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

<b>let </b><font color="Maroon">l</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">j</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>







<b>assume </b><a NAME="E1:68"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>set </b><font color="Maroon">x</font> = <font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font>;<br/>
<b>set </b><font color="Maroon">nx</font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">dx</font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">ny</font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">dy</font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span>;<br/>
<a NAME="E2:68"/><i><font color="Green">E14</font></i>: 
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#D4" target="_self">Def4</a>, <a class="ref" href="arytm_3.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E3:68"/><i><font color="Green">E15</font></i>: 
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">k</font>,<font color="Maroon">l</font> &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#D4" target="_self">Def4</a>, <a class="ref" href="arytm_3.html#T14" target="_self">Th14</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:68_1"/>
<font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font>
 ;<br/><a NAME="E2:68_1"/><b>then </b>
( <font color="Maroon">i</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">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">l</font> <a href="hidden.html#R1">=</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">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><font color="Maroon">i</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">l</font> = 
<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> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">j</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>
<br/>.= 
<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">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">j</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> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>
<br/>.= 
<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">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">j</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">j</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> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>
<br/>.= 
<font color="Maroon">j</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="arytm_3.html#D2" target="_self">Def2</a></i>
;<br/><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:68"/>
<font color="Maroon">i</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E6:68"/><b>then </b>
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">l</font></span>)</span> &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">l</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#D4" target="_self">Def4</a>, , </i>;<br/>
<a NAME="E7:68"/><b>then </b>
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span> &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#D4" target="_self">Def4</a>, , </i>;<br/>
<a NAME="E8:68"/><b>then </b>
<font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</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">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</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">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E9:68"/>
<font color="Maroon">i</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T10" target="_self">Th10</a></i>;<br/>


</div>
