<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>reconsider </b><font color="Maroon" title="c2">e</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c3">c</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:20"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c2">e</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">e</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Olive" title="b1">n</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c2">e</font> ) )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T14" target="_self" title="ARYTM_3:th.14">Th14</a></i>;<br/>
<a NAME="E3:20"/>
( <font color="Maroon" title="c1">a</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c3">c</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Maroon" title="c3">c</font> &amp; ( for <font color="Olive" title="b1">b</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c1">a</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">b</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c3">c</font> <a href="arytm_3.html#R2" title="ARYTM_3:pred.2">divides</a> <font color="Olive" title="b1">b</font> ) )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T14" target="_self" title="ARYTM_3:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E4:20"/>
( <font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">hcf</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="arytm_3.html#K2" title="ARYTM_3:func.2">lcm</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="txt" href="#E2:20"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="arytm_3.html#D4" target="_self" title="ARYTM_3:def.4">Def4</a>, <a class="ref" href="arytm_3.html#D5" target="_self" title="ARYTM_3:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
