<br/>
<a NAME="E8:2_1_1"/>
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( ( 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="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ex <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 <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:2_1_1_3"/><b>then </b><i><font color="Green">E3</font></i>: 
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#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a>  )
 
;<br/>
<b>thus </b><a NAME="E3:2_1_1_3"/>
for <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 <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_1_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:2_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:2_1_1_3"><i><font color="Green">E3</font></i></a>, <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/>
<b>hence </b><a NAME="E3:2_1_1_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E5:2_1_1_3"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:2_1_1_3"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:2_1_1_3"><i><font color="Green">E3</font></i></a>, <a class="ref" href="arytm_3.html#T101">ARYTM_3:101</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E7:2_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:2_1_1_3"><i><font color="Green">E5</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:2_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:2_1_1_3"><i><font color="Green">E4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:2_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <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 <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( 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>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <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 <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E10:2_1_1"/>
not  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:2_1_1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>


</div>
