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

<b>set </b><font color="Maroon">nx</font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">dx</font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font>;<br/>
<b>assume </b><a NAME="E1:80"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 ;<br/>

<a NAME="E2:80"/><b>then </b><i><font color="Green">E13</font></i>: 
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#D3" target="_self">Def3</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span>;<br/>

<a NAME="E3:80"/>
 <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font>, <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:80"/><b>then </b><i><font color="Green">E14</font></i>: 
(  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font> &amp;  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T19" target="_self">Th19</a>, <a class="ref" href="arytm_3.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T19" target="_self">Th19</a>, <a class="ref" href="ordinal3.html#T34">ORDINAL3:34</a></i>;<br/>
<b>thus </b><a NAME="E6:80"/>
<font color="Maroon">x</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T20" target="_self">Th20</a>, <a class="ref" href="arytm_3.html#T13" target="_self">Th13</a></i>;<br/>


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