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

<b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> , <font color="Maroon">B</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 ;<br/>

<b>set </b><font color="Maroon">DIF</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) </span>}</span> ;<br/>
<a NAME="E4:48"/>
not <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">s1</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="E6:48"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E7:48"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<a NAME="E8:48"/>
<font color="Maroon">s1</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T92">ARYTM_3:92</a></i>;<br/>
<a NAME="E9:48"/><b>then </b><i><font color="Green">E33</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">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E10:48"/>
<span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</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="48_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="E1:48_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp;  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:48_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">DIF</font> = <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) </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/>;<br/>
<a NAME="E12:48"/>
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">DIF</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">DIF</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">DIF</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="48_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="E1:48_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font>
 ;<br/>

<a NAME="E2:48_2"/><b>then </b>
 ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) )
 
;<br/>
<b>then consider </b><font color="Maroon">r0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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="E4:48_2"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:48_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:48_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">r0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 ;<br/>
<b>thus </b><a NAME="E7:48_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">DIF</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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="E1:48_2_1"/>
<font color="Maroon">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</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="E3:48_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">s</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/>
<a NAME="E4:48_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">p</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="arytm_3.html#T85">ARYTM_3:85</a></i>;<br/>
<a NAME="E5:48_2_1"/>
<font color="Maroon">t</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="arytm_3.html#T85">ARYTM_3:85</a></i>;<br/>
<a NAME="E6:48_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</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="E8:48_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">p</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/>
<a NAME="E9:48_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">s</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>;<br/>
<a NAME="E10:48_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">r0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T69">ARYTM_3:69</a></i>;<br/>
<a NAME="E11:48_2_1"/>
<font color="Maroon">q</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="arytm_3.html#T85">ARYTM_3:85</a></i>;<br/>
<a NAME="E12:48_2_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E18"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="arytm_2.html#E3:48"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:48_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a>, </i>;<br/>


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

<b>consider </b><font color="Maroon">s1</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="E9:48_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E10:48_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">s0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E18"><i><font color="Green">Lemma54</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">q</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="E12:48_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">s0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font>
;<br/>

<a NAME="E13:48_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">r0</font> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>;<br/>
<b>hence </b><a NAME="E14:48_2"/>
<font color="Maroon">t</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a></i>;<br/>

<a NAME="E15:48_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">t</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T85">ARYTM_3:85</a></i>;<br/>
<a NAME="E16:48_2"/>
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E17:48_2"/>
<font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:48"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">DIF</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/>
<a NAME="E14:48"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon">s2</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="E16:48"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T49">SUBSET_1:49</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3"/>
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font>
 ;<br/><a NAME="E2:48_3"/><b>then </b>
 ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp;  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) )
 ;<br/><b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">t</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="E4:48_3"/>
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:48_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:48_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">r</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E7:48_3"/>
<font color="Maroon">s2</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="arytm_3.html#T85">ARYTM_3:85</a></i>;<br/><b>hence </b><a NAME="E8:48_3"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E18"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="arytm_2.html#E3:48"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:48"/><b>then </b>
<font color="Maroon">DIF</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">DIF</font> = <font color="Maroon">DIF</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="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>take </b>
<font color="Maroon">DIF</font>
;<br/>

<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</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">t</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> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font> ) </span>}</span> ;<br/>
<a NAME="E20:48"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</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">t</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> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_4"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</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">t</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> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4_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="E1:48_4_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_4_1_1"/>
( <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or  not <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_1"><b>suppose </b></a><a NAME="E1:48_4_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:48_4_1_1_1"/>
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T92">ARYTM_3:92</a></i>;<br/><b>hence </b><a NAME="E3:48_4_1_1_1"/>
<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#K11">+</a> <font color="Olive">t</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">t</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> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font> ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2"><b>suppose </b></a><a NAME="E1:48_4_1_1_2"/><i><font color="Green">E87</font></i>: 
not <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><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="E3:48_4_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:48_4_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/><a NAME="E5:48_4_1_1_2"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a>, <a class="txt" href="arytm_2.html#E3:48"><i><font color="Green">E21</font></i></a></i>;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( not <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> ) </span>}</span> ;<br/><a NAME="E6:48_4_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">s0</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T92">ARYTM_3:92</a></i>;<br/><a NAME="E7:48_4_1_1_2"/>
for <font color="Olive">r2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r2</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s0</font> holds <br/> ex <font color="Olive">s</font>, <font color="Olive">t</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">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( not <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> ) </span>}</span>  &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r2</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="E1:48_4_1_1_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">r2</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s0</font>
 ;<br/>

