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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">i</font> <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="c1">i</font></span><br/>

<b>set </b><font color="Maroon" title="c3">a</font> = <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">i</font></span>)</span>;<br/>
<a NAME="E1:29"/><i><font color="Green" title="E22">E</font></i>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="int_6.html#T13" target="_self" title="INT_6:th.13">thabs0</a></i>;<br/>
<a NAME="E2:29"/><b>then </b><i><font color="Green" title="E23">D</font></i>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T2" title="INT_2:th.2">INT_2:2</a></i>;<br/>
<a NAME="E3:29"/>
for <font color="Olive" title="b1">m</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Olive" title="b1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> holds <br/><font color="Olive" title="b1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">m</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="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> implies <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">i</font> )</span><br/>

<b>assume </b><a NAME="E1:29_1"/>
( <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">i</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green" title="E24">H</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">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/>
<a NAME="E4:29_1"/>
<font color="Maroon" title="c1">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="int_6.html#T13" target="_self" title="INT_6:th.13">thabs0</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">l</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:29_1"/><i><font color="Green" title="E25">K</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">l</font> <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="c1">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/>
<a NAME="E7:29_1"/>
<font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">l</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="c1">i</font>
 
<b>by </b><i><a class="txt" href="int_6.html#E3:29_1"><i><font color="Green" title="E24">H</font></i></a>, <a class="txt" href="int_6.html#E6:29_1"><i><font color="Green" title="E25">K</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:29_1"/>
<font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">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>; <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="E4:29"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">i</font> <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="c1">i</font>
 <b>by </b><i><a class="txt" href="int_6.html#E2:29"><i><font color="Green" title="E23">D</font></i></a>, <a class="txt" href="int_6.html#E1:29"><i><font color="Green" title="E22">E</font></i></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</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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