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

<b>let </b><font color="Maroon">F</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V4">right-distributive</a> <a href="vectsp_1.html#V5">left-distributive</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="gcd_1.html#D11" target="_self">GCD_1:def 11</a><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> ) or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1"><b>case </b></a><a NAME="E1:51_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> )
 ;<br/><div class="add"><a NAME="E2:51_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/><a NAME="E3:51_1_1_1_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E4:51_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/><a NAME="E5:51_1_1_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E6:51_1_1_1_1_1"/>
for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:51_1_1_1_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">zz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> or <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:51_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">zz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">zz'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:51_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">zz</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz'</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a></i>;<br/><b>thus </b><a NAME="E4:51_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:51_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:51_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E4:51_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">zz</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:51_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, , , <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:51_1_1_1_1_1_1"/>
<font color="Maroon">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:51_1_1_1_1_1"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> st <br/>( <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Olive">z</font> ) )
 <b>by </b><i><a class="ref" href="gcd_1.html#D3" target="_self">Def3</a>, <a class="ref" href="gcd_1.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2"><b>case </b></a><a NAME="E1:51_1_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:51_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>;<br/><a NAME="E3:51_1_1_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">y</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E4:51_1_1_1_1_2"/>
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>  st <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font>
 ;<br/><b>hence </b><a NAME="E5:51_1_1_1_1_2"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> st <br/>( <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Olive">z</font> ) )
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, <a class="ref" href="gcd_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_3"><b>case </b></a><a NAME="E1:51_1_1_1_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:51_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>;<br/><a NAME="E3:51_1_1_1_1_3"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E4:51_1_1_1_1_3"/>
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>  st <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> holds <br/><font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E5:51_1_1_1_1_3"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> st <br/>( <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Olive">z</font> ) )
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a>, <a class="ref" href="gcd_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_1_1"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font> st <br/>( <font color="Olive">z</font> <a href="gcd_1.html#R1" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R1" target="_self">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R1" target="_self">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R1" target="_self">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R1" target="_self">divides</a> <font color="Olive">z</font> ) )
 ;<br/>


</div>
