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<b>let </b><font color="Maroon" title="c1">c</font>, <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">c</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies <font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">b</font> )</span><br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">c</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T39" target="_self" title="INT_2:th.39">Th39</a></i>;<br/>
<a NAME="E3:32"/>
<font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T12" target="_self" title="INT_2:th.12">Th12</a></i>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_2.html#E1:32"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>
<a NAME="E5:32"/>
( <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E2:32_1"/><b>then </b>
<font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="int_2.html#E2:32"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="int_2.html#E4:32"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:32_1"/>
<font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="int_2.html#T14" target="_self" title="INT_2:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon" title="c1">c</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="int_2.html#E2:32"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="int_2.html#E4:32"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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