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



<b>assume </b><a NAME="E1:65"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1"/>
(  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 or  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><b>suppose </b></a><a NAME="E1:65_1_1"/><i><font color="Green">E14</font></i>: 
 <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:65_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E3:65_1_1"/>
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T16" target="_self">Th16</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2"><b>suppose </b></a><a NAME="E1:65_1_2"/>
 <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:65_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 <b>by </b><i><a class="ref" href="arytm_3.html#T15" target="_self">Th15</a>, <a class="ref" href="arytm_3.html#T9" target="_self">Th9</a>, </i>;<br/><b>hence </b><a NAME="E3:65_1_2"/>
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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