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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_1.html#V11">Boolean</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>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="waybel15.html#K1" target="_self">ATOM</a> <font color="Maroon">L</font>
 ;<br/>

<b>thus </b><a NAME="E2:33"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies ( <font color="Maroon">x</font> is <a href="waybel15.html#V1" target="_self">atom</a> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>hence </b><a NAME="E2:33_1"/>
<font color="Maroon">x</font> is <a href="waybel15.html#V1" target="_self">atom</a>
 <b>by </b><i>, </i>;<br/>

<a NAME="E3:33_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


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

<b>thus </b><a NAME="E3:33"/>
( <font color="Maroon">x</font> is <a href="waybel15.html#V1" target="_self">atom</a> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:33_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> is <a href="waybel15.html#V1" target="_self">atom</a>
 <b>and </b><br/><a NAME="E2:33_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:33_2"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:33_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 ;<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="E3:33_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E4:33_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E5:33_2_1"/>
<font color="Maroon">y</font> is <a href="waybel15.html#V1" target="_self">atom</a>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:33_2_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<a NAME="E5:33_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel15.html#T1" target="_self">Th1</a>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> ,<font color="Maroon">L</font>
<b>by </b><i/>
;<br/>
<a NAME="E7:33_2"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E8:33_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T22">YELLOW_5:22</a></i>;<br/>
<b>hence </b><a NAME="E9:33_2"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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


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