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

<a NAME="E1:52_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font> holds <br/><font color="Olive">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_1_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font> )
 ;<br/>

<a NAME="E2:52_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">I</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:52_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E5:52_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">J</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:52_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>
<a NAME="E8:52_1_1_1"/>
<font color="Maroon">I</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">J</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:52_1_1_1"/><b>then </b>
<font color="Maroon">I</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E10:52_1_1_1"/>
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="yellow_2.html#E4:52_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="yellow_2.html#E7:52_1_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:52_1_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span> is <a href="lattice3.html#V2">with_infima</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T12">YELLOW_1:12</a></i>;<br/>


</div>
