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

<b>thus </b><a NAME="E1:7_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  implies  ex <font color="Olive">IT</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> &amp; <font color="Olive">IT</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <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">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">r</font> </span>}</span>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  ;<br/>
<b>set </b><font color="Maroon">IT</font> = <span class="p1">{<span class="default"> <font color="Olive">s</font> where <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">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> ;<br/>
<a NAME="E3:7_1_1_1"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   holds <br/>(  not <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or  not <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E4:7_1_1_1"/><b>then </b>
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <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">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<a NAME="E5:7_1_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">s</font> where <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">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
;<br/>
<a NAME="E6:7_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">s</font> where <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">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> </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="7_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:7_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <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">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/>

<a NAME="E2:7_1_1_1_1"/><b>then </b>
 ex <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="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<b>hence </b><a NAME="E3:7_1_1_1_1"/>
<font color="Maroon">e</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">IT</font> = <span class="p1">{<span class="default"> <font color="Olive">s</font> where <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">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  ;<br/>
<a NAME="E8:7_1_1_1"/>
for <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> ) &amp;  ex <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="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> &amp; <font color="Olive">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:7_1_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 ;<br/>

<a NAME="E2:7_1_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
 ex <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">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<b>thus </b><a NAME="E3:7_1_1_1_2"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:7_1_1_1_2_1"/>
<font color="Maroon">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E2:7_1_1_1_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_1_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 ;<br/>


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

<b>consider </b><font color="Maroon">s0</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="E5:7_1_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s0</font>
 <b>and </b><br/><a NAME="E6:7_1_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">s0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T101">ARYTM_3:101</a></i>;<br/>
<b>take </b>
<font color="Maroon">s0</font>
;<br/>

<b>thus </b><a NAME="E7:7_1_1_1_2"/>
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E8:7_1_1_1_2"/>
<font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s0</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:7_1_1_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <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="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E11:7_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <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">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  )
 ;<br/>


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

<b>assume </b><a NAME="E2:7_1_1"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/>

<a NAME="E3:7_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 
;<br/>
<a NAME="E4:7_1_1"/><b>then </b>
<font color="Maroon">x</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="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:7_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>


</div>
