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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>thus </b><a NAME="E1:6"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> implies for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">F</font> being  <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">T</font></span>)</span></span>)</span>  st <font color="Olive">x</font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a> <font color="Olive">F</font>,<font color="Maroon">T</font> holds <br/><font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#D5">WAYBEL_7:def 5</a></i>;<br/>

<b>assume </b><a NAME="E2:6"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">F</font> being  <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">T</font></span>)</span></span>)</span>  st <font color="Olive">x</font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a> <font color="Olive">F</font>,<font color="Maroon">T</font> holds <br/><font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>;<br/>
<b>assume </b><a NAME="E4:6"/>
not <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E5:6"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E6:6"/><b>then </b>
not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E7:6"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</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_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E8:6"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E10:6"/>
( <font color="Maroon">x</font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:6"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:6"/><b>then </b>
<font color="Maroon">A</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:6"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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