<a NAME="E2:48_4_1_1_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">r2</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s0</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s0</font> )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_4_1_1_2_1_1"/>
( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or  not <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_4_1_1_2_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">r2</font>
;<br/><b>take </b>
 <a href="arytm_3.html#K13">{}</a> 
;<br/><b>thus </b><a NAME="E2:48_4_1_1_2_1_1_1"/>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a></i>;<br/><b>thus </b><a NAME="E3:48_4_1_1_2_1_1_1"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( not <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E4:48_4_1_1_2_1_1_1"/>
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T92">ARYTM_3:92</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_4_1_1_2_1_1_2"/><i><font color="Green">E119</font></i>: 
not <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">q</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="E3:48_4_1_1_2_1_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">r2</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><a NAME="E4:48_4_1_1_2_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a>, , <a class="ref" href="arytm_3.html#T92">ARYTM_3:92</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E6:48_4_1_1_2_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>and </b><br/><a NAME="E7:48_4_1_1_2_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">s0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">n</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T103">ARYTM_3:103</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>;<br/><a NAME="E8:48_4_1_1_2_1_1_2"/><i><font color="Green">E123</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green">Lemma66</font></i></a>, <a class="txt" href="arytm_2.html#E3:48"><i><font color="Green">E21</font></i></a></i>;<br/><b>consider </b><font color="Maroon">y0</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="E10:48_4_1_1_2_1_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E11:48_4_1_1_2_1_1_2"/>
<a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>;<br/><a NAME="E12:48_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="arytm_3.html#K13">{}</a> ]
 <b>by </b><i><a class="txt" href="arytm_2.html#E28"><i><font color="Green">Lemma68</font></i></a>, <a class="txt" href="arytm_2.html#E6:48"><i><font color="Green">E22</font></i></a></i>;<br/><b>consider </b><font color="Maroon">n0</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="E14:48_4_1_1_2_1_1_2"/>
<font color="Maroon">n0</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>and </b><br/><a NAME="E15:48_4_1_1_2_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n0</font>]
 <b>and </b><br/><a NAME="E16:48_4_1_1_2_1_1_2"/><i><font color="Green">E127</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n0</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K14">one</a> ]
 <b>from </b><i><a class="ref" href="arytm_3.html#S2">ARYTM_3:sch 2</a>(<a class="txt" href="arytm_2.html#E10"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="arytm_2.html#T2" target="_self">Th2</a>, , <a class="txt" href="arytm_2.html#E29"><i><font color="Green">Lemma69</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green">Lemma67</font></i></a>);<br/></i><i><font color="Green">E128</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font></span>)</span> <a href="arytm_3.html#K11">+</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T63">ARYTM_3:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
;<br/><a NAME="E18:48_4_1_1_2_1_1_2"/>
<font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="arytm_2.html#E30"><i><font color="Green">Lemma70</font></i></a>, <a class="txt" href="arytm_2.html#E3:48"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E19:48_4_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s0</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/><b>then consider </b><font color="Maroon">t</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="E21:48_4_1_1_2_1_1_2"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><b>take </b>
<font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font>
;<br/><b>take </b>
<font color="Maroon">t</font>
;<br/><b>thus </b><a NAME="E22:48_4_1_1_2_1_1_2"/>
<font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E30"><i><font color="Green">Lemma70</font></i></a></i>;<br/><b>thus </b><a NAME="E23:48_4_1_1_2_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( not <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><a NAME="E24:48_4_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T57">ARYTM_3:57</a></i>;<br/><b>hence </b><a NAME="E25:48_4_1_1_2_1_1_2"/>
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">q</font></span>)</span> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T69">ARYTM_3:69</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">t</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="E9:48_4_1_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E10:48_4_1_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( not <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E11:48_4_1_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a></i>;<br/><a NAME="E12:48_4_1_1_2"/>
 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">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> &amp;  ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( not <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> ) )
 <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a></i>;<br/><a NAME="E13:48_4_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a></i>;<br/><b>hence </b><a NAME="E14:48_4_1_1_2"/>
<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#K11">+</a> <font color="Olive">t</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">t</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> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, <a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<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="E2:48_4"/>
<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#K11">+</a> <font color="Olive">t</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">t</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> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">s3</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">t3</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="E4:48_4"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s3</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t3</font>
 <b>and </b><br/><a NAME="E5:48_4"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">s3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:48_4"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">t3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DIF</font>
 ;<br/>
<a NAME="E7:48_4"/>
 ex <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">t3</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp;  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/>( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#K11">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> ) )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma57</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">r4</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">s4</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="E9:48_4"/><i><font color="Green">E136</font></i>: 
not <font color="Maroon">r4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E10:48_4"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">s4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E11:48_4"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">r4</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s4</font>
 ;<br/>
<a NAME="E12:48_4"/>
<font color="Maroon">s3</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r4</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a>, <a class="txt" href="arytm_2.html#E3:48"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E13:48_4"/><b>then </b>
<font color="Maroon">s3</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t3</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s4</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/>
<b>hence </b><a NAME="E14:48_4"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a>, , <a class="txt" href="arytm_2.html#E3:48"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:48"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">DIF</font>
 ;<br/>


</div>
