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

<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#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1"><b>suppose </b></a><a NAME="E1:72_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E2:72"><i><font color="Green">E62</font></i></a></i>;<br/><b>take </b>
 <a href="arytm_3.html#K13">{}</a> 
;<br/><b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/><b>thus </b><a NAME="E3:72_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:72_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><b>thus </b><a NAME="E4:72_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="arytm_3.html#K13">{}</a> 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:72_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="arytm_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_2"><b>suppose </b></a><a NAME="E1:72_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E1:72"><i><font color="Green">E61</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/><b>take </b>
 <a href="arytm_3.html#K13">{}</a> 
;<br/><b>thus </b><a NAME="E3:72_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>thus </b><a NAME="E4:72_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:72_1_2"><i><font color="Green">E63</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="arytm_3.html#K13">{}</a> 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:72_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="arytm_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:72_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>and </b><br/><a NAME="E2:72_1_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>6</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="E4:72_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:72_1_3"/><i><font color="Green">E65</font></i>: 
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <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> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:72"><i><font color="Green">E61</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self">Def3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</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="E7:72_1_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E8:72_1_3"/><i><font color="Green">E67</font></i>: 
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</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> <font color="Maroon">c<sub>7</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:72"><i><font color="Green">E62</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self">Def3</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/><b>thus </b><a NAME="E9:72_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:72_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="arytm_2.html#E7:72_1_3"><i><font color="Green">E66</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E10:72_1_3"/><i><font color="Green">E68</font></i>: 
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>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> iff <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>

<b>thus </b><a NAME="E1:72_1_3_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> implies <font color="Maroon">c<sub>10</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1_3_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:72_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , 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="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</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="E4:72_1_3_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E5:72_1_3_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:72_1_3_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E7:72_1_3_1_1"/><i><font color="Green">E72</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>11</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:72_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="arytm_2.html#E5:72_1_3_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E8:72_1_3_1_1"/><i><font color="Green">E73</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>12</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:72_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="arytm_2.html#E6:72_1_3_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E9:72_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<a NAME="E10:72_1_3_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:72_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="arytm_2.html#E4:72_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="arytm_3.html#T62">ARYTM_3:62</a></i>;<br/>
<a NAME="E11:72_1_3_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>12</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:72_1_3_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/>
<a NAME="E12:72_1_3_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>12</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:72_1_3_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/>
<a NAME="E13:72_1_3_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>12</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E7:72_1_3_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/>
<b>hence </b><a NAME="E14:72_1_3_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:72_1_3_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="arytm_2.html#E12:72_1_3_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/>


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

<b>assume </b><a NAME="E2:72_1_3_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</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="E4:72_1_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E5:72_1_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87">ARYTM_3:87</a></i>;<br/>
<a NAME="E6:72_1_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:72_1_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="arytm_2.html#E4:72_1_3_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E7:72_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:72_1_3_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</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="E9:72_1_3_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E10:72_1_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T101">ARYTM_3:101</a></i>;<br/>
<a NAME="E11:72_1_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>12</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4:72_1_3_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="arytm_2.html#E9:72_1_3_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</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="E13:72_1_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E14:72_1_3_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T87">ARYTM_3:87</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1_3_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:72_1_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:72_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="arytm_2.html#E4:72_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="arytm_2.html#E4:72_1_3_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="arytm_2.html#E13:72_1_3_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="arytm_3.html#T62">ARYTM_3:62</a></i>;<br/><b>hence </b><a NAME="E3:72_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E9:72_1_3_1"><i><font color="Green">E72</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:72_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E14:72_1_3_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/>
<a NAME="E17:72_1_3_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E5:72_1_3"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E18:72_1_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:72_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="arytm_2.html#E10:72_1_3_1"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E19:72_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , 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="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="arytm_2.html#E13:72_1_3_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="arytm_2.html#E17:72_1_3_1"><i><font color="Green">E76</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:72_1_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>10</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="E12:72_1_3"/><i><font color="Green">E69</font></i>: 
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E13:72_1_3"/><i><font color="Green">E70</font></i>: 
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>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> iff <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E14:72_1_3"/><i><font color="Green">E71</font></i>: 
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>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>

<a NAME="E1:72_1_3_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> iff <font color="Maroon">c<sub>11</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E10:72_1_3"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:72_1_3_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font> iff <font color="Maroon">c<sub>11</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E13:72_1_3"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K6" target="_self">*'</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="arytm_2.html#E12:72_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="arytm_2.html#E14:72_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma9</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
