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<b>let </b><font color="Maroon">a</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:29"/>
( <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="int_2.html#R1" target="_self">are_relative_prime</a>  &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="int_2.html#R1" target="_self">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">a</font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">b</font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E3:29"/><i><font color="Green">E11</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E12</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:29"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:29"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> 1
 
<b>by </b><i>, , <a class="ref" href="int_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:29"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1 )
 
<b>by </b><i><a class="txt" href="int_2.html#E2:29"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
<font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="int_2.html#R1" target="_self">are_relative_prime</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="int_1.html#R2">divides</a> 1
 
<b>by </b><i>, , <a class="ref" href="int_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E11:29"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1 )
 
<b>by </b><i><a class="txt" href="int_2.html#E2:29"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:29"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="int_2.html#R1" target="_self">are_relative_prime</a> 
 <b>by </b><i>, </i>;<br/>


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