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<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/>

<a NAME="E1:62"/><i><font color="Green">E53</font></i>: 
 <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E2:62"/>
 <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>3</sub></font>, <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>3</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#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E3:62"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <a href="arytm_3.html#K6" target="_self">RED</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>,<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="hidden.html#R1">=</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="arytm_3.html#K6" target="_self">RED</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>,<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="hidden.html#R1">=</a>  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T28" target="_self">Th28</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1"/>
(  <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 or  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>3</sub></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="62_1_1"><b>suppose </b></a><a NAME="E1:62_1_1"/><i><font color="Green">E55</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
 ;<br/><div class="add"><a NAME="E2:62_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <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#T42" target="_self">Th42</a></i>;<br/><a NAME="E3:62_1_1"/><b>then </b>
 <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E4:62_1_1"/>
<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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:62"><i><font color="Green">E54</font></i></a>, <a class="txt" href="arytm_3.html#E1:62_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="arytm_3.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_2"><b>suppose </b></a><a NAME="E1:62_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:62_1_2"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <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#D9" target="_self">Def9</a></i>;<br/><a NAME="E3:62_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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#K8" target="_self">numerator</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<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><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="arytm_3.html#T42" target="_self">Th42</a></i>;<br/><b>hence </b><a NAME="E4:62_1_2"/>
<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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:62"><i><font color="Green">E53</font></i></a>, <a class="txt" href="arytm_3.html#E3:62"><i><font color="Green">E54</font></i></a>, <a class="txt" href="arytm_3.html#E1:62_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="arytm_3.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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