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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">E80</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:96"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E3:96"/><i><font color="Green">E82</font></i>: 
(  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E1:96"><i><font color="Green">E80</font></i></a>, <a class="ref" href="arytm_3.html#D9" target="_self">Def9</a>, <a class="ref" href="arytm_3.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E4:96"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_3.html#E2:96"><i><font color="Green">E81</font></i></a>, <a class="txt" href="arytm_3.html#E3:96"><i><font color="Green">E82</font></i></a>, <a class="txt" href="arytm_3.html#E4:96"><i><font color="Green">E83</font></i></a>, <a class="ref" href="arytm_3.html#T51" target="_self">Th51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_3.html#E3:96"><i><font color="Green">E82</font></i></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:96"><i><font color="Green">E82</font></i></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>4</sub></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">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></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">c<sub>4</sub></font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#R2" target="_self">divides</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="ref" href="arytm_3.html#D3" target="_self">Def3</a></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">c<sub>4</sub></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">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font>, <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  )
 
<b>by </b><i><a class="ref" href="arytm_3.html#D5" target="_self">Def5</a>, <a class="ref" href="arytm_3.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E12:96"/><b>then </b>
 <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="arytm_3.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E13:96"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:96"><i><font color="Green">E81</font></i></a>, <a class="ref" href="arytm_3.html#T46" target="_self">Th46</a></i>;<br/>
<b>hence </b><a NAME="E14:96"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>


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