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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="tex_2.html#V1">proper</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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:29_1"/><b>then </b>
 <a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow19.html#T20" target="_self">Th20</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="yellow_6.html#K6">"</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="yellow_6.html#M2">subnet</a> of  <a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#T31">YELLOW_6:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E5:29_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T32">YELLOW_6:32</a></i>;<br/><a NAME="E6:29_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow19.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E7:29_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="yellow19.html#T18" target="_self">Th18</a>, <a class="ref" href="yellow_6.html#T41">YELLOW_6:41</a></i>;<br/><b>hence </b><a NAME="E8:29_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="waybel_9.html#T29">WAYBEL_9:29</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E3:29"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E5:29"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</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="yellow19.html#E3:29"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E2:29_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><a NAME="E3:29_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:29_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><a NAME="E4:29_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:29"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel_9.html#D9">WAYBEL_9:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E6:29_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a></i>;<br/><a NAME="E7:29_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>8</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="yellow19.html#E5:29"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yellow19.html#E6:29_2"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E8:29_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E6:29_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:29"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>


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