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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>thus </b><a NAME="E1:49"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V2">compact</a> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="yellow19.html#V1" target="_self">Cauchy</a> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="yellow_6.html#V3">convergent</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V2">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:49_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow19.html#D5" target="_self">YELLOW19:def 5</a><br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:49_1"/><i><font color="Green">E30</font></i>: 
<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>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:49_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow19.html#E37"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><a NAME="E1:49_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:49_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E2:49_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:49_1_1"/><i><font color="Green">E31</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>2</sub></font> st <font color="Maroon">c<sub>5</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>2</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>4</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/><a NAME="E5:49_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E4:49_1"><i><font color="Green">E30</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>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:49_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a></i>;<br/><a NAME="E8:49_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E4:49_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="yellow19.html#E7:49_1_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E9:49_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow19.html#E7:49_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:49_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> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D18">YELLOW_6:def 18</a></i>;<br/>
<b>hence </b><a NAME="E7:49_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V3">convergent</a>
 <b>by </b><i><a class="ref" href="yellow_6.html#D19">YELLOW_6:def 19</a></i>;<br/>


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

<b>assume </b><a NAME="E2:49"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="yellow19.html#V1" target="_self">Cauchy</a> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="yellow_6.html#V3">convergent</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_7.html#V3">ultra</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/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <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>2</sub></font></span>)</span>;<br/><a NAME="E2:49_2"/>
 <a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V3">convergent</a>
 <b>by </b><i><a class="txt" href="yellow19.html#E2:49"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E3:49_2"/><b>then </b><i><font color="Green">E29</font></i>: 
 <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>2</sub></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="yellow_6.html#D19">YELLOW_6:def 19</a></i>;<br/><a NAME="E4:49_2"/><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>2</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E6:49_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:49_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow19.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V2">compact</a>
 <b>by </b><i><a class="ref" href="yellow19.html#T33" target="_self">Th33</a></i>;<br/>


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