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

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> ;<br/>
<a NAME="E1:119_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">J</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E3:119_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:119_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:119_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 ;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:119_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:119_1_1_1"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 
<b>by </b><i><a class="ref" href="ideal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="ideal_1.html#T12" target="_self">Th12</a>, </i>;<br/>
<b>hence </b><a NAME="E7:119_1_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="ideal_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:119_1_1"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">J</font> is  <a href="ideal_1.html#V1" target="_self">add-closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div>
