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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel33.html#V1" target="_self">lim-inf</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_7.html#V3">ultra</a> <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D5">WAYBEL_7:def 5</a><br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:40"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_7.html#V2">prime</a>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T26">WAYBEL_7:26</a></i>;<br/>
<a NAME="E3:40"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_7.html#T25">WAYBEL_7:25</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E5:40"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="txt" href="waybel33.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel33.html#E3:40"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a>, <a class="ref" href="yellow19.html#T2">YELLOW19:2</a></i>;<br/>
<a NAME="E6:40"/><b>then </b>
 <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="waybel33.html#E3:40"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel33.html#E4:40"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel33.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E7:40"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel33.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>


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