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<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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><a NAME="E1:62"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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/>

<b>thus </b><a NAME="E2:62"/>
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
  <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="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:62_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E3:62_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E4:62_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:62_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E6:62_1"/>
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>7</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>  )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:62"><i><font color="Green">E58</font></i></a>, <a class="txt" href="arytm_2.html#E4:62_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E7:62_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>8</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/>
<a NAME="E8:62_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>8</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/>
<b>hence </b><a NAME="E9:62_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:62_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="arytm_2.html#E5:62_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green">Lemma21</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</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="E3:62"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E6:62"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E7:62"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:62"><i><font color="Green">E59</font></i></a>, <a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma11</font></i></a></i>;<br/>
<a NAME="E8:62"/>
<font color="Maroon">c<sub>8</sub></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#E7:62"><i><font color="Green">E61</font></i></a>, <a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</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="E10:62"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T61">ARYTM_3:61</a></i>;<br/>
<a NAME="E11:62"/>
<a href="arytm_3.html#K14">one</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:62"><i><font color="Green">E62</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/>
<a NAME="E12:62"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="arytm_3.html#R3">&lt;='</a>  <a href="arytm_3.html#K14">one</a> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E7:62"><i><font color="Green">E61</font></i></a>, <a class="ref" href="arytm_3.html#T91">ARYTM_3:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 ;<br/><a NAME="E2:62_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:62"><i><font color="Green">E62</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>;<br/><b>hence </b><a NAME="E3:62_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E7:62"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:62"/><b>then </b>
<font color="Maroon">c<sub>9</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#E12:62"><i><font color="Green">E63</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/>
<a NAME="E15:62"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:62"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:62"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#K6" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:62"><i><font color="Green">E60</font></i></a>, <a class="txt" href="arytm_2.html#E10:62"><i><font color="Green">E62</font></i></a></i>;<br/>


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