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<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="E1:80"/><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>thus </b><a NAME="E2:80"/>
( <font color="Maroon">B</font> is    <a href="waybel23.html#M1" target="_self">CLbasis</a> of <font color="Maroon">L</font> implies 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 not <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</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; not <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon">B</font> is    <a href="waybel23.html#M1" target="_self">CLbasis</a> of <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:80_1"/><b>then </b>
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> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:80_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">L</font>  st not <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</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; not <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> )
 <b>by </b><i>, <a class="txt" href="waybel23.html#E2"><i><font color="Green">Lemma39</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E3:80"/>
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 not <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</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; not <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> )
 ;<br/>

<a NAME="E4:80"/><b>then </b>
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 not <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</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; not <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">b</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">y</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:80"/>
<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/>;<br/>


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