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<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="E1:59_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E2:59_1"/><b>then </b><i><font color="Green">E52</font></i>: 
not <font color="Maroon">c<sub>3</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#D8" target="_self">Def8</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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="E4:59_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="ref" href="arytm_3.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E5:59_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:59_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="arytm_3.html#E2:59_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="arytm_3.html#E4:59_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="arytm_3.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E6:59_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:59_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="arytm_3.html#T7" target="_self">Th7</a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E2:59"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E3:59"/>
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="arytm_3.html#D8" target="_self">Def8</a></i>;<br/>


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