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<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> ;<br/>

<b>set </b><font color="Maroon" title="c4">nx</font> =  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font>;<br/>
<b>set </b><font color="Maroon" title="c5">dx</font> =  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font>;<br/>
<b>assume </b><a NAME="E1:69"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ;<br/>

<a NAME="E2:69"/><b>then </b><i><font color="Green" title="E63">A1</font></i>: 
(  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  &amp;  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T41" target="_self" title="ARYTM_3:th.41">Th41</a>, <a class="ref" href="arytm_3.html#T43" target="_self" title="ARYTM_3:th.43">Th43</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">y</font> = <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font></span>)</span>;<br/>

<a NAME="E3:69"/>
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font>, <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T40" target="_self" title="ARYTM_3:th.40">Th40</a></i>;<br/>
<a NAME="E4:69"/><b>then </b><i><font color="Green" title="E64">A2</font></i>: 
(  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font> &amp;  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:69"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="arytm_3.html#T49" target="_self" title="ARYTM_3:th.49">Th49</a></i>;<br/>
<a NAME="E5:69"/>
<span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:69"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></i>;<br/>
<b>hence </b><a NAME="E6:69"/>
<font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:69"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="arytm_3.html#T47" target="_self" title="ARYTM_3:th.47">Th47</a></i>;<br/>


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