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

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E2:27_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<a NAME="E3:27_1_1"/>
<font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>;<br/>
<a NAME="E5:27_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font> holds <br/> ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>;<br/>

<b>assume </b><a NAME="E1:27_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E2:27_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E24</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E4:27_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:27_1_1_1"/>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:27_1_1"/>
for <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>  st  ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> ) holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>;<br/>

<b>given </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><a NAME="E1:27_1_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:27_1_1_2"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:27_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T51">GCD_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T51">GCD_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T51">GCD_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T51">GCD_1:51</a></i>
;<br/>
<a NAME="E5:27_1_1_2"/>
<font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="gcd_1.html#R2">divides</a> <font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> 
 
;<br/>
<a NAME="E6:27_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="gcd_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7">GCD_1:7</a></i>;<br/>
<a NAME="E7:27_1_1_2"/>
<font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="gcd_1.html#R2">divides</a> <font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> 
 
;<br/>
<a NAME="E8:27_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="gcd_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7">GCD_1:7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="gcd_1.html#K1">/</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E5:27_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="gcd_1.html#T9">GCD_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="gcd_1.html#K1">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E5:27_1_1"><i><font color="Green">E23</font></i></a>, , <a class="txt" href="quofield.html#E1:27_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="gcd_1.html#T11">GCD_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="gcd_1.html#K1">/</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E5:27_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="quofield.html#T1" target="_self">Th1</a>, <a class="ref" href="gcd_1.html#T11">GCD_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E5:27_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="gcd_1.html#T9">GCD_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E10:27_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:27_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font> st <br/>for <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff  ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E2:27_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>


</div>
