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

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:99_1"/><i><font color="Green">E82</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">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>3</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7" target="_self">RAT+</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> <b> such that </b><br/><a NAME="E3:99_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 <b>by </b><i><a class="ref" href="arytm_3.html#T80" target="_self">Th80</a></i>;<br/><a NAME="E4:99_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">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>3</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:99_1"><i><font color="Green">E83</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">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>3</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><a NAME="E2:99_1_1"/><b>then </b>
<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" target="_self">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>3</sub></font> </span>}</span>  is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>hence </b><a NAME="E3:99_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:99_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> <b> such that </b><br/><a NAME="E7:99_1"/><i><font color="Green">E85</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">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>3</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:99_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="arytm_3.html#E3:99_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="arytm_3.html#T35" target="_self">Th35</a>, <a class="ref" href="arytm_3.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E8:99_1"/>
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71" target="_self">Th71</a></i>;<br/><a NAME="E9:99_1"/><b>then </b>
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:99_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="arytm_3.html#T73" target="_self">Th73</a></i>;<br/><a NAME="E10:99_1"/><b>then </b>
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">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>3</sub></font> </span>}</span> 
 ;<br/><a NAME="E11:99_1"/><b>then </b>
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="arytm_3.html#E7:99_1"><i><font color="Green">E85</font></i></a></i>;<br/><b>hence </b><a NAME="E12:99_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:99"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 ;<br/>

<a NAME="E3:99"/>
<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" target="_self">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>3</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:99_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">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>3</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:99_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</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>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:99_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:99"><i><font color="Green">E82</font></i></a>, <a class="ref" href="arytm_3.html#T71" target="_self">Th71</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:99"/><b>then </b>
<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" target="_self">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>3</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:99"/>
<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" target="_self">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>3</sub></font> </span>}</span>  <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7" target="_self">RAT+</a> 
 ;<br/>


</div>
