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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">c</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/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &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="hidden.html#R1">=</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>hence </b><a NAME="E3:34_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <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> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">d</font> = <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> 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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font>;<br/><a NAME="E3:34_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="E4:34_1_2"/><b>then </b>
( <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#E1:34_1_2"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E5:34_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <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="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <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="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/><a NAME="E6:34_1_2"/><b>then </b>
( <font color="Maroon">d</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> &amp; <font color="Maroon">d</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> )
 <b>by </b><i/>;<br/><a NAME="E7:34_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="arytm_3.html#R2" target="_self">divides</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E14</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">d</font> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_3.html#E1:34_1_2"><i><font color="Green">E13</font></i></a></i>
<br/>.= 
<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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>
;<br/><a NAME="E9:34_1_2"/><b>then </b>
( <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></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#K3" target="_self">*^</a> <font color="Maroon">c</font> &amp; <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></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> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> )
 <b>by </b><i/>;<br/><a NAME="E10:34_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></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#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p2">(<span class="default"><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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></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">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p2">(<span class="default"><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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:34_1_2"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E11:34_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <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#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T58">ORDINAL3:58</a></i>;<br/><a NAME="E12:34_1_2"/><b>then </b>
( <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></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#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p2">(<span class="default"><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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></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">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="ordinal3.html#K6">div^</a> <span class="p2">(<span class="default"><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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T3" target="_self">Th3</a>, <a class="ref" href="ordinal3.html#T36">ORDINAL3:36</a></i>;<br/><a NAME="E13:34_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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E14:34_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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span> <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="E15:34_1_2"/><b>then </b>
<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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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><a class="txt" href="arytm_3.html#E1:34_1_2"><i><font color="Green">E13</font></i></a></i>;<br/><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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">a</font> <a href="arytm_3.html#K5" target="_self">hcf</a> <font color="Maroon">b</font>
<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">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
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>
;<br/><a NAME="E17:34_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>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="ordinal3.html#T36">ORDINAL3:36</a></i>;<br/><a NAME="E18:34_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="arytm_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="ordinal3.html#T41">ORDINAL3:41</a></i>;<br/><b>hence </b><a NAME="E19:34_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <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> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_3"><b>suppose </b></a><a NAME="E1:34_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_3"/><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">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="arytm_3.html#T1" target="_self">Th1</a>, <a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>;<br/><b>hence </b><a NAME="E3:34_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <a href="arytm_3.html#K5" target="_self">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">c</font></span>)</span> <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> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
