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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">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>



<a NAME="E1:5"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>let </b><font color="Maroon">F</font> be  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</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>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a> <font color="Maroon">F</font>,<font color="Maroon">T</font>
 ;<br/><b>thus </b><a NAME="E2:5_1"/>
 <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> <b>by </b><i/>;<br/>
<a NAME="E3:5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E4:5_1_1"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a>, <a class="ref" href="waybel_7.html#D5">WAYBEL_7:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:5_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T11">WAYBEL_7:11</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:5"/><i><font color="Green">E37</font></i>: 
 <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</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="E4:5"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E5:5"/><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><a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<a NAME="E6:5"/><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>
 
;<br/>
<b>hence </b><a NAME="E7:5"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


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