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<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>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>assume </b><a NAME="E1:9"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</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/>

<b>then consider </b><font color="Maroon">t1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:9"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E4:9"/>
<font color="Maroon">t1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:9"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t1</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">t2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:9"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E8:9"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:9"/>
<font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <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="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


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