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



<b>assume </b><a NAME="E1:96"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</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>set </b><font color="Maroon">ny</font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font>;<br/>
<b>set </b><font color="Maroon">dy</font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font>;<br/>
<a NAME="E2:96"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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#K10" 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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:96"/><i><font color="Green">E15</font></i>: 
(  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</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> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T29" target="_self">Th29</a>, , </i>;<br/>
<a NAME="E4:96"/><b>then </b><i><font color="Green">E16</font></i>: 
<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#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</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="ordinal3.html#T34">ORDINAL3:34</a></i>;<br/>
<a NAME="E5:96"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T30" target="_self">Th30</a>, <a class="ref" href="arytm_3.html#T31" target="_self">Th31</a>, <a class="ref" href="arytm_3.html#T32" target="_self">Th32</a>, <a class="txt" href="arytm_3.html#E21"><i><font color="Green">Lemma58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T31" target="_self">Th31</a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>
;<br/>
<a NAME="E7:96"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T31" target="_self">Th31</a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<a NAME="E8:96"/><b>then </b>
 <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>;<br/>
<a NAME="E9:96"/><b>then </b>
 <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T76">ORDINAL3:76</a></i>;<br/>
<a NAME="E10:96"/><b>then </b>
(  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font> &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">m</font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">m</font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> &amp; <font color="Olive">m</font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font> holds <br/><font color="Olive">m</font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E11:96"/><b>then </b>
( <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> &amp;  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font>, <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:96"/><b>then </b>
 <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="arytm_3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E13:96"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T30" target="_self">Th30</a>, <a class="ref" href="arytm_3.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E14:96"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>


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