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





<b>assume </b><a NAME="E1:12"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">b1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:12"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E5:12"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E6:12"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i>, <a class="ref" href="int_2.html#T38">INT_2:38</a></i>;<br/>
<a NAME="E7:12"/>
( <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b1</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E8:12"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E9:12"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i>, </i>;<br/>


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