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

<b>let </b><font color="Maroon">M1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>, <font color="Maroon">M2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>;<br/>

<b>assume </b><a NAME="E1:28_1_2"/><i><font color="Green">E56</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> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</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#K2" target="_self">`1</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"><font color="Olive">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> ) )
 ;<br/>

<b>assume </b><a NAME="E2:28_1_2"/><i><font color="Green">E57</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> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</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#K2" target="_self">`1</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"><font color="Olive">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> ) )
 ;<br/>

<a NAME="E3:28_1_2"/><i><font color="Green">E58</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> <font color="Maroon">M1</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_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="E1:28_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font>
 ;<br/>

<a NAME="E2:28_1_2_1"/><b>then </b>
 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#K2" target="_self">`1</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="Olive">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:28_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:28_1_2"/>
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> <font color="Maroon">M2</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_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>assume </b><a NAME="E1:28_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font>
 ;<br/>

<a NAME="E2:28_1_2_2"/><b>then </b>
 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#K2" target="_self">`1</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="Olive">a</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:28_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:28_1_2"/>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>


</div>
