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

<b>let </b><font color="Maroon">R</font> be   <a href="gcd_1.html#NM3">gcdDomain</a>;<br/><b>let </b><font color="Maroon">Amp</font> be    <a href="gcd_1.html#M2" target="_self">AmpleSet</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">r2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:67"/><i><font color="Green">E20</font></i>: 
(  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>

<b>set </b><font color="Maroon">d</font> =  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font>;<br/>
<a NAME="E2:67"/><i><font color="Green">E21</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:67"/><i><font color="Green">E25</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:67"/><i><font color="Green">E26</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>set </b><font color="Maroon">r</font> = <font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">s</font> = <font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>;<br/>
<a NAME="E5:67"/><i><font color="Green">E53</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E6:67"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<a NAME="E1:67_1"/><i><font color="Green">E54</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E2:67_1"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:67_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>
 <b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">s2</font>
<b>by </b><i><a class="ref" href="gcd_1.html#T10" target="_self">Th10</a>, <a class="txt" href="gcd_1.html#E5:67"><i><font color="Green">E53</font></i></a>, </i>
;<br/>
<a NAME="E6:67_1"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">s2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:67_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T12" target="_self">Th12</a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E8:67_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/>
<a NAME="E9:67_1"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:67_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T9" target="_self">Th9</a>, <a class="ref" href="gcd_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E11:67_1"/><i><font color="Green">E64</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:67_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:67_1"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">s1</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T15" target="_self">Th15</a>, <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:67"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E8:67"/><i><font color="Green">E84</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E5:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#T6" target="_self">Th6</a></i>;<br/>
<i><font color="Green">E85</font></i>:  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font> = 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Amp</font>
<b>by </b><i>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="gcd_1.html#T12" target="_self">Th12</a>, , <a class="ref" href="gcd_1.html#D4" target="_self">Def4</a></i>
;<br/>
<a NAME="E10:67"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><b>proof </b></a><div class="add">

<a NAME="E1:67_3"/><i><font color="Green">E87</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r2</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r2</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r2</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E5:67"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r2</font></span>)</span> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E5:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T11" target="_self">Th11</a>, <a class="txt" href="gcd_1.html#E5:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a>, </i>
;<br/>
<b>hence </b><a NAME="E3:67_3"/>
<font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:67"/><i><font color="Green">E88</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E12:67"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4"><b>proof </b></a><div class="add">

<a NAME="E1:67_4"/><i><font color="Green">E89</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E2:67_4"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:67_4"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>
 <b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">r2</font>
<b>by </b><i><a class="ref" href="gcd_1.html#T11" target="_self">Th11</a>, <a class="txt" href="gcd_1.html#E5:67"><i><font color="Green">E53</font></i></a>, </i>
;<br/>
<a NAME="E6:67_4"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:67_4"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E8:67_4"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/>
<a NAME="E9:67_4"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:67_4"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T9" target="_self">Th9</a>, , </i>;<br/>
<a NAME="E11:67_4"/><i><font color="Green">E94</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:67_4"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:67_4"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">r1</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:67"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E14:67"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E5:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E15:67"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T17" target="_self">Th17</a>, <a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E16:67"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E17:67"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T14" target="_self">Th14</a>, <a class="ref" href="gcd_1.html#T15" target="_self">Th15</a>, <a class="ref" href="gcd_1.html#T5" target="_self">Th5</a></i>;<br/>


</div>
