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



<b>set </b><font color="Maroon">nr</font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">dr</font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">ns</font> =  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">ds</font> =  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">s</font>;<br/>
<a NAME="E1:91"/><i><font color="Green">E13</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">s</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> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">s</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/>;<br/>
<a NAME="E2:91"/><b>then </b><i><font color="Green">E14</font></i>: 
<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="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">s</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="E3:91"/><i><font color="Green">E15</font></i>: 
(  <a href="arytm_3.html#K9" target="_self">denominator</a> <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="arytm_3.html#K8" target="_self">numerator</a> <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 
<b>by </b><i>, , <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<b>assume </b><a NAME="E4:91"/>
<font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 ;<br/>

<a NAME="E5:91"/><b>then </b>
<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"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">s</font></span>)</span></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">s</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></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</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">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#K10" target="_self">/</a> 1
 
<b>by </b><i><a class="ref" href="arytm_3.html#T27" target="_self">Th27</a>, <a class="ref" href="arytm_3.html#T10" target="_self">Th10</a></i>;<br/>
<b>then </b><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"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">s</font></span>)</span></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">s</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></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">r</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">s</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <a href="arytm_3.html#K13" target="_self">{}</a> 
<b>by </b><i><a class="ref" href="arytm_3.html#T26" target="_self">Th26</a>, <a class="txt" href="arytm_3.html#E21"><i><font color="Green">Lemma58</font></i></a></i>
<br/>.= 
 <a href="arytm_3.html#K13" target="_self">{}</a> 
<b>by </b><i><a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>
;<br/>
<a NAME="E7:91"/><b>then </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"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="arytm_3.html#K2" 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">s</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> <a href="hidden.html#R1">=</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="E8:91"/><b>then </b>
<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="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">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 
<b>by </b><i><a class="ref" href="ordinal3.html#T29">ORDINAL3:29</a></i>;<br/>
<a NAME="E9:91"/><b>then </b>
 <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 
<b>by </b><i><a class="ref" href="arytm_3.html#T25" target="_self">Th25</a>, <a class="ref" href="ordinal3.html#T34">ORDINAL3:34</a></i>;<br/>
<b>hence </b><a NAME="E10:91"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</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/>


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