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

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> : <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> </span>}</span> ;<br/>
<a NAME="E1:12_1_1"/><i><font color="Green">E20</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> : <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> : <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> </span>}</span>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> : <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E3:12_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:12_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> : <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:12_1_1"/>
for <font color="Olive">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>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> : <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> </span>}</span>  holds <br/><span class="p1">(<span class="default"><font color="Olive">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:12_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> : <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E3:12_1_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:12_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:12_1_1"/><b>then </b>
for <font color="Olive">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> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> : <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> </span>}</span>  iff <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E4:12_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font></span>)</span> 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 <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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</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#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> )
 <b>by </b><i/>;<br/>


</div>
