<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c5">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E2:54_1_1"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c5">u</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">y</font> &amp; ( for <font color="Olive" title="b1">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">y</font> holds <br/><font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">u</font> ) )
 <b>by </b><i><a class="ref" href="gcd_1.html#D11" target="_self" title="GCD_1:def.11">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font><b> such that </b><br/><a NAME="E4:54_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c5">u</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></i>;<br/>
<a NAME="E5:54_1_1"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">u</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E4:54_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>;<br/>
<a NAME="E6:54_1_1"/><b>then </b><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:54_1_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="gcd_1.html#T2" target="_self" title="GCD_1:th.2">Th2</a></i>;<br/>
<a NAME="E7:54_1_1"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:54_1_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="gcd_1.html#E5:54_1_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="gcd_1.html#T2" target="_self" title="GCD_1:th.2">Th2</a></i>;<br/>
<a NAME="E8:54_1_1"/>
for <font color="Olive" title="b1">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">y</font> holds <br/><font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c6">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" title="c7">zz</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>

<b>assume </b><a NAME="E1:54_1_1_1"/>
( <font color="Maroon" title="c7">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c7">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">y</font> )
 ;<br/>

<a NAME="E2:54_1_1_1"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c7">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">u</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:54_1_1"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E3:54_1_1_1"/>
<font color="Maroon" title="c5">u</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c6">z</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E4:54_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1_1_1"/>
<font color="Maroon" title="c7">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:54_1_1_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="gcd_1.html#T2" target="_self" title="GCD_1:th.2">Th2</a></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" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">y</font> &amp; ( for <font color="Olive" title="b2">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">y</font> holds <br/><font color="Olive" title="b2">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="gcd_1.html#E6:54_1_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="gcd_1.html#E7:54_1_1"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/>


</div>
