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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<a NAME="E1:68"/>
( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font> implies ( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">M</font> = <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font>;<br/>
<b>assume </b><a NAME="E1:68_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:68_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Olive">l</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> holds <br/><font color="Olive">l</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font> ) )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:68_1"/>
( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="newton.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:68"/>
( ( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> ) iff <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>


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