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<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E2:54_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">u</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Maroon">u</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">u</font> ) )
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font><b> such that </b><br/><a NAME="E4:54_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">z</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:54_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D3" target="_self">Def3</a>, </i>;<br/>
<a NAME="E6:54_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E7:54_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E8:54_1_1"/>
for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">zz</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:54_1_1_1"/>
( <font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:54_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:54_1_1_1"/>
<font color="Maroon">u</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D3" target="_self">Def3</a>, </i>;<br/>
<b>hence </b><a NAME="E4:54_1_1_1"/>
<font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:54_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="gcd_1.html#T2" target="_self">Th2</a>, </i>;<br/>


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