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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:29"/>
( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R1" target="_self">are_relative_prime</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></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">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="int_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green">E21</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E22</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T31" target="_self">Th31</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">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_2.html#E3:29"><i><font color="Green">E21</font></i></a>, <a class="ref" href="int_2.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> 1
 
<b>by </b><i><a class="txt" href="int_2.html#E2:29"><i><font color="Green">E20</font></i></a>, <a class="txt" href="int_2.html#E4:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="int_2.html#T33" target="_self">Th33</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">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i><a class="ref" href="int_2.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R1" target="_self">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D4" target="_self">Def4</a>, <a class="ref" href="int_2.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="int_2.html#E3:29"><i><font color="Green">E21</font></i></a>, <a class="ref" href="int_2.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E10:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> 1
 
<b>by </b><i><a class="txt" href="int_2.html#E2:29"><i><font color="Green">E20</font></i></a>, <a class="txt" href="int_2.html#E3:29"><i><font color="Green">E21</font></i></a>, <a class="ref" href="int_2.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E11:29"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i><a class="ref" href="int_2.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E12:29"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R1" target="_self">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4" target="_self">Def4</a>, <a class="ref" href="int_2.html#T8" target="_self">Th8</a></i>;<br/>


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