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<b>let </b><font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>





<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1"/>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>suppose </b></a><a NAME="E1:56_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:56_1_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="arytm_3.html#D10" target="_self" title="ARYTM_3:def.10">Def10</a>, <a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D10" target="_self" title="ARYTM_3:def.10">Def10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2"><b>suppose </b></a><a NAME="E1:56_1_2"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:56_1_2"/><b>then </b><i><font color="Green" title="E55">A3</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:56"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></i>;<br/><b>then </b><i><font color="Green" title="E56">A4</font></i>:  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> = 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T48" target="_self" title="ARYTM_3:th.48">Th48</a></i>
<br/>.= 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="txt" href="arytm_3.html#E1:56"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="arytm_3.html#T33" target="_self" title="ARYTM_3:th.33">Th33</a></i>
<br/>.= 
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_3.html#E1:56_1_2"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="arytm_3.html#T48" target="_self" title="ARYTM_3:th.48">Th48</a></i>
;<br/> <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> = 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_3.html#E2:56_1_2"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="arytm_3.html#T48" target="_self" title="ARYTM_3:th.48">Th48</a></i>
<br/>.= 
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="arytm_3.html#E1:56"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="arytm_3.html#T33" target="_self" title="ARYTM_3:th.33">Th33</a></i>
<br/>.= 
 <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_3.html#E1:56_1_2"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="arytm_3.html#T48" target="_self" title="ARYTM_3:th.48">Th48</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <font color="Maroon" title="c3">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span>)</span></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#K7" title="ARYTM_3:func.7">denominator</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_3.html#E3:56_1_2"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="arytm_3.html#T45" target="_self" title="ARYTM_3:th.45">Th45</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T45" target="_self" title="ARYTM_3:th.45">Th45</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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