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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">B</font> be  <a href="waybel23.html#V2" target="_self">join-closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:77"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>assume </b><a NAME="E2:77"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">y</font> holds <br/> ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">y</font> )
 ;<br/>

<b>thus </b><a NAME="E3:77"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:77_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>

<a NAME="E2:77_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E4:77_1"/>
<font color="Maroon">z1</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E5:77_1"/><b>then </b>
<font color="Maroon">z1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:77_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:77_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E9:77_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">z1</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E10:77_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z1</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<b>hence </b><a NAME="E11:77_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="waybel23.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


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

<a NAME="E4:77"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">B</font> is    <a href="waybel23.html#M1" target="_self">CLbasis</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="waybel23.html#T2" target="_self">Th2</a>, </i>;<br/>
<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>;<br/>

<b>assume </b><a NAME="E5:77"/>
not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:77"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:77"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E9:77"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">b</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E10:77"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="waybel23.html#T3" target="_self">Th3</a>, , <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>


</div>
