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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">y</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c3">y</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">m</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:59"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E4:59"/>
<font color="Maroon" title="c3">y</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:59"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a>, <a class="txt" href="int_4.html#E3:59"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">r</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:59"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E7:59"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E3:59"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="int_4.html#E6:59"><i><font color="Green" title="E44">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:59"/>
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>


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