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<b>let </b><font color="Maroon">N</font> be  <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>, <font color="Maroon">Y</font> be  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>;<br/>



<b>thus </b><a NAME="E1:46"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="waybel30.html#K1" target="_self">^0</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="waybel30.html#K1" target="_self">^0</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="waybel30.html#K1" target="_self">^0</a> 
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>assume </b><a NAME="E2:46"/>
not <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="waybel30.html#K1" target="_self">^0</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="waybel30.html#K1" target="_self">^0</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="waybel30.html#K1" target="_self">^0</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:46"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="waybel30.html#K1" target="_self">^0</a> 
 <b>and </b><br/><a NAME="E5:46"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="waybel30.html#K1" target="_self">^0</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="waybel30.html#K1" target="_self">^0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:46"/><i><font color="Green">E35</font></i>: 
( not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="waybel30.html#K1" target="_self">^0</a>  &amp; not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="waybel30.html#K1" target="_self">^0</a>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">D</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E9:46"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> &amp; <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="waybel30.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">E</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E11:46"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">E</font> &amp; <font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="waybel30.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">xy</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E13:46"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xy</font> &amp; ( for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>  st <font color="Maroon">xy</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> holds <br/><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">D</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E14:46"/>
<font color="Maroon">s</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/>
<a NAME="E15:46"/><b>then </b>
<font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <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">E</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/>
<a NAME="E16:46"/><b>then </b>
<font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T51">WAYBEL_2:51</a></i>;<br/>
<a NAME="E17:46"/><b>then </b>
<font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:46"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E19:46"/>
( <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font> &amp; <font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font> )
 
<b>by </b><i>, , <a class="ref" href="yellow12.html#T21">YELLOW12:21</a></i>;<br/>
<a NAME="E20:46"/><b>then </b>
( <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E21:46"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">E</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>hence </b><a NAME="E22:46"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>


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