<?xml version="1.0"?>
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<b>let </b><font color="Maroon">t</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">r</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="E1:119"/>
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 ;<br/>

<a NAME="E2:119"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</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/>
<a NAME="E3:119"/>
<span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">t</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> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">t</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> <font color="Maroon">r</font></span>)</span> as  <a href="ordinal1.html#V3">ordinal</a>  <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E5:119"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>

<b>set </b><font color="Maroon">y</font> = <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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</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> <font color="Maroon">r</font></span>)</span>;<br/>
<b>take </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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p4">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</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> <font color="Maroon">r</font></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D13" target="_self">ARYTM_3:def 13</a><br/>

<a NAME="E6:119"/><i><font color="Green">E14</font></i>: 
(  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</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">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E7:119"/><b>then </b>
<span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</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">r</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="E8:119"/><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"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</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">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<a NAME="E9:119"/><b>then </b>
 <a href="arytm_3.html#K14" target="_self">one</a>  <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</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">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, </i>;<br/>
<a NAME="E10:119"/><b>then </b>
<span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</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">r</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="arytm_3.html#K2" target="_self">+^</a> <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">t</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">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal3.html#D6">ORDINAL3:def 6</a></i>;<br/>
<b>then </b><font color="Maroon">s</font> <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> <font color="Maroon">t</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">r</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">t</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> <font color="Maroon">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default">1 <a href="arytm_3.html#K2" target="_self">+^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> 1</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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T54">ORDINAL3:54</a></i>
;<br/>
<a NAME="E12:119"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <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> <font color="Maroon">t</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#K9" target="_self">denominator</a> <font color="Maroon">r</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">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> 1</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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">s</font> <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> <font color="Maroon">t</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> <font color="Maroon">t</font></span>)</span> = 
<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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> 1</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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></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> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, <a class="txt" href="arytm_3.html#E21"><i><font color="Green">Lemma58</font></i></a></i>
<br/>.= 
<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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> 1</span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11" target="_self">+</a> <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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<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">r</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="arytm_3.html#K11" target="_self">+</a> <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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T10" target="_self">Th10</a></i>
;<br/>
<b>then </b><font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <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> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default">1 <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">t</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K10" target="_self">/</a> 1</span>)</span> <a href="arytm_3.html#K12" 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">t</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">t</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">s</font> <a href="arytm_3.html#K12" 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">t</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">t</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T11" target="_self">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E15:119"/>
<font color="Maroon">s</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <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">r</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p5">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="ordinal3.html#K5">-^</a> 1</span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T10" target="_self">Th10</a></i>;<br/>


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