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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <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> holds   <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Olive" title="b1">F</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c1">L</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="waybel33.html#K1" title="WAYBEL33:func.1">lim_inf</a> <font color="Maroon" title="c2">F</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span><br/>



<span class="kw">set </span><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/>
<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_7.html#D5" title="WAYBEL_7:def.5">WAYBEL_7:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or  not  <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> or <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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:40"/>
<font color="Maroon" title="c2">F</font> is  <a href="waybel_7.html#V2" title="WAYBEL_7:attr.2">prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_7.html#T26" title="WAYBEL_7:th.26">WAYBEL_7:26</a></span>;<br/>
<a NAME="E5:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="waybel_7.html#T25" title="WAYBEL_7:th.25">WAYBEL_7:25</a></span>;<br/>
<a NAME="E6:40"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow19.html#T2" title="YELLOW19:th.2">YELLOW19:2</a></span>;<br/>
<a NAME="E7:40"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="waybel33.html#T18" target="_self" title="WAYBEL33:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:40"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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