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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel33.html#V1" title="WAYBEL33:attr.1">lim-inf</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font> be  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/>



<b>set </b><font color="Maroon" title="c3">x</font> =  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c2">F</font>;<br/>
<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D5" title="WAYBEL_7:def.5">WAYBEL_7:def 5</a><br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; not <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )
 ;<br/>

<a NAME="E2:40"/>
<font color="Maroon" title="c2">F</font> is <a href="waybel_7.html#V2" title="WAYBEL_7:attr.2">prime</a>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T26" title="WAYBEL_7:th.26">WAYBEL_7:26</a></i>;<br/>
<a NAME="E3:40"/><b>then </b><i><font color="Green" title="E32">A2</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="waybel_7.html#T25" title="WAYBEL_7:th.25">WAYBEL_7:25</a></i>;<br/>
<a NAME="E4:40"/>
( <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="waybel33.html#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="waybel33.html#E3:40"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a>, <a class="ref" href="yellow19.html#T2" title="YELLOW19:th.2">YELLOW19:2</a></i>;<br/>
<a NAME="E5:40"/><b>then </b>
 <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="waybel33.html#E3:40"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="waybel33.html#T18" target="_self" title="WAYBEL33:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E6:40"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel33.html#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


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