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<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">i</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">k</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1"/>
( ( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  ) or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  or <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1"><b>suppose </b></a><a NAME="E1:75_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:75_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
( <font color="Maroon">i</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp; <font color="Maroon">k</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">l</font> <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="E3:75_1_1"/>
(  <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#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">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> &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#T14" target="_self">Th14</a></i>;<br/><b>hence </b><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="arytm_3.html#K12" target="_self">*'</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">i</font>,<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#K6" target="_self">RED</a> <font color="Maroon">k</font>,<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">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#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<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>
<b>by </b><i><a class="ref" href="arytm_3.html#T17" target="_self">Th17</a>, <a class="ref" href="arytm_3.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></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></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<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>
<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#K6" target="_self">RED</a> <font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">i</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<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>
<b>by </b><i><a class="ref" href="arytm_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">i</font></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#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">j</font>,<font color="Maroon">i</font></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></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#K6" target="_self">RED</a> <font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">i</font></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#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">i</font></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> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></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="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="arytm_3.html#T17" target="_self">Th17</a>, <a class="ref" href="arytm_3.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<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></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></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="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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></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="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="arytm_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" target="_self">RED</a> <font color="Maroon">l</font>,<font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p3">(<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></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</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>
<b>by </b><i><a class="ref" href="arytm_3.html#D2" target="_self">Def2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_2"><b>suppose </b></a><a NAME="E1:75_1_2"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  or <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:75_1_2"/><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>  <a href="arytm_3.html#K13" target="_self">{}</a>  or <font color="Maroon">k</font> <a href="arytm_3.html#K10" target="_self">/</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  ) &amp; <font color="Maroon">j</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">l</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#T9" target="_self">Th9</a>, <a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>;<br/><a NAME="E3:75_1_2"/><b>then </b>
( <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="arytm_3.html#K12" target="_self">*'</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#K13" target="_self">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</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> <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#T9" target="_self">Th9</a>, </i>;<br/><b>hence </b><a NAME="E4:75_1_2"/>
<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="arytm_3.html#K12" target="_self">*'</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> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">k</font></span>)</span> <a href="arytm_3.html#K10" target="_self">/</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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
