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<b>let </b><font color="Maroon">a</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">b</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:36"/><i><font color="Green">E14</font></i>: 
( 1 <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; 1 <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<b>set </b><font color="Maroon">ab</font> = <font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>suppose </b></a><a NAME="E1:36_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:36_1_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i><a class="ref" href="arytm_3.html#T4" target="_self">Th4</a>, <a class="ref" href="arytm_3.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_3.html#T1" target="_self">Th1</a>, <a class="ref" href="ordinal3.html#T81">ORDINAL3:81</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon">a</font> <a href="ordinal3.html#K6">div^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">b</font> <a href="ordinal3.html#K6">div^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><b>suppose </b></a><a NAME="E1:36_1_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:36_1_2"/>
( <font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E3:36_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="ordinal3.html#K6">div^</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="ordinal3.html#K6">div^</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:36"><i><font color="Green">E13</font></i></a></i>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">d1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">d2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D2" target="_self">ARYTM_3:def 2</a><br/><b>assume </b><a NAME="E4:36_1_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">a</font> <a href="ordinal3.html#K6">div^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">b</font> <a href="ordinal3.html#K6">div^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">d2</font> )
 ;<br/><a NAME="E5:36_1_2"/>
( <font color="Maroon">a</font> <a href="ordinal3.html#K6">div^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="ordinal3.html#K6">div^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <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="ordinal2.html#T52">ORDINAL2:52</a></i>;<br/><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font>, <font color="Maroon">d1</font> = <font color="Maroon">d1</font>, <font color="Maroon">d2</font> = <font color="Maroon">d2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E7:36_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">d2</font> )
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T5" target="_self">Th5</a>, <a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/><a NAME="E8:36_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E9:36_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E10:36_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:36"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E11:36_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/><a NAME="E12:36_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</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="ordinal2.html#T52">ORDINAL2:52</a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/><a NAME="E13:36_1_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal3.html#T36">ORDINAL3:36</a></i>;<br/><b>hence </b><a NAME="E14:36_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="ordinal3.html#T41">ORDINAL3:41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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