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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="gcd_1.html#NM3">gcdDomain</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="gcd_1.html#M2" target="_self">AmpleSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:67"/><i><font color="Green">E50</font></i>: 
(  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:67"/><i><font color="Green">E51</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E3:67"/><i><font color="Green">E52</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:67"/><i><font color="Green">E53</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:67"><i><font color="Green">E50</font></i></a>, <a class="ref" href="gcd_1.html#T34" target="_self">Th34</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E5:67"/><i><font color="Green">E54</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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T40" target="_self">Th40</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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></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">E55</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self">Def12</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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self">Def12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:67_1"/><i><font color="Green">E56</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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E4:67_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="gcd_1.html#E2:67"><i><font color="Green">E51</font></i></a>, <a class="txt" href="gcd_1.html#E4:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:67_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:67_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gcd_1.html#D12" target="_self">Def12</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">c<sub>1</sub></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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:67_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:67"><i><font color="Green">E50</font></i></a>, <a class="txt" href="gcd_1.html#E7:67_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E11:67_1"/><i><font color="Green">E59</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E12:67_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D8" target="_self">Def8</a></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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E10:67_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gcd_1.html#E11:67_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="gcd_1.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:67"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E8:67"/><i><font color="Green">E56</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E4:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#T39" target="_self">Th39</a></i>;<br/>
<i><font color="Green">E57</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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> = 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="gcd_1.html#E5:67"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gcd_1.html#T30" target="_self">Th30</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="gcd_1.html#E7:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gcd_1.html#E8:67"><i><font color="Green">E56</font></i></a>, <a class="ref" href="gcd_1.html#T30" target="_self">Th30</a></i>
;<br/>
<a NAME="E10:67"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">E59</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T6" target="_self">Th6</a></i>;<br/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E4:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#T9" target="_self">Th9</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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E4:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gcd_1.html#E1:67_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="gcd_1.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E3:67"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gcd_1.html#E4:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gcd_1.html#E1:67_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="gcd_1.html#T11" target="_self">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E3:67_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:67"/><i><font color="Green">E59</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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T40" target="_self">Th40</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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></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">E60</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self">Def12</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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self">Def12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:67_4"/><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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E4:67_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="gcd_1.html#E3:67"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gcd_1.html#E4:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:67_4"/><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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:67_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="gcd_1.html#D12" target="_self">Def12</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">c<sub>1</sub></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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:67_4"/><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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:67"><i><font color="Green">E50</font></i></a>, <a class="txt" href="gcd_1.html#E7:67_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E11:67_4"/><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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E12:67_4"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D8" target="_self">Def8</a></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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E10:67_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gcd_1.html#E11:67_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="gcd_1.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:67"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T38" target="_self">Th38</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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E4:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gcd_1.html#T39" target="_self">Th39</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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E11:67"><i><font color="Green">E59</font></i></a>, <a class="txt" href="gcd_1.html#E13:67"><i><font color="Green">E60</font></i></a>, <a class="ref" href="gcd_1.html#T30" target="_self">Th30</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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T38" target="_self">Th38</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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E9:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="gcd_1.html#E10:67"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gcd_1.html#T38" target="_self">Th38</a></i>;<br/>


</div>
