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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font>,<font color="Maroon">k</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:8"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E3:8"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T6">EULER_1:6</a></i>;<br/>
<a NAME="E4:8"/>
<font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<b>hence </b><a NAME="E5:8"/>
<font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="newton.html#T63">NEWTON:63</a></i>;<br/>


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