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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><a NAME="E1:4"/>
(  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> is  <a href="waybel_6.html#V1">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font> implies <font color="Maroon">x</font> is <a href="waybel_3.html#V1">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E31</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> is  <a href="waybel_6.html#V1">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:4_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E3:4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:4_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="waybel_6.html#D1">WAYBEL_6:def 1</a></i>;<br/>
<a NAME="E6:4_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E7:4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/>
<b>hence </b><a NAME="E8:4_1"/>
<font color="Maroon">x</font> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> is <a href="waybel_3.html#V1">compact</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 ;<br/><b>take </b><font color="Maroon">x2</font> = <font color="Maroon">x</font>;<br/><a NAME="E2:4_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x2</font>
 ;<br/><b>hence </b><a NAME="E3:4_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E4:4_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E5:4_2"/>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/><b>hence </b><a NAME="E6:4_2"/>
<font color="Maroon">x2</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> is  <a href="waybel_6.html#V1">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_6.html#D1">WAYBEL_6:def 1</a></i>;<br/>


</div>
