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

<b>let </b><font color="Maroon">L</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">x</font> is  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/><b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> ;<br/><a NAME="E4:34_1"/>
<font color="Maroon">Y</font> is <a href="waybel_0.html#V2">filtered</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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 <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a><br/>

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:34_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font> <a href="lattice3.html#K8">~</a> 
;<br/>

<a NAME="E4:34_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="lattice3.html#K8">~</a>  &amp;  <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="lattice3.html#K8">~</a>  )
 
;<br/>
<b>hence </b><a NAME="E5:34_1_1"/>
( <font color="Maroon">z</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">z</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:34_1"/>
<font color="Maroon">x</font> is  <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:34"/>
<font color="Maroon">x</font> is  <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E5:34"/>
<font color="Maroon">Y</font> is <a href="waybel_0.html#V1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><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">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E3:34_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="lattice3.html#K8">~</a>  )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a></i>;<br/>
<b>take </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">z</font>
;<br/>

<a NAME="E4:34_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">z</font> )
 
;<br/>
<b>hence </b><a NAME="E5:34_2"/>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
<font color="Maroon">x</font> is  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 ;<br/>


</div>
