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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="E1:28"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#R1" target="_self">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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> <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="hidden.html#R1">=</a>  <a href="int_2.html#K1" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E3:28"/>
<font color="Maroon">c<sub>1</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>
 
<b>by </b><i><a class="ref" href="int_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green">E22</font></i>: 
<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>2</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> <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>
 
<b>by </b><i><a class="txt" href="int_2.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="int_2.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E5:28"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="int_2.html#E2:28"><i><font color="Green">E21</font></i></a>, <a class="txt" href="int_2.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:28_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="int_2.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:28"/>
<font color="Maroon">c<sub>1</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#E2:28"><i><font color="Green">E21</font></i></a>, <a class="txt" href="int_2.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>


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